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

Optimal. Leaf size=306 \[ \frac {2 d x^{3/2} \left (a^2 d^2-3 a b c d+3 b^2 c^2\right )}{3 b^3}+\frac {(b c-a d)^3 \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} \sqrt [4]{a} b^{15/4}}-\frac {(b c-a d)^3 \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} \sqrt [4]{a} b^{15/4}}-\frac {(b c-a d)^3 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} b^{15/4}}+\frac {(b c-a d)^3 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} b^{15/4}}+\frac {2 d^2 x^{7/2} (3 b c-a d)}{7 b^2}+\frac {2 d^3 x^{11/2}}{11 b} \]

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Rubi [A]  time = 0.25, antiderivative size = 306, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {461, 329, 297, 1162, 617, 204, 1165, 628} \begin {gather*} \frac {2 d x^{3/2} \left (a^2 d^2-3 a b c d+3 b^2 c^2\right )}{3 b^3}+\frac {2 d^2 x^{7/2} (3 b c-a d)}{7 b^2}+\frac {(b c-a d)^3 \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} \sqrt [4]{a} b^{15/4}}-\frac {(b c-a d)^3 \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} \sqrt [4]{a} b^{15/4}}-\frac {(b c-a d)^3 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} b^{15/4}}+\frac {(b c-a d)^3 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} b^{15/4}}+\frac {2 d^3 x^{11/2}}{11 b} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*d*(3*b^2*c^2 - 3*a*b*c*d + a^2*d^2)*x^(3/2))/(3*b^3) + (2*d^2*(3*b*c - a*d)*x^(7/2))/(7*b^2) + (2*d^3*x^(11
/2))/(11*b) - ((b*c - a*d)^3*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(Sqrt[2]*a^(1/4)*b^(15/4)) + ((b*c
- a*d)^3*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(Sqrt[2]*a^(1/4)*b^(15/4)) + ((b*c - a*d)^3*Log[Sqrt[
a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*a^(1/4)*b^(15/4)) - ((b*c - a*d)^3*Log[Sqrt[a] +
Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*a^(1/4)*b^(15/4))

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 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 461

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

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 {\sqrt {x} \left (c+d x^2\right )^3}{a+b x^2} \, dx &=\int \left (\frac {d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) \sqrt {x}}{b^3}+\frac {d^2 (3 b c-a d) x^{5/2}}{b^2}+\frac {d^3 x^{9/2}}{b}+\frac {\left (b^3 c^3-3 a b^2 c^2 d+3 a^2 b c d^2-a^3 d^3\right ) \sqrt {x}}{b^3 \left (a+b x^2\right )}\right ) \, dx\\ &=\frac {2 d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) x^{3/2}}{3 b^3}+\frac {2 d^2 (3 b c-a d) x^{7/2}}{7 b^2}+\frac {2 d^3 x^{11/2}}{11 b}+\frac {(b c-a d)^3 \int \frac {\sqrt {x}}{a+b x^2} \, dx}{b^3}\\ &=\frac {2 d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) x^{3/2}}{3 b^3}+\frac {2 d^2 (3 b c-a d) x^{7/2}}{7 b^2}+\frac {2 d^3 x^{11/2}}{11 b}+\frac {\left (2 (b c-a d)^3\right ) \operatorname {Subst}\left (\int \frac {x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{b^3}\\ &=\frac {2 d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) x^{3/2}}{3 b^3}+\frac {2 d^2 (3 b c-a d) x^{7/2}}{7 b^2}+\frac {2 d^3 x^{11/2}}{11 b}-\frac {(b c-a d)^3 \operatorname {Subst}\left (\int \frac {\sqrt {a}-\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{b^{7/2}}+\frac {(b c-a d)^3 \operatorname {Subst}\left (\int \frac {\sqrt {a}+\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{b^{7/2}}\\ &=\frac {2 d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) x^{3/2}}{3 b^3}+\frac {2 d^2 (3 b c-a d) x^{7/2}}{7 b^2}+\frac {2 d^3 x^{11/2}}{11 b}+\frac {(b c-a d)^3 \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 )}{2 b^4}+\frac {(b c-a d)^3 \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 )}{2 b^4}+\frac {(b c-a d)^3 \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 )}{2 \sqrt {2} \sqrt [4]{a} b^{15/4}}+\frac {(b c-a d)^3 \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 )}{2 \sqrt {2} \sqrt [4]{a} b^{15/4}}\\ &=\frac {2 d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) x^{3/2}}{3 b^3}+\frac {2 d^2 (3 b c-a d) x^{7/2}}{7 b^2}+\frac {2 d^3 x^{11/2}}{11 b}+\frac {(b c-a d)^3 \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} \sqrt [4]{a} b^{15/4}}-\frac {(b c-a d)^3 \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} \sqrt [4]{a} b^{15/4}}+\frac {(b c-a d)^3 \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} b^{15/4}}-\frac {(b c-a d)^3 \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} b^{15/4}}\\ &=\frac {2 d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) x^{3/2}}{3 b^3}+\frac {2 d^2 (3 b c-a d) x^{7/2}}{7 b^2}+\frac {2 d^3 x^{11/2}}{11 b}-\frac {(b c-a d)^3 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} b^{15/4}}+\frac {(b c-a d)^3 \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} b^{15/4}}+\frac {(b c-a d)^3 \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} \sqrt [4]{a} b^{15/4}}-\frac {(b c-a d)^3 \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} \sqrt [4]{a} b^{15/4}}\\ \end {align*}

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Mathematica [C]  time = 0.38, size = 97, normalized size = 0.32 \begin {gather*} \frac {2 x^{3/2} \left (a d \left (77 a^2 d^2-33 a b d \left (7 c+d x^2\right )+3 b^2 \left (77 c^2+33 c d x^2+7 d^2 x^4\right )\right )+77 (b c-a d)^3 \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};-\frac {b x^2}{a}\right )\right )}{231 a b^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x^(3/2)*(a*d*(77*a^2*d^2 - 33*a*b*d*(7*c + d*x^2) + 3*b^2*(77*c^2 + 33*c*d*x^2 + 7*d^2*x^4)) + 77*(b*c - a*
d)^3*Hypergeometric2F1[3/4, 1, 7/4, -((b*x^2)/a)]))/(231*a*b^3)

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IntegrateAlgebraic [A]  time = 0.25, size = 199, normalized size = 0.65 \begin {gather*} \frac {2 d x^{3/2} \left (77 a^2 d^2-231 a b c d-33 a b d^2 x^2+231 b^2 c^2+99 b^2 c d x^2+21 b^2 d^2 x^4\right )}{231 b^3}+\frac {(a d-b c)^3 \tan ^{-1}\left (\frac {\frac {\sqrt [4]{a}}{\sqrt {2} \sqrt [4]{b}}-\frac {\sqrt [4]{b} x}{\sqrt {2} \sqrt [4]{a}}}{\sqrt {x}}\right )}{\sqrt {2} \sqrt [4]{a} b^{15/4}}+\frac {(a d-b c)^3 \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{\sqrt {2} \sqrt [4]{a} b^{15/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*d*x^(3/2)*(231*b^2*c^2 - 231*a*b*c*d + 77*a^2*d^2 + 99*b^2*c*d*x^2 - 33*a*b*d^2*x^2 + 21*b^2*d^2*x^4))/(231
*b^3) + ((-(b*c) + a*d)^3*ArcTan[(a^(1/4)/(Sqrt[2]*b^(1/4)) - (b^(1/4)*x)/(Sqrt[2]*a^(1/4)))/Sqrt[x]])/(Sqrt[2
]*a^(1/4)*b^(15/4)) + ((-(b*c) + a*d)^3*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/(Sqr
t[2]*a^(1/4)*b^(15/4))

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fricas [B]  time = 1.60, size = 2441, normalized size = 7.98

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^2+c)^3*x^(1/2)/(b*x^2+a),x, algorithm="fricas")

[Out]

1/462*(924*b^3*(-(b^12*c^12 - 12*a*b^11*c^11*d + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*
d^4 - 792*a^5*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*b^3*c^3*
d^9 + 66*a^10*b^2*c^2*d^10 - 12*a^11*b*c*d^11 + a^12*d^12)/(a*b^15))^(1/4)*arctan((sqrt((b^18*c^18 - 18*a*b^17
*c^17*d + 153*a^2*b^16*c^16*d^2 - 816*a^3*b^15*c^15*d^3 + 3060*a^4*b^14*c^14*d^4 - 8568*a^5*b^13*c^13*d^5 + 18
564*a^6*b^12*c^12*d^6 - 31824*a^7*b^11*c^11*d^7 + 43758*a^8*b^10*c^10*d^8 - 48620*a^9*b^9*c^9*d^9 + 43758*a^10
*b^8*c^8*d^10 - 31824*a^11*b^7*c^7*d^11 + 18564*a^12*b^6*c^6*d^12 - 8568*a^13*b^5*c^5*d^13 + 3060*a^14*b^4*c^4
*d^14 - 816*a^15*b^3*c^3*d^15 + 153*a^16*b^2*c^2*d^16 - 18*a^17*b*c*d^17 + a^18*d^18)*x - (a*b^19*c^12 - 12*a^
2*b^18*c^11*d + 66*a^3*b^17*c^10*d^2 - 220*a^4*b^16*c^9*d^3 + 495*a^5*b^15*c^8*d^4 - 792*a^6*b^14*c^7*d^5 + 92
4*a^7*b^13*c^6*d^6 - 792*a^8*b^12*c^5*d^7 + 495*a^9*b^11*c^4*d^8 - 220*a^10*b^10*c^3*d^9 + 66*a^11*b^9*c^2*d^1
0 - 12*a^12*b^8*c*d^11 + a^13*b^7*d^12)*sqrt(-(b^12*c^12 - 12*a*b^11*c^11*d + 66*a^2*b^10*c^10*d^2 - 220*a^3*b
^9*c^9*d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b
^4*c^4*d^8 - 220*a^9*b^3*c^3*d^9 + 66*a^10*b^2*c^2*d^10 - 12*a^11*b*c*d^11 + a^12*d^12)/(a*b^15)))*b^4*(-(b^12
*c^12 - 12*a*b^11*c^11*d + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*
d^5 + 924*a^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*b^3*c^3*d^9 + 66*a^10*b^2*c^2*
d^10 - 12*a^11*b*c*d^11 + a^12*d^12)/(a*b^15))^(1/4) + (b^13*c^9 - 9*a*b^12*c^8*d + 36*a^2*b^11*c^7*d^2 - 84*a
^3*b^10*c^6*d^3 + 126*a^4*b^9*c^5*d^4 - 126*a^5*b^8*c^4*d^5 + 84*a^6*b^7*c^3*d^6 - 36*a^7*b^6*c^2*d^7 + 9*a^8*
b^5*c*d^8 - a^9*b^4*d^9)*sqrt(x)*(-(b^12*c^12 - 12*a*b^11*c^11*d + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^3
+ 495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8
- 220*a^9*b^3*c^3*d^9 + 66*a^10*b^2*c^2*d^10 - 12*a^11*b*c*d^11 + a^12*d^12)/(a*b^15))^(1/4))/(b^12*c^12 - 12*
a*b^11*c^11*d + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^5 + 924*a
^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*b^3*c^3*d^9 + 66*a^10*b^2*c^2*d^10 - 12*a
^11*b*c*d^11 + a^12*d^12)) - 231*b^3*(-(b^12*c^12 - 12*a*b^11*c^11*d + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*
d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*
d^8 - 220*a^9*b^3*c^3*d^9 + 66*a^10*b^2*c^2*d^10 - 12*a^11*b*c*d^11 + a^12*d^12)/(a*b^15))^(1/4)*log(a*b^11*(-
(b^12*c^12 - 12*a*b^11*c^11*d + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*b^7
*c^7*d^5 + 924*a^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*b^3*c^3*d^9 + 66*a^10*b^2
*c^2*d^10 - 12*a^11*b*c*d^11 + a^12*d^12)/(a*b^15))^(3/4) - (b^9*c^9 - 9*a*b^8*c^8*d + 36*a^2*b^7*c^7*d^2 - 84
*a^3*b^6*c^6*d^3 + 126*a^4*b^5*c^5*d^4 - 126*a^5*b^4*c^4*d^5 + 84*a^6*b^3*c^3*d^6 - 36*a^7*b^2*c^2*d^7 + 9*a^8
*b*c*d^8 - a^9*d^9)*sqrt(x)) + 231*b^3*(-(b^12*c^12 - 12*a*b^11*c^11*d + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^
9*d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^
4*d^8 - 220*a^9*b^3*c^3*d^9 + 66*a^10*b^2*c^2*d^10 - 12*a^11*b*c*d^11 + a^12*d^12)/(a*b^15))^(1/4)*log(-a*b^11
*(-(b^12*c^12 - 12*a*b^11*c^11*d + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*
b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*b^3*c^3*d^9 + 66*a^10*
b^2*c^2*d^10 - 12*a^11*b*c*d^11 + a^12*d^12)/(a*b^15))^(3/4) - (b^9*c^9 - 9*a*b^8*c^8*d + 36*a^2*b^7*c^7*d^2 -
84*a^3*b^6*c^6*d^3 + 126*a^4*b^5*c^5*d^4 - 126*a^5*b^4*c^4*d^5 + 84*a^6*b^3*c^3*d^6 - 36*a^7*b^2*c^2*d^7 + 9*
a^8*b*c*d^8 - a^9*d^9)*sqrt(x)) + 4*(21*b^2*d^3*x^5 + 33*(3*b^2*c*d^2 - a*b*d^3)*x^3 + 77*(3*b^2*c^2*d - 3*a*b
*c*d^2 + a^2*d^3)*x)*sqrt(x))/b^3

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giac [B]  time = 0.49, size = 490, normalized size = 1.60 \begin {gather*} \frac {\sqrt {2} {\left (\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 + 3 \, \left (a b^{3}\right )^{\frac {3}{4}} a^{2} b c d^{2} - \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 )}{2 \, a b^{6}} + \frac {\sqrt {2} {\left (\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 + 3 \, \left (a b^{3}\right )^{\frac {3}{4}} a^{2} b c d^{2} - \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 )}{2 \, a b^{6}} - \frac {\sqrt {2} {\left (\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 + 3 \, \left (a b^{3}\right )^{\frac {3}{4}} a^{2} b c d^{2} - \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 )}{4 \, a b^{6}} + \frac {\sqrt {2} {\left (\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 + 3 \, \left (a b^{3}\right )^{\frac {3}{4}} a^{2} b c d^{2} - \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 )}{4 \, a b^{6}} + \frac {2 \, {\left (21 \, b^{10} d^{3} x^{\frac {11}{2}} + 99 \, b^{10} c d^{2} x^{\frac {7}{2}} - 33 \, a b^{9} d^{3} x^{\frac {7}{2}} + 231 \, b^{10} c^{2} d x^{\frac {3}{2}} - 231 \, a b^{9} c d^{2} x^{\frac {3}{2}} + 77 \, a^{2} b^{8} d^{3} x^{\frac {3}{2}}\right )}}{231 \, b^{11}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^2+c)^3*x^(1/2)/(b*x^2+a),x, algorithm="giac")

[Out]

1/2*sqrt(2)*((a*b^3)^(3/4)*b^3*c^3 - 3*(a*b^3)^(3/4)*a*b^2*c^2*d + 3*(a*b^3)^(3/4)*a^2*b*c*d^2 - (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*b^6) + 1/2*sqrt(2)*((a*b^3)^(3/
4)*b^3*c^3 - 3*(a*b^3)^(3/4)*a*b^2*c^2*d + 3*(a*b^3)^(3/4)*a^2*b*c*d^2 - (a*b^3)^(3/4)*a^3*d^3)*arctan(-1/2*sq
rt(2)*(sqrt(2)*(a/b)^(1/4) - 2*sqrt(x))/(a/b)^(1/4))/(a*b^6) - 1/4*sqrt(2)*((a*b^3)^(3/4)*b^3*c^3 - 3*(a*b^3)^
(3/4)*a*b^2*c^2*d + 3*(a*b^3)^(3/4)*a^2*b*c*d^2 - (a*b^3)^(3/4)*a^3*d^3)*log(sqrt(2)*sqrt(x)*(a/b)^(1/4) + x +
sqrt(a/b))/(a*b^6) + 1/4*sqrt(2)*((a*b^3)^(3/4)*b^3*c^3 - 3*(a*b^3)^(3/4)*a*b^2*c^2*d + 3*(a*b^3)^(3/4)*a^2*b
*c*d^2 - (a*b^3)^(3/4)*a^3*d^3)*log(-sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(a*b^6) + 2/231*(21*b^10*d^3
*x^(11/2) + 99*b^10*c*d^2*x^(7/2) - 33*a*b^9*d^3*x^(7/2) + 231*b^10*c^2*d*x^(3/2) - 231*a*b^9*c*d^2*x^(3/2) +
77*a^2*b^8*d^3*x^(3/2))/b^11

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maple [B]  time = 0.01, size = 659, normalized size = 2.15 \begin {gather*} \frac {2 d^{3} x^{\frac {11}{2}}}{11 b}-\frac {2 a \,d^{3} x^{\frac {7}{2}}}{7 b^{2}}+\frac {6 c \,d^{2} x^{\frac {7}{2}}}{7 b}+\frac {2 a^{2} d^{3} x^{\frac {3}{2}}}{3 b^{3}}-\frac {2 a c \,d^{2} x^{\frac {3}{2}}}{b^{2}}+\frac {2 c^{2} d \,x^{\frac {3}{2}}}{b}-\frac {\sqrt {2}\, a^{3} d^{3} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )}{2 \left (\frac {a}{b}\right )^{\frac {1}{4}} b^{4}}-\frac {\sqrt {2}\, a^{3} d^{3} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )}{2 \left (\frac {a}{b}\right )^{\frac {1}{4}} b^{4}}-\frac {\sqrt {2}\, a^{3} 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 )}{4 \left (\frac {a}{b}\right )^{\frac {1}{4}} b^{4}}+\frac {3 \sqrt {2}\, a^{2} c \,d^{2} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )}{2 \left (\frac {a}{b}\right )^{\frac {1}{4}} b^{3}}+\frac {3 \sqrt {2}\, a^{2} c \,d^{2} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )}{2 \left (\frac {a}{b}\right )^{\frac {1}{4}} b^{3}}+\frac {3 \sqrt {2}\, a^{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 )}{4 \left (\frac {a}{b}\right )^{\frac {1}{4}} b^{3}}-\frac {3 \sqrt {2}\, a \,c^{2} d \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )}{2 \left (\frac {a}{b}\right )^{\frac {1}{4}} b^{2}}-\frac {3 \sqrt {2}\, a \,c^{2} d \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )}{2 \left (\frac {a}{b}\right )^{\frac {1}{4}} b^{2}}-\frac {3 \sqrt {2}\, a \,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 )}{4 \left (\frac {a}{b}\right )^{\frac {1}{4}} b^{2}}+\frac {\sqrt {2}\, c^{3} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )}{2 \left (\frac {a}{b}\right )^{\frac {1}{4}} b}+\frac {\sqrt {2}\, c^{3} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )}{2 \left (\frac {a}{b}\right )^{\frac {1}{4}} b}+\frac {\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 )}{4 \left (\frac {a}{b}\right )^{\frac {1}{4}} b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x^2+c)^3*x^(1/2)/(b*x^2+a),x)

[Out]

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

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maxima [A]  time = 2.42, size = 282, normalized size = 0.92 \begin {gather*} \frac {{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - 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 )}}{4 \, b^{3}} + \frac {2 \, {\left (21 \, b^{2} d^{3} x^{\frac {11}{2}} + 33 \, {\left (3 \, b^{2} c d^{2} - a b d^{3}\right )} x^{\frac {7}{2}} + 77 \, {\left (3 \, b^{2} c^{2} d - 3 \, a b c d^{2} + a^{2} d^{3}\right )} x^{\frac {3}{2}}\right )}}{231 \, b^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^2+c)^3*x^(1/2)/(b*x^2+a),x, algorithm="maxima")

[Out]

1/4*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - 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(sqrt(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)))/b^3 + 2/231*(21*b^2*d^3*x^(11/2) + 33*(3*b^2*c*d^2 - a
*b*d^3)*x^(7/2) + 77*(3*b^2*c^2*d - 3*a*b*c*d^2 + a^2*d^3)*x^(3/2))/b^3

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mupad [B]  time = 0.16, size = 574, normalized size = 1.88 \begin {gather*} x^{3/2}\,\left (\frac {2\,c^2\,d}{b}+\frac {a\,\left (\frac {2\,a\,d^3}{b^2}-\frac {6\,c\,d^2}{b}\right )}{3\,b}\right )-x^{7/2}\,\left (\frac {2\,a\,d^3}{7\,b^2}-\frac {6\,c\,d^2}{7\,b}\right )+\frac {2\,d^3\,x^{11/2}}{11\,b}-\frac {\mathrm {atan}\left (\frac {b^{1/4}\,\sqrt {x}\,{\left (a\,d-b\,c\right )}^3\,\left (a^7\,d^6-6\,a^6\,b\,c\,d^5+15\,a^5\,b^2\,c^2\,d^4-20\,a^4\,b^3\,c^3\,d^3+15\,a^3\,b^4\,c^4\,d^2-6\,a^2\,b^5\,c^5\,d+a\,b^6\,c^6\right )}{{\left (-a\right )}^{1/4}\,\left (a^{10}\,d^9-9\,a^9\,b\,c\,d^8+36\,a^8\,b^2\,c^2\,d^7-84\,a^7\,b^3\,c^3\,d^6+126\,a^6\,b^4\,c^4\,d^5-126\,a^5\,b^5\,c^5\,d^4+84\,a^4\,b^6\,c^6\,d^3-36\,a^3\,b^7\,c^7\,d^2+9\,a^2\,b^8\,c^8\,d-a\,b^9\,c^9\right )}\right )\,{\left (a\,d-b\,c\right )}^3}{{\left (-a\right )}^{1/4}\,b^{15/4}}-\frac {\mathrm {atan}\left (\frac {b^{1/4}\,\sqrt {x}\,{\left (a\,d-b\,c\right )}^3\,\left (a^7\,d^6-6\,a^6\,b\,c\,d^5+15\,a^5\,b^2\,c^2\,d^4-20\,a^4\,b^3\,c^3\,d^3+15\,a^3\,b^4\,c^4\,d^2-6\,a^2\,b^5\,c^5\,d+a\,b^6\,c^6\right )\,1{}\mathrm {i}}{{\left (-a\right )}^{1/4}\,\left (a^{10}\,d^9-9\,a^9\,b\,c\,d^8+36\,a^8\,b^2\,c^2\,d^7-84\,a^7\,b^3\,c^3\,d^6+126\,a^6\,b^4\,c^4\,d^5-126\,a^5\,b^5\,c^5\,d^4+84\,a^4\,b^6\,c^6\,d^3-36\,a^3\,b^7\,c^7\,d^2+9\,a^2\,b^8\,c^8\,d-a\,b^9\,c^9\right )}\right )\,{\left (a\,d-b\,c\right )}^3\,1{}\mathrm {i}}{{\left (-a\right )}^{1/4}\,b^{15/4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^(1/2)*(c + d*x^2)^3)/(a + b*x^2),x)

[Out]

x^(3/2)*((2*c^2*d)/b + (a*((2*a*d^3)/b^2 - (6*c*d^2)/b))/(3*b)) - x^(7/2)*((2*a*d^3)/(7*b^2) - (6*c*d^2)/(7*b)
) + (2*d^3*x^(11/2))/(11*b) - (atan((b^(1/4)*x^(1/2)*(a*d - b*c)^3*(a^7*d^6 + a*b^6*c^6 - 6*a^2*b^5*c^5*d + 15
*a^3*b^4*c^4*d^2 - 20*a^4*b^3*c^3*d^3 + 15*a^5*b^2*c^2*d^4 - 6*a^6*b*c*d^5))/((-a)^(1/4)*(a^10*d^9 - a*b^9*c^9
+ 9*a^2*b^8*c^8*d - 36*a^3*b^7*c^7*d^2 + 84*a^4*b^6*c^6*d^3 - 126*a^5*b^5*c^5*d^4 + 126*a^6*b^4*c^4*d^5 - 84*
a^7*b^3*c^3*d^6 + 36*a^8*b^2*c^2*d^7 - 9*a^9*b*c*d^8)))*(a*d - b*c)^3)/((-a)^(1/4)*b^(15/4)) - (atan((b^(1/4)*
x^(1/2)*(a*d - b*c)^3*(a^7*d^6 + a*b^6*c^6 - 6*a^2*b^5*c^5*d + 15*a^3*b^4*c^4*d^2 - 20*a^4*b^3*c^3*d^3 + 15*a^
5*b^2*c^2*d^4 - 6*a^6*b*c*d^5)*1i)/((-a)^(1/4)*(a^10*d^9 - a*b^9*c^9 + 9*a^2*b^8*c^8*d - 36*a^3*b^7*c^7*d^2 +
84*a^4*b^6*c^6*d^3 - 126*a^5*b^5*c^5*d^4 + 126*a^6*b^4*c^4*d^5 - 84*a^7*b^3*c^3*d^6 + 36*a^8*b^2*c^2*d^7 - 9*a
^9*b*c*d^8)))*(a*d - b*c)^3*1i)/((-a)^(1/4)*b^(15/4))

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sympy [A]  time = 93.47, size = 874, normalized size = 2.86

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x**2+c)**3*x**(1/2)/(b*x**2+a),x)

[Out]

Piecewise((zoo*(-2*c**3/sqrt(x) + 2*c**2*d*x**(3/2) + 6*c*d**2*x**(7/2)/7 + 2*d**3*x**(11/2)/11), Eq(a, 0) & E
q(b, 0)), ((2*c**3*x**(3/2)/3 + 6*c**2*d*x**(7/2)/7 + 6*c*d**2*x**(11/2)/11 + 2*d**3*x**(15/2)/15)/a, Eq(b, 0)
), ((-2*c**3/sqrt(x) + 2*c**2*d*x**(3/2) + 6*c*d**2*x**(7/2)/7 + 2*d**3*x**(11/2)/11)/b, Eq(a, 0)), ((-1)**(3/
4)*a**(11/4)*d**3*(1/b)**(3/4)*log(-(-1)**(1/4)*a**(1/4)*(1/b)**(1/4) + sqrt(x))/(2*b**3) - (-1)**(3/4)*a**(11
/4)*d**3*(1/b)**(3/4)*log((-1)**(1/4)*a**(1/4)*(1/b)**(1/4) + sqrt(x))/(2*b**3) - (-1)**(3/4)*a**(11/4)*d**3*(
1/b)**(3/4)*atan((-1)**(3/4)*sqrt(x)/(a**(1/4)*(1/b)**(1/4)))/b**3 - 3*(-1)**(3/4)*a**(7/4)*c*d**2*(1/b)**(3/4
)*log(-(-1)**(1/4)*a**(1/4)*(1/b)**(1/4) + sqrt(x))/(2*b**2) + 3*(-1)**(3/4)*a**(7/4)*c*d**2*(1/b)**(3/4)*log(
(-1)**(1/4)*a**(1/4)*(1/b)**(1/4) + sqrt(x))/(2*b**2) + 3*(-1)**(3/4)*a**(7/4)*c*d**2*(1/b)**(3/4)*atan((-1)**
(3/4)*sqrt(x)/(a**(1/4)*(1/b)**(1/4)))/b**2 + 3*(-1)**(3/4)*a**(3/4)*c**2*d*(1/b)**(3/4)*log(-(-1)**(1/4)*a**(
1/4)*(1/b)**(1/4) + sqrt(x))/(2*b) - 3*(-1)**(3/4)*a**(3/4)*c**2*d*(1/b)**(3/4)*log((-1)**(1/4)*a**(1/4)*(1/b)
**(1/4) + sqrt(x))/(2*b) - 3*(-1)**(3/4)*a**(3/4)*c**2*d*(1/b)**(3/4)*atan((-1)**(3/4)*sqrt(x)/(a**(1/4)*(1/b)
**(1/4)))/b + 2*a**2*d**3*x**(3/2)/(3*b**3) - 2*a*c*d**2*x**(3/2)/b**2 - 2*a*d**3*x**(7/2)/(7*b**2) + 2*c**2*d
*x**(3/2)/b + 6*c*d**2*x**(7/2)/(7*b) + 2*d**3*x**(11/2)/(11*b) - (-1)**(3/4)*c**3*(1/b)**(3/4)*log(-(-1)**(1/
4)*a**(1/4)*(1/b)**(1/4) + sqrt(x))/(2*a**(1/4)) + (-1)**(3/4)*c**3*(1/b)**(3/4)*log((-1)**(1/4)*a**(1/4)*(1/b
)**(1/4) + sqrt(x))/(2*a**(1/4)) + (-1)**(3/4)*c**3*(1/b)**(3/4)*atan((-1)**(3/4)*sqrt(x)/(a**(1/4)*(1/b)**(1/
4)))/a**(1/4), True))

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