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

Optimal. Leaf size=368 \[ -\frac {(b c-a d)^2 (7 a d+5 b c) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} a^{9/4} b^{11/4}}+\frac {(b c-a d)^2 (7 a d+5 b c) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} a^{9/4} b^{11/4}}+\frac {(b c-a d)^2 (7 a d+5 b c) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{9/4} b^{11/4}}-\frac {(b c-a d)^2 (7 a d+5 b c) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt {2} a^{9/4} b^{11/4}}-\frac {c^2 (5 b c-a d)}{2 a^2 b \sqrt {x}}-\frac {d^2 x^{3/2} (3 b c-7 a d)}{6 a b^2}+\frac {\left (c+d x^2\right )^2 (b c-a d)}{2 a b \sqrt {x} \left (a+b x^2\right )} \]

[Out]

-1/6*d^2*(-7*a*d+3*b*c)*x^(3/2)/a/b^2+1/8*(-a*d+b*c)^2*(7*a*d+5*b*c)*arctan(1-b^(1/4)*2^(1/2)*x^(1/2)/a^(1/4))
/a^(9/4)/b^(11/4)*2^(1/2)-1/8*(-a*d+b*c)^2*(7*a*d+5*b*c)*arctan(1+b^(1/4)*2^(1/2)*x^(1/2)/a^(1/4))/a^(9/4)/b^(
11/4)*2^(1/2)-1/16*(-a*d+b*c)^2*(7*a*d+5*b*c)*ln(a^(1/2)+x*b^(1/2)-a^(1/4)*b^(1/4)*2^(1/2)*x^(1/2))/a^(9/4)/b^
(11/4)*2^(1/2)+1/16*(-a*d+b*c)^2*(7*a*d+5*b*c)*ln(a^(1/2)+x*b^(1/2)+a^(1/4)*b^(1/4)*2^(1/2)*x^(1/2))/a^(9/4)/b
^(11/4)*2^(1/2)-1/2*c^2*(-a*d+5*b*c)/a^2/b/x^(1/2)+1/2*(-a*d+b*c)*(d*x^2+c)^2/a/b/(b*x^2+a)/x^(1/2)

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Rubi [A]  time = 0.43, antiderivative size = 368, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {466, 468, 570, 297, 1162, 617, 204, 1165, 628} \[ -\frac {(b c-a d)^2 (7 a d+5 b c) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} a^{9/4} b^{11/4}}+\frac {(b c-a d)^2 (7 a d+5 b c) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} a^{9/4} b^{11/4}}+\frac {(b c-a d)^2 (7 a d+5 b c) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{9/4} b^{11/4}}-\frac {(b c-a d)^2 (7 a d+5 b c) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt {2} a^{9/4} b^{11/4}}-\frac {c^2 (5 b c-a d)}{2 a^2 b \sqrt {x}}-\frac {d^2 x^{3/2} (3 b c-7 a d)}{6 a b^2}+\frac {\left (c+d x^2\right )^2 (b c-a d)}{2 a b \sqrt {x} \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(c^2*(5*b*c - a*d))/(2*a^2*b*Sqrt[x]) - (d^2*(3*b*c - 7*a*d)*x^(3/2))/(6*a*b^2) + ((b*c - a*d)*(c + d*x^2)^2)
/(2*a*b*Sqrt[x]*(a + b*x^2)) + ((b*c - a*d)^2*(5*b*c + 7*a*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(
4*Sqrt[2]*a^(9/4)*b^(11/4)) - ((b*c - a*d)^2*(5*b*c + 7*a*d)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4
*Sqrt[2]*a^(9/4)*b^(11/4)) - ((b*c - a*d)^2*(5*b*c + 7*a*d)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sq
rt[b]*x])/(8*Sqrt[2]*a^(9/4)*b^(11/4)) + ((b*c - a*d)^2*(5*b*c + 7*a*d)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*
Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(9/4)*b^(11/4))

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 297

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

Rule 466

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> With[{k = Deno
minator[m]}, Dist[k/e, Subst[Int[x^(k*(m + 1) - 1)*(a + (b*x^(k*n))/e^n)^p*(c + (d*x^(k*n))/e^n)^q, x], x, (e*
x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && FractionQ[m] && Intege
rQ[p]

Rule 468

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

Rule 570

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_))^
(r_.), x_Symbol] :> Int[ExpandIntegrand[(g*x)^m*(a + b*x^n)^p*(c + d*x^n)^q*(e + f*x^n)^r, x], x] /; FreeQ[{a,
b, c, d, e, f, g, m, n}, x] && IGtQ[p, -2] && IGtQ[q, 0] && IGtQ[r, 0]

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 {\left (c+d x^2\right )^3}{x^{3/2} \left (a+b x^2\right )^2} \, dx &=2 \operatorname {Subst}\left (\int \frac {\left (c+d x^4\right )^3}{x^2 \left (a+b x^4\right )^2} \, dx,x,\sqrt {x}\right )\\ &=\frac {(b c-a d) \left (c+d x^2\right )^2}{2 a b \sqrt {x} \left (a+b x^2\right )}-\frac {\operatorname {Subst}\left (\int \frac {\left (c+d x^4\right ) \left (-c (5 b c-a d)+d (3 b c-7 a d) x^4\right )}{x^2 \left (a+b x^4\right )} \, dx,x,\sqrt {x}\right )}{2 a b}\\ &=\frac {(b c-a d) \left (c+d x^2\right )^2}{2 a b \sqrt {x} \left (a+b x^2\right )}-\frac {\operatorname {Subst}\left (\int \left (\frac {c^2 (-5 b c+a d)}{a x^2}+\frac {d^2 (3 b c-7 a d) x^2}{b}+\frac {(-b c+a d)^2 (5 b c+7 a d) x^2}{a b \left (a+b x^4\right )}\right ) \, dx,x,\sqrt {x}\right )}{2 a b}\\ &=-\frac {c^2 (5 b c-a d)}{2 a^2 b \sqrt {x}}-\frac {d^2 (3 b c-7 a d) x^{3/2}}{6 a b^2}+\frac {(b c-a d) \left (c+d x^2\right )^2}{2 a b \sqrt {x} \left (a+b x^2\right )}-\frac {\left ((b c-a d)^2 (5 b c+7 a d)\right ) \operatorname {Subst}\left (\int \frac {x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{2 a^2 b^2}\\ &=-\frac {c^2 (5 b c-a d)}{2 a^2 b \sqrt {x}}-\frac {d^2 (3 b c-7 a d) x^{3/2}}{6 a b^2}+\frac {(b c-a d) \left (c+d x^2\right )^2}{2 a b \sqrt {x} \left (a+b x^2\right )}+\frac {\left ((b c-a d)^2 (5 b c+7 a d)\right ) \operatorname {Subst}\left (\int \frac {\sqrt {a}-\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{4 a^2 b^{5/2}}-\frac {\left ((b c-a d)^2 (5 b c+7 a d)\right ) \operatorname {Subst}\left (\int \frac {\sqrt {a}+\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{4 a^2 b^{5/2}}\\ &=-\frac {c^2 (5 b c-a d)}{2 a^2 b \sqrt {x}}-\frac {d^2 (3 b c-7 a d) x^{3/2}}{6 a b^2}+\frac {(b c-a d) \left (c+d x^2\right )^2}{2 a b \sqrt {x} \left (a+b x^2\right )}-\frac {\left ((b c-a d)^2 (5 b c+7 a d)\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt {x}\right )}{8 a^2 b^3}-\frac {\left ((b c-a d)^2 (5 b c+7 a d)\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt {x}\right )}{8 a^2 b^3}-\frac {\left ((b c-a d)^2 (5 b c+7 a d)\right ) \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{b}}+2 x}{-\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx,x,\sqrt {x}\right )}{8 \sqrt {2} a^{9/4} b^{11/4}}-\frac {\left ((b c-a d)^2 (5 b c+7 a d)\right ) \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{b}}-2 x}{-\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx,x,\sqrt {x}\right )}{8 \sqrt {2} a^{9/4} b^{11/4}}\\ &=-\frac {c^2 (5 b c-a d)}{2 a^2 b \sqrt {x}}-\frac {d^2 (3 b c-7 a d) x^{3/2}}{6 a b^2}+\frac {(b c-a d) \left (c+d x^2\right )^2}{2 a b \sqrt {x} \left (a+b x^2\right )}-\frac {(b c-a d)^2 (5 b c+7 a d) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{9/4} b^{11/4}}+\frac {(b c-a d)^2 (5 b c+7 a d) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{9/4} b^{11/4}}-\frac {\left ((b c-a d)^2 (5 b c+7 a d)\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{9/4} b^{11/4}}+\frac {\left ((b c-a d)^2 (5 b c+7 a d)\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{9/4} b^{11/4}}\\ &=-\frac {c^2 (5 b c-a d)}{2 a^2 b \sqrt {x}}-\frac {d^2 (3 b c-7 a d) x^{3/2}}{6 a b^2}+\frac {(b c-a d) \left (c+d x^2\right )^2}{2 a b \sqrt {x} \left (a+b x^2\right )}+\frac {(b c-a d)^2 (5 b c+7 a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{9/4} b^{11/4}}-\frac {(b c-a d)^2 (5 b c+7 a d) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{9/4} b^{11/4}}-\frac {(b c-a d)^2 (5 b c+7 a d) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{9/4} b^{11/4}}+\frac {(b c-a d)^2 (5 b c+7 a d) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{9/4} b^{11/4}}\\ \end {align*}

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Mathematica [C]  time = 2.39, size = 355, normalized size = 0.96 \[ -\frac {32768 b^3 x^6 \left (c+d x^2\right )^3 \, _5F_4\left (\frac {3}{4},2,2,2,2;1,1,1,\frac {19}{4};-\frac {b x^2}{a}\right )+55 \left (a \left (7 a^2 \left (14641 c^3+43923 c^2 d x^2+43923 c d^2 x^4+11953 d^3 x^6\right )+18 a b x^2 \left (361 c^3+1083 c^2 d x^2+2427 c d^2 x^4+809 d^3 x^6\right )-21 b^2 x^4 \left (-1919 c^3+3 c^2 d x^2+3 c d^2 x^4+d^3 x^6\right )\right )-7 \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};-\frac {b x^2}{a}\right ) \left (a^3 \left (14641 c^3+43923 c^2 d x^2+43923 c d^2 x^4+11953 d^3 x^6\right )+3 a^2 b x^2 \left (2401 c^3+7203 c^2 d x^2+8355 c d^2 x^4+2401 d^3 x^6\right )+9 a b^2 x^4 \left (27 c^3+209 c^2 d x^2+81 c d^2 x^4+27 d^3 x^6\right )+b^3 x^6 \left (-1919 c^3+3 c^2 d x^2+3 c d^2 x^4+d^3 x^6\right )\right )\right )}{887040 a^3 b^2 x^{9/2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-1/887040*(55*(a*(-21*b^2*x^4*(-1919*c^3 + 3*c^2*d*x^2 + 3*c*d^2*x^4 + d^3*x^6) + 18*a*b*x^2*(361*c^3 + 1083*c
^2*d*x^2 + 2427*c*d^2*x^4 + 809*d^3*x^6) + 7*a^2*(14641*c^3 + 43923*c^2*d*x^2 + 43923*c*d^2*x^4 + 11953*d^3*x^
6)) - 7*(b^3*x^6*(-1919*c^3 + 3*c^2*d*x^2 + 3*c*d^2*x^4 + d^3*x^6) + 9*a*b^2*x^4*(27*c^3 + 209*c^2*d*x^2 + 81*
c*d^2*x^4 + 27*d^3*x^6) + 3*a^2*b*x^2*(2401*c^3 + 7203*c^2*d*x^2 + 8355*c*d^2*x^4 + 2401*d^3*x^6) + a^3*(14641
*c^3 + 43923*c^2*d*x^2 + 43923*c*d^2*x^4 + 11953*d^3*x^6))*Hypergeometric2F1[3/4, 1, 7/4, -((b*x^2)/a)]) + 327
68*b^3*x^6*(c + d*x^2)^3*HypergeometricPFQ[{3/4, 2, 2, 2, 2}, {1, 1, 1, 19/4}, -((b*x^2)/a)])/(a^3*b^2*x^(9/2)
)

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fricas [B]  time = 0.61, size = 2547, normalized size = 6.92 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*(12*(a^2*b^3*x^3 + a^3*b^2*x)*(-(625*b^12*c^12 - 1500*a*b^11*c^11*d - 3150*a^2*b^10*c^10*d^2 + 11060*a^3*
b^9*c^9*d^3 + 1071*a^4*b^8*c^8*d^4 - 28728*a^5*b^7*c^7*d^5 + 19068*a^6*b^6*c^6*d^6 + 27144*a^7*b^5*c^5*d^7 - 3
7665*a^8*b^4*c^4*d^8 + 2324*a^9*b^3*c^3*d^9 + 19698*a^10*b^2*c^2*d^10 - 12348*a^11*b*c*d^11 + 2401*a^12*d^12)/
(a^9*b^11))^(1/4)*arctan((sqrt((15625*b^18*c^18 - 56250*a*b^17*c^17*d - 84375*a^2*b^16*c^16*d^2 + 570000*a^3*b
^15*c^15*d^3 - 211500*a^4*b^14*c^14*d^4 - 2174040*a^5*b^13*c^13*d^5 + 2720004*a^6*b^12*c^12*d^6 + 3321072*a^7*
b^11*c^11*d^7 - 8368866*a^8*b^10*c^10*d^8 + 640420*a^9*b^9*c^9*d^9 + 11255310*a^10*b^8*c^8*d^10 - 8509968*a^11
*b^7*c^7*d^11 - 4831644*a^12*b^6*c^6*d^12 + 9537192*a^13*b^5*c^5*d^13 - 3095820*a^14*b^4*c^4*d^14 - 2551920*a^
15*b^3*c^3*d^15 + 2614689*a^16*b^2*c^2*d^16 - 907578*a^17*b*c*d^17 + 117649*a^18*d^18)*x - (625*a^5*b^17*c^12
- 1500*a^6*b^16*c^11*d - 3150*a^7*b^15*c^10*d^2 + 11060*a^8*b^14*c^9*d^3 + 1071*a^9*b^13*c^8*d^4 - 28728*a^10*
b^12*c^7*d^5 + 19068*a^11*b^11*c^6*d^6 + 27144*a^12*b^10*c^5*d^7 - 37665*a^13*b^9*c^4*d^8 + 2324*a^14*b^8*c^3*
d^9 + 19698*a^15*b^7*c^2*d^10 - 12348*a^16*b^6*c*d^11 + 2401*a^17*b^5*d^12)*sqrt(-(625*b^12*c^12 - 1500*a*b^11
*c^11*d - 3150*a^2*b^10*c^10*d^2 + 11060*a^3*b^9*c^9*d^3 + 1071*a^4*b^8*c^8*d^4 - 28728*a^5*b^7*c^7*d^5 + 1906
8*a^6*b^6*c^6*d^6 + 27144*a^7*b^5*c^5*d^7 - 37665*a^8*b^4*c^4*d^8 + 2324*a^9*b^3*c^3*d^9 + 19698*a^10*b^2*c^2*
d^10 - 12348*a^11*b*c*d^11 + 2401*a^12*d^12)/(a^9*b^11)))*a^2*b^3*(-(625*b^12*c^12 - 1500*a*b^11*c^11*d - 3150
*a^2*b^10*c^10*d^2 + 11060*a^3*b^9*c^9*d^3 + 1071*a^4*b^8*c^8*d^4 - 28728*a^5*b^7*c^7*d^5 + 19068*a^6*b^6*c^6*
d^6 + 27144*a^7*b^5*c^5*d^7 - 37665*a^8*b^4*c^4*d^8 + 2324*a^9*b^3*c^3*d^9 + 19698*a^10*b^2*c^2*d^10 - 12348*a
^11*b*c*d^11 + 2401*a^12*d^12)/(a^9*b^11))^(1/4) - (125*a^2*b^12*c^9 - 225*a^3*b^11*c^8*d - 540*a^4*b^10*c^7*d
^2 + 1308*a^5*b^9*c^6*d^3 + 342*a^6*b^8*c^5*d^4 - 2430*a^7*b^7*c^4*d^5 + 1140*a^8*b^6*c^3*d^6 + 1260*a^9*b^5*c
^2*d^7 - 1323*a^10*b^4*c*d^8 + 343*a^11*b^3*d^9)*sqrt(x)*(-(625*b^12*c^12 - 1500*a*b^11*c^11*d - 3150*a^2*b^10
*c^10*d^2 + 11060*a^3*b^9*c^9*d^3 + 1071*a^4*b^8*c^8*d^4 - 28728*a^5*b^7*c^7*d^5 + 19068*a^6*b^6*c^6*d^6 + 271
44*a^7*b^5*c^5*d^7 - 37665*a^8*b^4*c^4*d^8 + 2324*a^9*b^3*c^3*d^9 + 19698*a^10*b^2*c^2*d^10 - 12348*a^11*b*c*d
^11 + 2401*a^12*d^12)/(a^9*b^11))^(1/4))/(625*b^12*c^12 - 1500*a*b^11*c^11*d - 3150*a^2*b^10*c^10*d^2 + 11060*
a^3*b^9*c^9*d^3 + 1071*a^4*b^8*c^8*d^4 - 28728*a^5*b^7*c^7*d^5 + 19068*a^6*b^6*c^6*d^6 + 27144*a^7*b^5*c^5*d^7
- 37665*a^8*b^4*c^4*d^8 + 2324*a^9*b^3*c^3*d^9 + 19698*a^10*b^2*c^2*d^10 - 12348*a^11*b*c*d^11 + 2401*a^12*d^
12)) - 3*(a^2*b^3*x^3 + a^3*b^2*x)*(-(625*b^12*c^12 - 1500*a*b^11*c^11*d - 3150*a^2*b^10*c^10*d^2 + 11060*a^3*
b^9*c^9*d^3 + 1071*a^4*b^8*c^8*d^4 - 28728*a^5*b^7*c^7*d^5 + 19068*a^6*b^6*c^6*d^6 + 27144*a^7*b^5*c^5*d^7 - 3
7665*a^8*b^4*c^4*d^8 + 2324*a^9*b^3*c^3*d^9 + 19698*a^10*b^2*c^2*d^10 - 12348*a^11*b*c*d^11 + 2401*a^12*d^12)/
(a^9*b^11))^(1/4)*log(a^7*b^8*(-(625*b^12*c^12 - 1500*a*b^11*c^11*d - 3150*a^2*b^10*c^10*d^2 + 11060*a^3*b^9*c
^9*d^3 + 1071*a^4*b^8*c^8*d^4 - 28728*a^5*b^7*c^7*d^5 + 19068*a^6*b^6*c^6*d^6 + 27144*a^7*b^5*c^5*d^7 - 37665*
a^8*b^4*c^4*d^8 + 2324*a^9*b^3*c^3*d^9 + 19698*a^10*b^2*c^2*d^10 - 12348*a^11*b*c*d^11 + 2401*a^12*d^12)/(a^9*
b^11))^(3/4) + (125*b^9*c^9 - 225*a*b^8*c^8*d - 540*a^2*b^7*c^7*d^2 + 1308*a^3*b^6*c^6*d^3 + 342*a^4*b^5*c^5*d
^4 - 2430*a^5*b^4*c^4*d^5 + 1140*a^6*b^3*c^3*d^6 + 1260*a^7*b^2*c^2*d^7 - 1323*a^8*b*c*d^8 + 343*a^9*d^9)*sqrt
(x)) + 3*(a^2*b^3*x^3 + a^3*b^2*x)*(-(625*b^12*c^12 - 1500*a*b^11*c^11*d - 3150*a^2*b^10*c^10*d^2 + 11060*a^3*
b^9*c^9*d^3 + 1071*a^4*b^8*c^8*d^4 - 28728*a^5*b^7*c^7*d^5 + 19068*a^6*b^6*c^6*d^6 + 27144*a^7*b^5*c^5*d^7 - 3
7665*a^8*b^4*c^4*d^8 + 2324*a^9*b^3*c^3*d^9 + 19698*a^10*b^2*c^2*d^10 - 12348*a^11*b*c*d^11 + 2401*a^12*d^12)/
(a^9*b^11))^(1/4)*log(-a^7*b^8*(-(625*b^12*c^12 - 1500*a*b^11*c^11*d - 3150*a^2*b^10*c^10*d^2 + 11060*a^3*b^9*
c^9*d^3 + 1071*a^4*b^8*c^8*d^4 - 28728*a^5*b^7*c^7*d^5 + 19068*a^6*b^6*c^6*d^6 + 27144*a^7*b^5*c^5*d^7 - 37665
*a^8*b^4*c^4*d^8 + 2324*a^9*b^3*c^3*d^9 + 19698*a^10*b^2*c^2*d^10 - 12348*a^11*b*c*d^11 + 2401*a^12*d^12)/(a^9
*b^11))^(3/4) + (125*b^9*c^9 - 225*a*b^8*c^8*d - 540*a^2*b^7*c^7*d^2 + 1308*a^3*b^6*c^6*d^3 + 342*a^4*b^5*c^5*
d^4 - 2430*a^5*b^4*c^4*d^5 + 1140*a^6*b^3*c^3*d^6 + 1260*a^7*b^2*c^2*d^7 - 1323*a^8*b*c*d^8 + 343*a^9*d^9)*sqr
t(x)) + 4*(4*a^2*b*d^3*x^4 - 12*a*b^2*c^3 - (15*b^3*c^3 - 9*a*b^2*c^2*d + 9*a^2*b*c*d^2 - 7*a^3*d^3)*x^2)*sqrt
(x))/(a^2*b^3*x^3 + a^3*b^2*x)

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giac [A]  time = 0.50, size = 504, normalized size = 1.37 \[ \frac {2 \, d^{3} x^{\frac {3}{2}}}{3 \, b^{2}} - \frac {5 \, b^{3} c^{3} x^{2} - 3 \, a b^{2} c^{2} d x^{2} + 3 \, a^{2} b c d^{2} x^{2} - a^{3} d^{3} x^{2} + 4 \, a b^{2} c^{3}}{2 \, {\left (b x^{\frac {5}{2}} + a \sqrt {x}\right )} a^{2} b^{2}} - \frac {\sqrt {2} {\left (5 \, \left (a b^{3}\right )^{\frac {3}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac {3}{4}} a b^{2} c^{2} d - 9 \, \left (a b^{3}\right )^{\frac {3}{4}} a^{2} b c d^{2} + 7 \, \left (a b^{3}\right )^{\frac {3}{4}} a^{3} d^{3}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{8 \, a^{3} b^{5}} - \frac {\sqrt {2} {\left (5 \, \left (a b^{3}\right )^{\frac {3}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac {3}{4}} a b^{2} c^{2} d - 9 \, \left (a b^{3}\right )^{\frac {3}{4}} a^{2} b c d^{2} + 7 \, \left (a b^{3}\right )^{\frac {3}{4}} a^{3} d^{3}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{8 \, a^{3} b^{5}} + \frac {\sqrt {2} {\left (5 \, \left (a b^{3}\right )^{\frac {3}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac {3}{4}} a b^{2} c^{2} d - 9 \, \left (a b^{3}\right )^{\frac {3}{4}} a^{2} b c d^{2} + 7 \, \left (a b^{3}\right )^{\frac {3}{4}} a^{3} d^{3}\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{16 \, a^{3} b^{5}} - \frac {\sqrt {2} {\left (5 \, \left (a b^{3}\right )^{\frac {3}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac {3}{4}} a b^{2} c^{2} d - 9 \, \left (a b^{3}\right )^{\frac {3}{4}} a^{2} b c d^{2} + 7 \, \left (a b^{3}\right )^{\frac {3}{4}} a^{3} d^{3}\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{16 \, a^{3} b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*d^3*x^(3/2)/b^2 - 1/2*(5*b^3*c^3*x^2 - 3*a*b^2*c^2*d*x^2 + 3*a^2*b*c*d^2*x^2 - a^3*d^3*x^2 + 4*a*b^2*c^3)/
((b*x^(5/2) + a*sqrt(x))*a^2*b^2) - 1/8*sqrt(2)*(5*(a*b^3)^(3/4)*b^3*c^3 - 3*(a*b^3)^(3/4)*a*b^2*c^2*d - 9*(a*
b^3)^(3/4)*a^2*b*c*d^2 + 7*(a*b^3)^(3/4)*a^3*d^3)*arctan(1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) + 2*sqrt(x))/(a/b)^(
1/4))/(a^3*b^5) - 1/8*sqrt(2)*(5*(a*b^3)^(3/4)*b^3*c^3 - 3*(a*b^3)^(3/4)*a*b^2*c^2*d - 9*(a*b^3)^(3/4)*a^2*b*c
*d^2 + 7*(a*b^3)^(3/4)*a^3*d^3)*arctan(-1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) - 2*sqrt(x))/(a/b)^(1/4))/(a^3*b^5) +
1/16*sqrt(2)*(5*(a*b^3)^(3/4)*b^3*c^3 - 3*(a*b^3)^(3/4)*a*b^2*c^2*d - 9*(a*b^3)^(3/4)*a^2*b*c*d^2 + 7*(a*b^3)
^(3/4)*a^3*d^3)*log(sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(a^3*b^5) - 1/16*sqrt(2)*(5*(a*b^3)^(3/4)*b^3
*c^3 - 3*(a*b^3)^(3/4)*a*b^2*c^2*d - 9*(a*b^3)^(3/4)*a^2*b*c*d^2 + 7*(a*b^3)^(3/4)*a^3*d^3)*log(-sqrt(2)*sqrt(
x)*(a/b)^(1/4) + x + sqrt(a/b))/(a^3*b^5)

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maple [B]  time = 0.02, size = 682, normalized size = 1.85 \[ \frac {a \,d^{3} x^{\frac {3}{2}}}{2 \left (b \,x^{2}+a \right ) b^{2}}+\frac {3 c^{2} d \,x^{\frac {3}{2}}}{2 \left (b \,x^{2}+a \right ) a}-\frac {b \,c^{3} x^{\frac {3}{2}}}{2 \left (b \,x^{2}+a \right ) a^{2}}-\frac {3 c \,d^{2} x^{\frac {3}{2}}}{2 \left (b \,x^{2}+a \right ) b}+\frac {2 d^{3} x^{\frac {3}{2}}}{3 b^{2}}-\frac {7 \sqrt {2}\, a \,d^{3} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )}{8 \left (\frac {a}{b}\right )^{\frac {1}{4}} b^{3}}-\frac {7 \sqrt {2}\, a \,d^{3} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )}{8 \left (\frac {a}{b}\right )^{\frac {1}{4}} b^{3}}-\frac {7 \sqrt {2}\, a \,d^{3} \ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}\right )}{16 \left (\frac {a}{b}\right )^{\frac {1}{4}} b^{3}}+\frac {3 \sqrt {2}\, c^{2} d \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )}{8 \left (\frac {a}{b}\right )^{\frac {1}{4}} a b}+\frac {3 \sqrt {2}\, c^{2} d \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )}{8 \left (\frac {a}{b}\right )^{\frac {1}{4}} a b}+\frac {3 \sqrt {2}\, c^{2} d \ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}\right )}{16 \left (\frac {a}{b}\right )^{\frac {1}{4}} a b}-\frac {5 \sqrt {2}\, c^{3} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )}{8 \left (\frac {a}{b}\right )^{\frac {1}{4}} a^{2}}-\frac {5 \sqrt {2}\, c^{3} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )}{8 \left (\frac {a}{b}\right )^{\frac {1}{4}} a^{2}}-\frac {5 \sqrt {2}\, c^{3} \ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}\right )}{16 \left (\frac {a}{b}\right )^{\frac {1}{4}} a^{2}}+\frac {9 \sqrt {2}\, c \,d^{2} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )}{8 \left (\frac {a}{b}\right )^{\frac {1}{4}} b^{2}}+\frac {9 \sqrt {2}\, c \,d^{2} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )}{8 \left (\frac {a}{b}\right )^{\frac {1}{4}} b^{2}}+\frac {9 \sqrt {2}\, c \,d^{2} \ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}\right )}{16 \left (\frac {a}{b}\right )^{\frac {1}{4}} b^{2}}-\frac {2 c^{3}}{a^{2} \sqrt {x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*d^3*x^(3/2)/b^2+1/2*a/b^2*x^(3/2)/(b*x^2+a)*d^3-3/2/b*x^(3/2)/(b*x^2+a)*c*d^2+3/2/a*x^(3/2)/(b*x^2+a)*c^2*
d-1/2/a^2*b*x^(3/2)/(b*x^2+a)*c^3-7/16*a/b^3/(a/b)^(1/4)*2^(1/2)*d^3*ln((x-(a/b)^(1/4)*2^(1/2)*x^(1/2)+(a/b)^(
1/2))/(x+(a/b)^(1/4)*2^(1/2)*x^(1/2)+(a/b)^(1/2)))-7/8*a/b^3/(a/b)^(1/4)*2^(1/2)*d^3*arctan(2^(1/2)/(a/b)^(1/4
)*x^(1/2)+1)-7/8*a/b^3/(a/b)^(1/4)*2^(1/2)*d^3*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)-1)+3/16/a/b/(a/b)^(1/4)*2^(1
/2)*c^2*d*ln((x-(a/b)^(1/4)*2^(1/2)*x^(1/2)+(a/b)^(1/2))/(x+(a/b)^(1/4)*2^(1/2)*x^(1/2)+(a/b)^(1/2)))+3/8/a/b/
(a/b)^(1/4)*2^(1/2)*c^2*d*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)+3/8/a/b/(a/b)^(1/4)*2^(1/2)*c^2*d*arctan(2^(1/
2)/(a/b)^(1/4)*x^(1/2)-1)-5/16/a^2/(a/b)^(1/4)*2^(1/2)*c^3*ln((x-(a/b)^(1/4)*2^(1/2)*x^(1/2)+(a/b)^(1/2))/(x+(
a/b)^(1/4)*2^(1/2)*x^(1/2)+(a/b)^(1/2)))-5/8/a^2/(a/b)^(1/4)*2^(1/2)*c^3*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)
-5/8/a^2/(a/b)^(1/4)*2^(1/2)*c^3*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)-1)+9/16/b^2/(a/b)^(1/4)*2^(1/2)*c*d^2*ln((
x-(a/b)^(1/4)*2^(1/2)*x^(1/2)+(a/b)^(1/2))/(x+(a/b)^(1/4)*2^(1/2)*x^(1/2)+(a/b)^(1/2)))+9/8/b^2/(a/b)^(1/4)*2^
(1/2)*c*d^2*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)+9/8/b^2/(a/b)^(1/4)*2^(1/2)*c*d^2*arctan(2^(1/2)/(a/b)^(1/4)
*x^(1/2)-1)-2*c^3/a^2/x^(1/2)

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maxima [A]  time = 2.48, size = 304, normalized size = 0.83 \[ \frac {2 \, d^{3} x^{\frac {3}{2}}}{3 \, b^{2}} - \frac {4 \, a b^{2} c^{3} + {\left (5 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} x^{2}}{2 \, {\left (a^{2} b^{3} x^{\frac {5}{2}} + a^{3} b^{2} \sqrt {x}\right )}} - \frac {{\left (5 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 9 \, a^{2} b c d^{2} + 7 \, a^{3} d^{3}\right )} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} - \frac {\sqrt {2} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}}\right )}}{16 \, a^{2} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*d^3*x^(3/2)/b^2 - 1/2*(4*a*b^2*c^3 + (5*b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*x^2)/(a^2*b^3*x
^(5/2) + a^3*b^2*sqrt(x)) - 1/16*(5*b^3*c^3 - 3*a*b^2*c^2*d - 9*a^2*b*c*d^2 + 7*a^3*d^3)*(2*sqrt(2)*arctan(1/2
*sqrt(2)*(sqrt(2)*a^(1/4)*b^(1/4) + 2*sqrt(b)*sqrt(x))/sqrt(sqrt(a)*sqrt(b)))/(sqrt(sqrt(a)*sqrt(b))*sqrt(b))
+ 2*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2)*a^(1/4)*b^(1/4) - 2*sqrt(b)*sqrt(x))/sqrt(sqrt(a)*sqrt(b)))/(sqrt(sqr
t(a)*sqrt(b))*sqrt(b)) - sqrt(2)*log(sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a))/(a^(1/4)*b^(3/4))
+ sqrt(2)*log(-sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a))/(a^(1/4)*b^(3/4)))/(a^2*b^2)

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mupad [B]  time = 0.41, size = 657, normalized size = 1.79 \[ \frac {\frac {x^2\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-5\,b^3\,c^3\right )}{2\,a^2}-\frac {2\,b^2\,c^3}{a}}{b^3\,x^{5/2}+a\,b^2\,\sqrt {x}}+\frac {2\,d^3\,x^{3/2}}{3\,b^2}-\frac {\mathrm {atan}\left (\frac {\sqrt {x}\,{\left (a\,d-b\,c\right )}^2\,\left (7\,a\,d+5\,b\,c\right )\,\left (1568\,a^{13}\,b^8\,d^6-4032\,a^{12}\,b^9\,c\,d^5+1248\,a^{11}\,b^{10}\,c^2\,d^4+3968\,a^{10}\,b^{11}\,c^3\,d^3-2592\,a^9\,b^{12}\,c^4\,d^2-960\,a^8\,b^{13}\,c^5\,d+800\,a^7\,b^{14}\,c^6\right )}{4\,{\left (-a\right )}^{9/4}\,b^{11/4}\,\left (2744\,a^{14}\,b^5\,d^9-10584\,a^{13}\,b^6\,c\,d^8+10080\,a^{12}\,b^7\,c^2\,d^7+9120\,a^{11}\,b^8\,c^3\,d^6-19440\,a^{10}\,b^9\,c^4\,d^5+2736\,a^9\,b^{10}\,c^5\,d^4+10464\,a^8\,b^{11}\,c^6\,d^3-4320\,a^7\,b^{12}\,c^7\,d^2-1800\,a^6\,b^{13}\,c^8\,d+1000\,a^5\,b^{14}\,c^9\right )}\right )\,{\left (a\,d-b\,c\right )}^2\,\left (7\,a\,d+5\,b\,c\right )}{4\,{\left (-a\right )}^{9/4}\,b^{11/4}}-\frac {\mathrm {atan}\left (\frac {\sqrt {x}\,{\left (a\,d-b\,c\right )}^2\,\left (7\,a\,d+5\,b\,c\right )\,\left (1568\,a^{13}\,b^8\,d^6-4032\,a^{12}\,b^9\,c\,d^5+1248\,a^{11}\,b^{10}\,c^2\,d^4+3968\,a^{10}\,b^{11}\,c^3\,d^3-2592\,a^9\,b^{12}\,c^4\,d^2-960\,a^8\,b^{13}\,c^5\,d+800\,a^7\,b^{14}\,c^6\right )\,1{}\mathrm {i}}{4\,{\left (-a\right )}^{9/4}\,b^{11/4}\,\left (2744\,a^{14}\,b^5\,d^9-10584\,a^{13}\,b^6\,c\,d^8+10080\,a^{12}\,b^7\,c^2\,d^7+9120\,a^{11}\,b^8\,c^3\,d^6-19440\,a^{10}\,b^9\,c^4\,d^5+2736\,a^9\,b^{10}\,c^5\,d^4+10464\,a^8\,b^{11}\,c^6\,d^3-4320\,a^7\,b^{12}\,c^7\,d^2-1800\,a^6\,b^{13}\,c^8\,d+1000\,a^5\,b^{14}\,c^9\right )}\right )\,{\left (a\,d-b\,c\right )}^2\,\left (7\,a\,d+5\,b\,c\right )\,1{}\mathrm {i}}{4\,{\left (-a\right )}^{9/4}\,b^{11/4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((x^2*(a^3*d^3 - 5*b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2))/(2*a^2) - (2*b^2*c^3)/a)/(b^3*x^(5/2) + a*b^2*x^(
1/2)) + (2*d^3*x^(3/2))/(3*b^2) - (atan((x^(1/2)*(a*d - b*c)^2*(7*a*d + 5*b*c)*(800*a^7*b^14*c^6 + 1568*a^13*b
^8*d^6 - 960*a^8*b^13*c^5*d - 4032*a^12*b^9*c*d^5 - 2592*a^9*b^12*c^4*d^2 + 3968*a^10*b^11*c^3*d^3 + 1248*a^11
*b^10*c^2*d^4))/(4*(-a)^(9/4)*b^(11/4)*(1000*a^5*b^14*c^9 + 2744*a^14*b^5*d^9 - 1800*a^6*b^13*c^8*d - 10584*a^
13*b^6*c*d^8 - 4320*a^7*b^12*c^7*d^2 + 10464*a^8*b^11*c^6*d^3 + 2736*a^9*b^10*c^5*d^4 - 19440*a^10*b^9*c^4*d^5
+ 9120*a^11*b^8*c^3*d^6 + 10080*a^12*b^7*c^2*d^7)))*(a*d - b*c)^2*(7*a*d + 5*b*c))/(4*(-a)^(9/4)*b^(11/4)) -
(atan((x^(1/2)*(a*d - b*c)^2*(7*a*d + 5*b*c)*(800*a^7*b^14*c^6 + 1568*a^13*b^8*d^6 - 960*a^8*b^13*c^5*d - 4032
*a^12*b^9*c*d^5 - 2592*a^9*b^12*c^4*d^2 + 3968*a^10*b^11*c^3*d^3 + 1248*a^11*b^10*c^2*d^4)*1i)/(4*(-a)^(9/4)*b
^(11/4)*(1000*a^5*b^14*c^9 + 2744*a^14*b^5*d^9 - 1800*a^6*b^13*c^8*d - 10584*a^13*b^6*c*d^8 - 4320*a^7*b^12*c^
7*d^2 + 10464*a^8*b^11*c^6*d^3 + 2736*a^9*b^10*c^5*d^4 - 19440*a^10*b^9*c^4*d^5 + 9120*a^11*b^8*c^3*d^6 + 1008
0*a^12*b^7*c^2*d^7)))*(a*d - b*c)^2*(7*a*d + 5*b*c)*1i)/(4*(-a)^(9/4)*b^(11/4))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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