3.5.15 \(\int \frac {x^{7/2} (a+b x^2)^2}{(c+d x^2)^3} \, dx\)

Optimal. Leaf size=440 \[ -\frac {\sqrt {x} \left (5 a^2 d^2-90 a b c d+117 b^2 c^2\right )}{16 c d^4}+\frac {x^{5/2} \left (5 a^2 d^2-90 a b c d+117 b^2 c^2\right )}{80 c^2 d^3}-\frac {\left (5 a^2 d^2-90 a b c d+117 b^2 c^2\right ) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{64 \sqrt {2} c^{3/4} d^{17/4}}+\frac {\left (5 a^2 d^2-90 a b c d+117 b^2 c^2\right ) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{64 \sqrt {2} c^{3/4} d^{17/4}}-\frac {\left (5 a^2 d^2-90 a b c d+117 b^2 c^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{3/4} d^{17/4}}+\frac {\left (5 a^2 d^2-90 a b c d+117 b^2 c^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{32 \sqrt {2} c^{3/4} d^{17/4}}-\frac {x^{9/2} (b c-a d) (17 b c-a d)}{16 c^2 d^2 \left (c+d x^2\right )}+\frac {x^{9/2} (b c-a d)^2}{4 c d^2 \left (c+d x^2\right )^2} \]

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Rubi [A]  time = 0.38, antiderivative size = 440, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 10, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {463, 457, 321, 329, 211, 1165, 628, 1162, 617, 204} \begin {gather*} \frac {x^{5/2} \left (5 a^2 d^2-90 a b c d+117 b^2 c^2\right )}{80 c^2 d^3}-\frac {\sqrt {x} \left (5 a^2 d^2-90 a b c d+117 b^2 c^2\right )}{16 c d^4}-\frac {\left (5 a^2 d^2-90 a b c d+117 b^2 c^2\right ) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{64 \sqrt {2} c^{3/4} d^{17/4}}+\frac {\left (5 a^2 d^2-90 a b c d+117 b^2 c^2\right ) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{64 \sqrt {2} c^{3/4} d^{17/4}}-\frac {\left (5 a^2 d^2-90 a b c d+117 b^2 c^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{3/4} d^{17/4}}+\frac {\left (5 a^2 d^2-90 a b c d+117 b^2 c^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{32 \sqrt {2} c^{3/4} d^{17/4}}-\frac {x^{9/2} (b c-a d) (17 b c-a d)}{16 c^2 d^2 \left (c+d x^2\right )}+\frac {x^{9/2} (b c-a d)^2}{4 c d^2 \left (c+d x^2\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^(7/2)*(a + b*x^2)^2)/(c + d*x^2)^3,x]

[Out]

-((117*b^2*c^2 - 90*a*b*c*d + 5*a^2*d^2)*Sqrt[x])/(16*c*d^4) + ((117*b^2*c^2 - 90*a*b*c*d + 5*a^2*d^2)*x^(5/2)
)/(80*c^2*d^3) + ((b*c - a*d)^2*x^(9/2))/(4*c*d^2*(c + d*x^2)^2) - ((b*c - a*d)*(17*b*c - a*d)*x^(9/2))/(16*c^
2*d^2*(c + d*x^2)) - ((117*b^2*c^2 - 90*a*b*c*d + 5*a^2*d^2)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(3
2*Sqrt[2]*c^(3/4)*d^(17/4)) + ((117*b^2*c^2 - 90*a*b*c*d + 5*a^2*d^2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(
1/4)])/(32*Sqrt[2]*c^(3/4)*d^(17/4)) - ((117*b^2*c^2 - 90*a*b*c*d + 5*a^2*d^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d
^(1/4)*Sqrt[x] + Sqrt[d]*x])/(64*Sqrt[2]*c^(3/4)*d^(17/4)) + ((117*b^2*c^2 - 90*a*b*c*d + 5*a^2*d^2)*Log[Sqrt[
c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(64*Sqrt[2]*c^(3/4)*d^(17/4))

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 211

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Di
st[1/(2*r), Int[(r - s*x^2)/(a + b*x^4), x], x] + Dist[1/(2*r), Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[
{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ, b
]]))

Rule 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 329

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + (b*x^(k*n))/c^n)^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 457

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> -Simp[((b*c - a*d
)*(e*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a*b*e*n*(p + 1)), x] - Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*b
*n*(p + 1)), Int[(e*x)^m*(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] &
& LtQ[p, -1] && (( !IntegerQ[p + 1/2] && NeQ[p, -5/4]) ||  !RationalQ[m] || (IGtQ[n, 0] && ILtQ[p + 1/2, 0] &&
LeQ[-1, m, -(n*(p + 1))]))

Rule 463

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^2, x_Symbol] :> -Simp[((b*c - a*
d)^2*(e*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a*b^2*e*n*(p + 1)), x] + Dist[1/(a*b^2*n*(p + 1)), Int[(e*x)^m*(a + b
*x^n)^(p + 1)*Simp[(b*c - a*d)^2*(m + 1) + b^2*c^2*n*(p + 1) + a*b*d^2*n*(p + 1)*x^n, x], x], x] /; FreeQ[{a,
b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[p, -1]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 1162

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(2*d)/e, 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 1165

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(-2*d)/e, 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rubi steps

\begin {align*} \int \frac {x^{7/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^3} \, dx &=\frac {(b c-a d)^2 x^{9/2}}{4 c d^2 \left (c+d x^2\right )^2}-\frac {\int \frac {x^{7/2} \left (\frac {1}{2} \left (-8 a^2 d^2+9 (b c-a d)^2\right )-4 b^2 c d x^2\right )}{\left (c+d x^2\right )^2} \, dx}{4 c d^2}\\ &=\frac {(b c-a d)^2 x^{9/2}}{4 c d^2 \left (c+d x^2\right )^2}-\frac {(b c-a d) (17 b c-a d) x^{9/2}}{16 c^2 d^2 \left (c+d x^2\right )}+\frac {\left (117 b^2 c^2-90 a b c d+5 a^2 d^2\right ) \int \frac {x^{7/2}}{c+d x^2} \, dx}{32 c^2 d^2}\\ &=\frac {\left (117 b^2 c^2-90 a b c d+5 a^2 d^2\right ) x^{5/2}}{80 c^2 d^3}+\frac {(b c-a d)^2 x^{9/2}}{4 c d^2 \left (c+d x^2\right )^2}-\frac {(b c-a d) (17 b c-a d) x^{9/2}}{16 c^2 d^2 \left (c+d x^2\right )}-\frac {\left (117 b^2 c^2-90 a b c d+5 a^2 d^2\right ) \int \frac {x^{3/2}}{c+d x^2} \, dx}{32 c d^3}\\ &=-\frac {\left (117 b^2 c^2-90 a b c d+5 a^2 d^2\right ) \sqrt {x}}{16 c d^4}+\frac {\left (117 b^2 c^2-90 a b c d+5 a^2 d^2\right ) x^{5/2}}{80 c^2 d^3}+\frac {(b c-a d)^2 x^{9/2}}{4 c d^2 \left (c+d x^2\right )^2}-\frac {(b c-a d) (17 b c-a d) x^{9/2}}{16 c^2 d^2 \left (c+d x^2\right )}+\frac {\left (117 b^2 c^2-90 a b c d+5 a^2 d^2\right ) \int \frac {1}{\sqrt {x} \left (c+d x^2\right )} \, dx}{32 d^4}\\ &=-\frac {\left (117 b^2 c^2-90 a b c d+5 a^2 d^2\right ) \sqrt {x}}{16 c d^4}+\frac {\left (117 b^2 c^2-90 a b c d+5 a^2 d^2\right ) x^{5/2}}{80 c^2 d^3}+\frac {(b c-a d)^2 x^{9/2}}{4 c d^2 \left (c+d x^2\right )^2}-\frac {(b c-a d) (17 b c-a d) x^{9/2}}{16 c^2 d^2 \left (c+d x^2\right )}+\frac {\left (117 b^2 c^2-90 a b c d+5 a^2 d^2\right ) \operatorname {Subst}\left (\int \frac {1}{c+d x^4} \, dx,x,\sqrt {x}\right )}{16 d^4}\\ &=-\frac {\left (117 b^2 c^2-90 a b c d+5 a^2 d^2\right ) \sqrt {x}}{16 c d^4}+\frac {\left (117 b^2 c^2-90 a b c d+5 a^2 d^2\right ) x^{5/2}}{80 c^2 d^3}+\frac {(b c-a d)^2 x^{9/2}}{4 c d^2 \left (c+d x^2\right )^2}-\frac {(b c-a d) (17 b c-a d) x^{9/2}}{16 c^2 d^2 \left (c+d x^2\right )}+\frac {\left (117 b^2 c^2-90 a b c d+5 a^2 d^2\right ) \operatorname {Subst}\left (\int \frac {\sqrt {c}-\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{32 \sqrt {c} d^4}+\frac {\left (117 b^2 c^2-90 a b c d+5 a^2 d^2\right ) \operatorname {Subst}\left (\int \frac {\sqrt {c}+\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{32 \sqrt {c} d^4}\\ &=-\frac {\left (117 b^2 c^2-90 a b c d+5 a^2 d^2\right ) \sqrt {x}}{16 c d^4}+\frac {\left (117 b^2 c^2-90 a b c d+5 a^2 d^2\right ) x^{5/2}}{80 c^2 d^3}+\frac {(b c-a d)^2 x^{9/2}}{4 c d^2 \left (c+d x^2\right )^2}-\frac {(b c-a d) (17 b c-a d) x^{9/2}}{16 c^2 d^2 \left (c+d x^2\right )}+\frac {\left (117 b^2 c^2-90 a b c d+5 a^2 d^2\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt {c}}{\sqrt {d}}-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}+x^2} \, dx,x,\sqrt {x}\right )}{64 \sqrt {c} d^{9/2}}+\frac {\left (117 b^2 c^2-90 a b c d+5 a^2 d^2\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt {c}}{\sqrt {d}}+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}+x^2} \, dx,x,\sqrt {x}\right )}{64 \sqrt {c} d^{9/2}}-\frac {\left (117 b^2 c^2-90 a b c d+5 a^2 d^2\right ) \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{c}}{\sqrt [4]{d}}+2 x}{-\frac {\sqrt {c}}{\sqrt {d}}-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}-x^2} \, dx,x,\sqrt {x}\right )}{64 \sqrt {2} c^{3/4} d^{17/4}}-\frac {\left (117 b^2 c^2-90 a b c d+5 a^2 d^2\right ) \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{c}}{\sqrt [4]{d}}-2 x}{-\frac {\sqrt {c}}{\sqrt {d}}+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}-x^2} \, dx,x,\sqrt {x}\right )}{64 \sqrt {2} c^{3/4} d^{17/4}}\\ &=-\frac {\left (117 b^2 c^2-90 a b c d+5 a^2 d^2\right ) \sqrt {x}}{16 c d^4}+\frac {\left (117 b^2 c^2-90 a b c d+5 a^2 d^2\right ) x^{5/2}}{80 c^2 d^3}+\frac {(b c-a d)^2 x^{9/2}}{4 c d^2 \left (c+d x^2\right )^2}-\frac {(b c-a d) (17 b c-a d) x^{9/2}}{16 c^2 d^2 \left (c+d x^2\right )}-\frac {\left (117 b^2 c^2-90 a b c d+5 a^2 d^2\right ) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{3/4} d^{17/4}}+\frac {\left (117 b^2 c^2-90 a b c d+5 a^2 d^2\right ) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{3/4} d^{17/4}}+\frac {\left (117 b^2 c^2-90 a b c d+5 a^2 d^2\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{3/4} d^{17/4}}-\frac {\left (117 b^2 c^2-90 a b c d+5 a^2 d^2\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{3/4} d^{17/4}}\\ &=-\frac {\left (117 b^2 c^2-90 a b c d+5 a^2 d^2\right ) \sqrt {x}}{16 c d^4}+\frac {\left (117 b^2 c^2-90 a b c d+5 a^2 d^2\right ) x^{5/2}}{80 c^2 d^3}+\frac {(b c-a d)^2 x^{9/2}}{4 c d^2 \left (c+d x^2\right )^2}-\frac {(b c-a d) (17 b c-a d) x^{9/2}}{16 c^2 d^2 \left (c+d x^2\right )}-\frac {\left (117 b^2 c^2-90 a b c d+5 a^2 d^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{3/4} d^{17/4}}+\frac {\left (117 b^2 c^2-90 a b c d+5 a^2 d^2\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{3/4} d^{17/4}}-\frac {\left (117 b^2 c^2-90 a b c d+5 a^2 d^2\right ) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{3/4} d^{17/4}}+\frac {\left (117 b^2 c^2-90 a b c d+5 a^2 d^2\right ) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{3/4} d^{17/4}}\\ \end {align*}

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Mathematica [A]  time = 0.47, size = 383, normalized size = 0.87 \begin {gather*} \frac {-\frac {40 \sqrt [4]{d} \sqrt {x} \left (9 a^2 d^2-34 a b c d+25 b^2 c^2\right )}{c+d x^2}-\frac {5 \sqrt {2} \left (5 a^2 d^2-90 a b c d+117 b^2 c^2\right ) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{c^{3/4}}+\frac {5 \sqrt {2} \left (5 a^2 d^2-90 a b c d+117 b^2 c^2\right ) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{c^{3/4}}-\frac {10 \sqrt {2} \left (5 a^2 d^2-90 a b c d+117 b^2 c^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{c^{3/4}}+\frac {10 \sqrt {2} \left (5 a^2 d^2-90 a b c d+117 b^2 c^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{c^{3/4}}+\frac {160 c \sqrt [4]{d} \sqrt {x} (b c-a d)^2}{\left (c+d x^2\right )^2}-1280 b \sqrt [4]{d} \sqrt {x} (3 b c-2 a d)+256 b^2 d^{5/4} x^{5/2}}{640 d^{17/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x^(7/2)*(a + b*x^2)^2)/(c + d*x^2)^3,x]

[Out]

(-1280*b*d^(1/4)*(3*b*c - 2*a*d)*Sqrt[x] + 256*b^2*d^(5/4)*x^(5/2) + (160*c*d^(1/4)*(b*c - a*d)^2*Sqrt[x])/(c
+ d*x^2)^2 - (40*d^(1/4)*(25*b^2*c^2 - 34*a*b*c*d + 9*a^2*d^2)*Sqrt[x])/(c + d*x^2) - (10*Sqrt[2]*(117*b^2*c^2
- 90*a*b*c*d + 5*a^2*d^2)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/c^(3/4) + (10*Sqrt[2]*(117*b^2*c^2 -
90*a*b*c*d + 5*a^2*d^2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/c^(3/4) - (5*Sqrt[2]*(117*b^2*c^2 - 90
*a*b*c*d + 5*a^2*d^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/c^(3/4) + (5*Sqrt[2]*(117*b^
2*c^2 - 90*a*b*c*d + 5*a^2*d^2)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/c^(3/4))/(640*d^(1
7/4))

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IntegrateAlgebraic [A]  time = 0.91, size = 272, normalized size = 0.62 \begin {gather*} -\frac {\left (5 a^2 d^2-90 a b c d+117 b^2 c^2\right ) \tan ^{-1}\left (\frac {\sqrt {c}-\sqrt {d} x}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}\right )}{32 \sqrt {2} c^{3/4} d^{17/4}}+\frac {\left (5 a^2 d^2-90 a b c d+117 b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}{\sqrt {c}+\sqrt {d} x}\right )}{32 \sqrt {2} c^{3/4} d^{17/4}}+\frac {\sqrt {x} \left (-25 a^2 c d^2-45 a^2 d^3 x^2+450 a b c^2 d+810 a b c d^2 x^2+320 a b d^3 x^4-585 b^2 c^3-1053 b^2 c^2 d x^2-416 b^2 c d^2 x^4+32 b^2 d^3 x^6\right )}{80 d^4 \left (c+d x^2\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(x^(7/2)*(a + b*x^2)^2)/(c + d*x^2)^3,x]

[Out]

(Sqrt[x]*(-585*b^2*c^3 + 450*a*b*c^2*d - 25*a^2*c*d^2 - 1053*b^2*c^2*d*x^2 + 810*a*b*c*d^2*x^2 - 45*a^2*d^3*x^
2 - 416*b^2*c*d^2*x^4 + 320*a*b*d^3*x^4 + 32*b^2*d^3*x^6))/(80*d^4*(c + d*x^2)^2) - ((117*b^2*c^2 - 90*a*b*c*d
+ 5*a^2*d^2)*ArcTan[(Sqrt[c] - Sqrt[d]*x)/(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])])/(32*Sqrt[2]*c^(3/4)*d^(17/4)) +
((117*b^2*c^2 - 90*a*b*c*d + 5*a^2*d^2)*ArcTanh[(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])/(Sqrt[c] + Sqrt[d]*x)])/(32
*Sqrt[2]*c^(3/4)*d^(17/4))

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fricas [B]  time = 1.44, size = 1427, normalized size = 3.24

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(7/2)*(b*x^2+a)^2/(d*x^2+c)^3,x, algorithm="fricas")

[Out]

1/320*(20*(d^6*x^4 + 2*c*d^5*x^2 + c^2*d^4)*(-(187388721*b^8*c^8 - 576580680*a*b^7*c^7*d + 697317660*a^2*b^6*c
^6*d^2 - 415092600*a^3*b^5*c^5*d^3 + 124525350*a^4*b^4*c^4*d^4 - 17739000*a^5*b^3*c^3*d^5 + 1273500*a^6*b^2*c^
2*d^6 - 45000*a^7*b*c*d^7 + 625*a^8*d^8)/(c^3*d^17))^(1/4)*arctan((sqrt(c^2*d^8*sqrt(-(187388721*b^8*c^8 - 576
580680*a*b^7*c^7*d + 697317660*a^2*b^6*c^6*d^2 - 415092600*a^3*b^5*c^5*d^3 + 124525350*a^4*b^4*c^4*d^4 - 17739
000*a^5*b^3*c^3*d^5 + 1273500*a^6*b^2*c^2*d^6 - 45000*a^7*b*c*d^7 + 625*a^8*d^8)/(c^3*d^17)) + (13689*b^4*c^4
- 21060*a*b^3*c^3*d + 9270*a^2*b^2*c^2*d^2 - 900*a^3*b*c*d^3 + 25*a^4*d^4)*x)*c^2*d^13*(-(187388721*b^8*c^8 -
576580680*a*b^7*c^7*d + 697317660*a^2*b^6*c^6*d^2 - 415092600*a^3*b^5*c^5*d^3 + 124525350*a^4*b^4*c^4*d^4 - 17
739000*a^5*b^3*c^3*d^5 + 1273500*a^6*b^2*c^2*d^6 - 45000*a^7*b*c*d^7 + 625*a^8*d^8)/(c^3*d^17))^(3/4) - (117*b
^2*c^4*d^13 - 90*a*b*c^3*d^14 + 5*a^2*c^2*d^15)*sqrt(x)*(-(187388721*b^8*c^8 - 576580680*a*b^7*c^7*d + 6973176
60*a^2*b^6*c^6*d^2 - 415092600*a^3*b^5*c^5*d^3 + 124525350*a^4*b^4*c^4*d^4 - 17739000*a^5*b^3*c^3*d^5 + 127350
0*a^6*b^2*c^2*d^6 - 45000*a^7*b*c*d^7 + 625*a^8*d^8)/(c^3*d^17))^(3/4))/(187388721*b^8*c^8 - 576580680*a*b^7*c
^7*d + 697317660*a^2*b^6*c^6*d^2 - 415092600*a^3*b^5*c^5*d^3 + 124525350*a^4*b^4*c^4*d^4 - 17739000*a^5*b^3*c^
3*d^5 + 1273500*a^6*b^2*c^2*d^6 - 45000*a^7*b*c*d^7 + 625*a^8*d^8)) + 5*(d^6*x^4 + 2*c*d^5*x^2 + c^2*d^4)*(-(1
87388721*b^8*c^8 - 576580680*a*b^7*c^7*d + 697317660*a^2*b^6*c^6*d^2 - 415092600*a^3*b^5*c^5*d^3 + 124525350*a
^4*b^4*c^4*d^4 - 17739000*a^5*b^3*c^3*d^5 + 1273500*a^6*b^2*c^2*d^6 - 45000*a^7*b*c*d^7 + 625*a^8*d^8)/(c^3*d^
17))^(1/4)*log(c*d^4*(-(187388721*b^8*c^8 - 576580680*a*b^7*c^7*d + 697317660*a^2*b^6*c^6*d^2 - 415092600*a^3*
b^5*c^5*d^3 + 124525350*a^4*b^4*c^4*d^4 - 17739000*a^5*b^3*c^3*d^5 + 1273500*a^6*b^2*c^2*d^6 - 45000*a^7*b*c*d
^7 + 625*a^8*d^8)/(c^3*d^17))^(1/4) + (117*b^2*c^2 - 90*a*b*c*d + 5*a^2*d^2)*sqrt(x)) - 5*(d^6*x^4 + 2*c*d^5*x
^2 + c^2*d^4)*(-(187388721*b^8*c^8 - 576580680*a*b^7*c^7*d + 697317660*a^2*b^6*c^6*d^2 - 415092600*a^3*b^5*c^5
*d^3 + 124525350*a^4*b^4*c^4*d^4 - 17739000*a^5*b^3*c^3*d^5 + 1273500*a^6*b^2*c^2*d^6 - 45000*a^7*b*c*d^7 + 62
5*a^8*d^8)/(c^3*d^17))^(1/4)*log(-c*d^4*(-(187388721*b^8*c^8 - 576580680*a*b^7*c^7*d + 697317660*a^2*b^6*c^6*d
^2 - 415092600*a^3*b^5*c^5*d^3 + 124525350*a^4*b^4*c^4*d^4 - 17739000*a^5*b^3*c^3*d^5 + 1273500*a^6*b^2*c^2*d^
6 - 45000*a^7*b*c*d^7 + 625*a^8*d^8)/(c^3*d^17))^(1/4) + (117*b^2*c^2 - 90*a*b*c*d + 5*a^2*d^2)*sqrt(x)) + 4*(
32*b^2*d^3*x^6 - 585*b^2*c^3 + 450*a*b*c^2*d - 25*a^2*c*d^2 - 32*(13*b^2*c*d^2 - 10*a*b*d^3)*x^4 - 9*(117*b^2*
c^2*d - 90*a*b*c*d^2 + 5*a^2*d^3)*x^2)*sqrt(x))/(d^6*x^4 + 2*c*d^5*x^2 + c^2*d^4)

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giac [A]  time = 0.44, size = 451, normalized size = 1.02 \begin {gather*} \frac {\sqrt {2} {\left (117 \, \left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} - 90 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d + 5 \, \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{64 \, c d^{5}} + \frac {\sqrt {2} {\left (117 \, \left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} - 90 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d + 5 \, \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{2}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{64 \, c d^{5}} + \frac {\sqrt {2} {\left (117 \, \left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} - 90 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d + 5 \, \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{2}\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{128 \, c d^{5}} - \frac {\sqrt {2} {\left (117 \, \left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} - 90 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d + 5 \, \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{2}\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{128 \, c d^{5}} - \frac {25 \, b^{2} c^{2} d x^{\frac {5}{2}} - 34 \, a b c d^{2} x^{\frac {5}{2}} + 9 \, a^{2} d^{3} x^{\frac {5}{2}} + 21 \, b^{2} c^{3} \sqrt {x} - 26 \, a b c^{2} d \sqrt {x} + 5 \, a^{2} c d^{2} \sqrt {x}}{16 \, {\left (d x^{2} + c\right )}^{2} d^{4}} + \frac {2 \, {\left (b^{2} d^{12} x^{\frac {5}{2}} - 15 \, b^{2} c d^{11} \sqrt {x} + 10 \, a b d^{12} \sqrt {x}\right )}}{5 \, d^{15}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(7/2)*(b*x^2+a)^2/(d*x^2+c)^3,x, algorithm="giac")

[Out]

1/64*sqrt(2)*(117*(c*d^3)^(1/4)*b^2*c^2 - 90*(c*d^3)^(1/4)*a*b*c*d + 5*(c*d^3)^(1/4)*a^2*d^2)*arctan(1/2*sqrt(
2)*(sqrt(2)*(c/d)^(1/4) + 2*sqrt(x))/(c/d)^(1/4))/(c*d^5) + 1/64*sqrt(2)*(117*(c*d^3)^(1/4)*b^2*c^2 - 90*(c*d^
3)^(1/4)*a*b*c*d + 5*(c*d^3)^(1/4)*a^2*d^2)*arctan(-1/2*sqrt(2)*(sqrt(2)*(c/d)^(1/4) - 2*sqrt(x))/(c/d)^(1/4))
/(c*d^5) + 1/128*sqrt(2)*(117*(c*d^3)^(1/4)*b^2*c^2 - 90*(c*d^3)^(1/4)*a*b*c*d + 5*(c*d^3)^(1/4)*a^2*d^2)*log(
sqrt(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt(c/d))/(c*d^5) - 1/128*sqrt(2)*(117*(c*d^3)^(1/4)*b^2*c^2 - 90*(c*d^3)^(
1/4)*a*b*c*d + 5*(c*d^3)^(1/4)*a^2*d^2)*log(-sqrt(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt(c/d))/(c*d^5) - 1/16*(25*b
^2*c^2*d*x^(5/2) - 34*a*b*c*d^2*x^(5/2) + 9*a^2*d^3*x^(5/2) + 21*b^2*c^3*sqrt(x) - 26*a*b*c^2*d*sqrt(x) + 5*a^
2*c*d^2*sqrt(x))/((d*x^2 + c)^2*d^4) + 2/5*(b^2*d^12*x^(5/2) - 15*b^2*c*d^11*sqrt(x) + 10*a*b*d^12*sqrt(x))/d^
15

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maple [A]  time = 0.02, size = 590, normalized size = 1.34 \begin {gather*} -\frac {9 a^{2} x^{\frac {5}{2}}}{16 \left (d \,x^{2}+c \right )^{2} d}+\frac {17 a b c \,x^{\frac {5}{2}}}{8 \left (d \,x^{2}+c \right )^{2} d^{2}}-\frac {25 b^{2} c^{2} x^{\frac {5}{2}}}{16 \left (d \,x^{2}+c \right )^{2} d^{3}}-\frac {5 a^{2} c \sqrt {x}}{16 \left (d \,x^{2}+c \right )^{2} d^{2}}+\frac {13 a b \,c^{2} \sqrt {x}}{8 \left (d \,x^{2}+c \right )^{2} d^{3}}-\frac {21 b^{2} c^{3} \sqrt {x}}{16 \left (d \,x^{2}+c \right )^{2} d^{4}}+\frac {2 b^{2} x^{\frac {5}{2}}}{5 d^{3}}+\frac {5 \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, a^{2} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )}{64 c \,d^{2}}+\frac {5 \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, a^{2} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )}{64 c \,d^{2}}+\frac {5 \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, a^{2} \ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {c}{d}}}\right )}{128 c \,d^{2}}-\frac {45 \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, a b \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )}{32 d^{3}}-\frac {45 \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, a b \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )}{32 d^{3}}-\frac {45 \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, a b \ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {c}{d}}}\right )}{64 d^{3}}+\frac {117 \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, b^{2} c \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )}{64 d^{4}}+\frac {117 \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, b^{2} c \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )}{64 d^{4}}+\frac {117 \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, b^{2} c \ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {c}{d}}}\right )}{128 d^{4}}+\frac {4 a b \sqrt {x}}{d^{3}}-\frac {6 b^{2} c \sqrt {x}}{d^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(7/2)*(b*x^2+a)^2/(d*x^2+c)^3,x)

[Out]

2/5*b^2/d^3*x^(5/2)+4*b/d^3*a*x^(1/2)-6*b^2/d^4*c*x^(1/2)-9/16/d/(d*x^2+c)^2*x^(5/2)*a^2+17/8/d^2/(d*x^2+c)^2*
x^(5/2)*a*b*c-25/16/d^3/(d*x^2+c)^2*x^(5/2)*b^2*c^2-5/16/d^2/(d*x^2+c)^2*x^(1/2)*a^2*c+13/8/d^3/(d*x^2+c)^2*x^
(1/2)*a*b*c^2-21/16/d^4/(d*x^2+c)^2*x^(1/2)*b^2*c^3+5/64/d^2*(c/d)^(1/4)/c*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*
x^(1/2)-1)*a^2-45/32/d^3*(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1)*a*b+117/64/d^4*(c/d)^(1/4)*
c*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1)*b^2+5/128/d^2*(c/d)^(1/4)/c*2^(1/2)*ln((x+(c/d)^(1/4)*2^(1/2)*
x^(1/2)+(c/d)^(1/2))/(x-(c/d)^(1/4)*2^(1/2)*x^(1/2)+(c/d)^(1/2)))*a^2-45/64/d^3*(c/d)^(1/4)*2^(1/2)*ln((x+(c/d
)^(1/4)*2^(1/2)*x^(1/2)+(c/d)^(1/2))/(x-(c/d)^(1/4)*2^(1/2)*x^(1/2)+(c/d)^(1/2)))*a*b+117/128/d^4*(c/d)^(1/4)*
c*2^(1/2)*ln((x+(c/d)^(1/4)*2^(1/2)*x^(1/2)+(c/d)^(1/2))/(x-(c/d)^(1/4)*2^(1/2)*x^(1/2)+(c/d)^(1/2)))*b^2+5/64
/d^2*(c/d)^(1/4)/c*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)+1)*a^2-45/32/d^3*(c/d)^(1/4)*2^(1/2)*arctan(2^(1
/2)/(c/d)^(1/4)*x^(1/2)+1)*a*b+117/64/d^4*(c/d)^(1/4)*c*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)+1)*b^2

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maxima [A]  time = 2.44, size = 389, normalized size = 0.88 \begin {gather*} -\frac {{\left (25 \, b^{2} c^{2} d - 34 \, a b c d^{2} + 9 \, a^{2} d^{3}\right )} x^{\frac {5}{2}} + {\left (21 \, b^{2} c^{3} - 26 \, a b c^{2} d + 5 \, a^{2} c d^{2}\right )} \sqrt {x}}{16 \, {\left (d^{6} x^{4} + 2 \, c d^{5} x^{2} + c^{2} d^{4}\right )}} + \frac {2 \, {\left (b^{2} d x^{\frac {5}{2}} - 5 \, {\left (3 \, b^{2} c - 2 \, a b d\right )} \sqrt {x}\right )}}{5 \, d^{4}} + \frac {\frac {2 \, \sqrt {2} {\left (117 \, b^{2} c^{2} - 90 \, a b c d + 5 \, a^{2} d^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} + 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {c} \sqrt {\sqrt {c} \sqrt {d}}} + \frac {2 \, \sqrt {2} {\left (117 \, b^{2} c^{2} - 90 \, a b c d + 5 \, a^{2} d^{2}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} - 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {c} \sqrt {\sqrt {c} \sqrt {d}}} + \frac {\sqrt {2} {\left (117 \, b^{2} c^{2} - 90 \, a b c d + 5 \, a^{2} d^{2}\right )} \log \left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {3}{4}} d^{\frac {1}{4}}} - \frac {\sqrt {2} {\left (117 \, b^{2} c^{2} - 90 \, a b c d + 5 \, a^{2} d^{2}\right )} \log \left (-\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {3}{4}} d^{\frac {1}{4}}}}{128 \, d^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(7/2)*(b*x^2+a)^2/(d*x^2+c)^3,x, algorithm="maxima")

[Out]

-1/16*((25*b^2*c^2*d - 34*a*b*c*d^2 + 9*a^2*d^3)*x^(5/2) + (21*b^2*c^3 - 26*a*b*c^2*d + 5*a^2*c*d^2)*sqrt(x))/
(d^6*x^4 + 2*c*d^5*x^2 + c^2*d^4) + 2/5*(b^2*d*x^(5/2) - 5*(3*b^2*c - 2*a*b*d)*sqrt(x))/d^4 + 1/128*(2*sqrt(2)
*(117*b^2*c^2 - 90*a*b*c*d + 5*a^2*d^2)*arctan(1/2*sqrt(2)*(sqrt(2)*c^(1/4)*d^(1/4) + 2*sqrt(d)*sqrt(x))/sqrt(
sqrt(c)*sqrt(d)))/(sqrt(c)*sqrt(sqrt(c)*sqrt(d))) + 2*sqrt(2)*(117*b^2*c^2 - 90*a*b*c*d + 5*a^2*d^2)*arctan(-1
/2*sqrt(2)*(sqrt(2)*c^(1/4)*d^(1/4) - 2*sqrt(d)*sqrt(x))/sqrt(sqrt(c)*sqrt(d)))/(sqrt(c)*sqrt(sqrt(c)*sqrt(d))
) + sqrt(2)*(117*b^2*c^2 - 90*a*b*c*d + 5*a^2*d^2)*log(sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x + sqrt(c))/
(c^(3/4)*d^(1/4)) - sqrt(2)*(117*b^2*c^2 - 90*a*b*c*d + 5*a^2*d^2)*log(-sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt
(d)*x + sqrt(c))/(c^(3/4)*d^(1/4)))/d^4

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mupad [B]  time = 0.48, size = 1426, normalized size = 3.24

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^(7/2)*(a + b*x^2)^2)/(c + d*x^2)^3,x)

[Out]

(2*b^2*x^(5/2))/(5*d^3) - (x^(1/2)*((21*b^2*c^3)/16 + (5*a^2*c*d^2)/16 - (13*a*b*c^2*d)/8) + x^(5/2)*((9*a^2*d
^3)/16 + (25*b^2*c^2*d)/16 - (17*a*b*c*d^2)/8))/(c^2*d^4 + d^6*x^4 + 2*c*d^5*x^2) - x^(1/2)*((6*b^2*c)/d^4 - (
4*a*b)/d^3) + (atan(((((x^(1/2)*(25*a^4*d^4 + 13689*b^4*c^4 + 9270*a^2*b^2*c^2*d^2 - 21060*a*b^3*c^3*d - 900*a
^3*b*c*d^3))/(64*d^5) - ((5*a^2*d^2 + 117*b^2*c^2 - 90*a*b*c*d)*(117*b^2*c^3 + 5*a^2*c*d^2 - 90*a*b*c^2*d))/(6
4*(-c)^(3/4)*d^(21/4)))*(5*a^2*d^2 + 117*b^2*c^2 - 90*a*b*c*d)*1i)/(64*(-c)^(3/4)*d^(17/4)) + (((x^(1/2)*(25*a
^4*d^4 + 13689*b^4*c^4 + 9270*a^2*b^2*c^2*d^2 - 21060*a*b^3*c^3*d - 900*a^3*b*c*d^3))/(64*d^5) + ((5*a^2*d^2 +
117*b^2*c^2 - 90*a*b*c*d)*(117*b^2*c^3 + 5*a^2*c*d^2 - 90*a*b*c^2*d))/(64*(-c)^(3/4)*d^(21/4)))*(5*a^2*d^2 +
117*b^2*c^2 - 90*a*b*c*d)*1i)/(64*(-c)^(3/4)*d^(17/4)))/((((x^(1/2)*(25*a^4*d^4 + 13689*b^4*c^4 + 9270*a^2*b^2
*c^2*d^2 - 21060*a*b^3*c^3*d - 900*a^3*b*c*d^3))/(64*d^5) - ((5*a^2*d^2 + 117*b^2*c^2 - 90*a*b*c*d)*(117*b^2*c
^3 + 5*a^2*c*d^2 - 90*a*b*c^2*d))/(64*(-c)^(3/4)*d^(21/4)))*(5*a^2*d^2 + 117*b^2*c^2 - 90*a*b*c*d))/(64*(-c)^(
3/4)*d^(17/4)) - (((x^(1/2)*(25*a^4*d^4 + 13689*b^4*c^4 + 9270*a^2*b^2*c^2*d^2 - 21060*a*b^3*c^3*d - 900*a^3*b
*c*d^3))/(64*d^5) + ((5*a^2*d^2 + 117*b^2*c^2 - 90*a*b*c*d)*(117*b^2*c^3 + 5*a^2*c*d^2 - 90*a*b*c^2*d))/(64*(-
c)^(3/4)*d^(21/4)))*(5*a^2*d^2 + 117*b^2*c^2 - 90*a*b*c*d))/(64*(-c)^(3/4)*d^(17/4))))*(5*a^2*d^2 + 117*b^2*c^
2 - 90*a*b*c*d)*1i)/(32*(-c)^(3/4)*d^(17/4)) + (atan(((((x^(1/2)*(25*a^4*d^4 + 13689*b^4*c^4 + 9270*a^2*b^2*c^
2*d^2 - 21060*a*b^3*c^3*d - 900*a^3*b*c*d^3))/(64*d^5) - ((5*a^2*d^2 + 117*b^2*c^2 - 90*a*b*c*d)*(117*b^2*c^3
+ 5*a^2*c*d^2 - 90*a*b*c^2*d)*1i)/(64*(-c)^(3/4)*d^(21/4)))*(5*a^2*d^2 + 117*b^2*c^2 - 90*a*b*c*d))/(64*(-c)^(
3/4)*d^(17/4)) + (((x^(1/2)*(25*a^4*d^4 + 13689*b^4*c^4 + 9270*a^2*b^2*c^2*d^2 - 21060*a*b^3*c^3*d - 900*a^3*b
*c*d^3))/(64*d^5) + ((5*a^2*d^2 + 117*b^2*c^2 - 90*a*b*c*d)*(117*b^2*c^3 + 5*a^2*c*d^2 - 90*a*b*c^2*d)*1i)/(64
*(-c)^(3/4)*d^(21/4)))*(5*a^2*d^2 + 117*b^2*c^2 - 90*a*b*c*d))/(64*(-c)^(3/4)*d^(17/4)))/((((x^(1/2)*(25*a^4*d
^4 + 13689*b^4*c^4 + 9270*a^2*b^2*c^2*d^2 - 21060*a*b^3*c^3*d - 900*a^3*b*c*d^3))/(64*d^5) - ((5*a^2*d^2 + 117
*b^2*c^2 - 90*a*b*c*d)*(117*b^2*c^3 + 5*a^2*c*d^2 - 90*a*b*c^2*d)*1i)/(64*(-c)^(3/4)*d^(21/4)))*(5*a^2*d^2 + 1
17*b^2*c^2 - 90*a*b*c*d)*1i)/(64*(-c)^(3/4)*d^(17/4)) - (((x^(1/2)*(25*a^4*d^4 + 13689*b^4*c^4 + 9270*a^2*b^2*
c^2*d^2 - 21060*a*b^3*c^3*d - 900*a^3*b*c*d^3))/(64*d^5) + ((5*a^2*d^2 + 117*b^2*c^2 - 90*a*b*c*d)*(117*b^2*c^
3 + 5*a^2*c*d^2 - 90*a*b*c^2*d)*1i)/(64*(-c)^(3/4)*d^(21/4)))*(5*a^2*d^2 + 117*b^2*c^2 - 90*a*b*c*d)*1i)/(64*(
-c)^(3/4)*d^(17/4))))*(5*a^2*d^2 + 117*b^2*c^2 - 90*a*b*c*d))/(32*(-c)^(3/4)*d^(17/4))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(7/2)*(b*x**2+a)**2/(d*x**2+c)**3,x)

[Out]

Timed out

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