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

Optimal. Leaf size=326 \[ \frac{2 d x^{5/2} \left (a^2 d^2-3 a b c d+3 b^2 c^2\right )}{5 b^3}+\frac{\sqrt [4]{a} (b c-a d)^3 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} b^{17/4}}-\frac{\sqrt [4]{a} (b c-a d)^3 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} b^{17/4}}+\frac{\sqrt [4]{a} (b c-a d)^3 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} b^{17/4}}-\frac{\sqrt [4]{a} (b c-a d)^3 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt{2} b^{17/4}}+\frac{2 \sqrt{x} (b c-a d)^3}{b^4}+\frac{2 d^2 x^{9/2} (3 b c-a d)}{9 b^2}+\frac{2 d^3 x^{13/2}}{13 b} \]

[Out]

(2*(b*c - a*d)^3*Sqrt[x])/b^4 + (2*d*(3*b^2*c^2 - 3*a*b*c*d + a^2*d^2)*x^(5/2))/
(5*b^3) + (2*d^2*(3*b*c - a*d)*x^(9/2))/(9*b^2) + (2*d^3*x^(13/2))/(13*b) + (a^(
1/4)*(b*c - a*d)^3*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(Sqrt[2]*b^(17
/4)) - (a^(1/4)*(b*c - a*d)^3*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(Sq
rt[2]*b^(17/4)) + (a^(1/4)*(b*c - a*d)^3*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*S
qrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*b^(17/4)) - (a^(1/4)*(b*c - a*d)^3*Log[Sqrt[a] +
Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*b^(17/4))

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Rubi [A]  time = 0.575506, antiderivative size = 326, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 9, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375 \[ \frac{2 d x^{5/2} \left (a^2 d^2-3 a b c d+3 b^2 c^2\right )}{5 b^3}+\frac{\sqrt [4]{a} (b c-a d)^3 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} b^{17/4}}-\frac{\sqrt [4]{a} (b c-a d)^3 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} b^{17/4}}+\frac{\sqrt [4]{a} (b c-a d)^3 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} b^{17/4}}-\frac{\sqrt [4]{a} (b c-a d)^3 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt{2} b^{17/4}}+\frac{2 \sqrt{x} (b c-a d)^3}{b^4}+\frac{2 d^2 x^{9/2} (3 b c-a d)}{9 b^2}+\frac{2 d^3 x^{13/2}}{13 b} \]

Antiderivative was successfully verified.

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

[Out]

(2*(b*c - a*d)^3*Sqrt[x])/b^4 + (2*d*(3*b^2*c^2 - 3*a*b*c*d + a^2*d^2)*x^(5/2))/
(5*b^3) + (2*d^2*(3*b*c - a*d)*x^(9/2))/(9*b^2) + (2*d^3*x^(13/2))/(13*b) + (a^(
1/4)*(b*c - a*d)^3*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(Sqrt[2]*b^(17
/4)) - (a^(1/4)*(b*c - a*d)^3*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(Sq
rt[2]*b^(17/4)) + (a^(1/4)*(b*c - a*d)^3*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*S
qrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*b^(17/4)) - (a^(1/4)*(b*c - a*d)^3*Log[Sqrt[a] +
Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*b^(17/4))

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Rubi in Sympy [A]  time = 105.181, size = 309, normalized size = 0.95 \[ - \frac{\sqrt{2} \sqrt [4]{a} \left (a d - b c\right )^{3} \log{\left (- \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x} + \sqrt{a} + \sqrt{b} x \right )}}{4 b^{\frac{17}{4}}} + \frac{\sqrt{2} \sqrt [4]{a} \left (a d - b c\right )^{3} \log{\left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x} + \sqrt{a} + \sqrt{b} x \right )}}{4 b^{\frac{17}{4}}} - \frac{\sqrt{2} \sqrt [4]{a} \left (a d - b c\right )^{3} \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}}{2 b^{\frac{17}{4}}} + \frac{\sqrt{2} \sqrt [4]{a} \left (a d - b c\right )^{3} \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}}{2 b^{\frac{17}{4}}} + \frac{2 d^{3} x^{\frac{13}{2}}}{13 b} - \frac{2 d^{2} x^{\frac{9}{2}} \left (a d - 3 b c\right )}{9 b^{2}} + \frac{2 d x^{\frac{5}{2}} \left (a^{2} d^{2} - 3 a b c d + 3 b^{2} c^{2}\right )}{5 b^{3}} - \frac{2 \sqrt{x} \left (a d - b c\right )^{3}}{b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(3/2)*(d*x**2+c)**3/(b*x**2+a),x)

[Out]

-sqrt(2)*a**(1/4)*(a*d - b*c)**3*log(-sqrt(2)*a**(1/4)*b**(1/4)*sqrt(x) + sqrt(a
) + sqrt(b)*x)/(4*b**(17/4)) + sqrt(2)*a**(1/4)*(a*d - b*c)**3*log(sqrt(2)*a**(1
/4)*b**(1/4)*sqrt(x) + sqrt(a) + sqrt(b)*x)/(4*b**(17/4)) - sqrt(2)*a**(1/4)*(a*
d - b*c)**3*atan(1 - sqrt(2)*b**(1/4)*sqrt(x)/a**(1/4))/(2*b**(17/4)) + sqrt(2)*
a**(1/4)*(a*d - b*c)**3*atan(1 + sqrt(2)*b**(1/4)*sqrt(x)/a**(1/4))/(2*b**(17/4)
) + 2*d**3*x**(13/2)/(13*b) - 2*d**2*x**(9/2)*(a*d - 3*b*c)/(9*b**2) + 2*d*x**(5
/2)*(a**2*d**2 - 3*a*b*c*d + 3*b**2*c**2)/(5*b**3) - 2*sqrt(x)*(a*d - b*c)**3/b*
*4

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Mathematica [A]  time = 0.219743, size = 314, normalized size = 0.96 \[ \frac{936 b^{5/4} d x^{5/2} \left (a^2 d^2-3 a b c d+3 b^2 c^2\right )+520 b^{9/4} d^2 x^{9/2} (3 b c-a d)+4680 \sqrt [4]{b} \sqrt{x} (b c-a d)^3-585 \sqrt{2} \sqrt [4]{a} (a d-b c)^3 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )+585 \sqrt{2} \sqrt [4]{a} (a d-b c)^3 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )-1170 \sqrt{2} \sqrt [4]{a} (a d-b c)^3 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )+1170 \sqrt{2} \sqrt [4]{a} (a d-b c)^3 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )+360 b^{13/4} d^3 x^{13/2}}{2340 b^{17/4}} \]

Antiderivative was successfully verified.

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

[Out]

(4680*b^(1/4)*(b*c - a*d)^3*Sqrt[x] + 936*b^(5/4)*d*(3*b^2*c^2 - 3*a*b*c*d + a^2
*d^2)*x^(5/2) + 520*b^(9/4)*d^2*(3*b*c - a*d)*x^(9/2) + 360*b^(13/4)*d^3*x^(13/2
) - 1170*Sqrt[2]*a^(1/4)*(-(b*c) + a*d)^3*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a
^(1/4)] + 1170*Sqrt[2]*a^(1/4)*(-(b*c) + a*d)^3*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt
[x])/a^(1/4)] - 585*Sqrt[2]*a^(1/4)*(-(b*c) + a*d)^3*Log[Sqrt[a] - Sqrt[2]*a^(1/
4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x] + 585*Sqrt[2]*a^(1/4)*(-(b*c) + a*d)^3*Log[Sqrt[
a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2340*b^(17/4))

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Maple [B]  time = 0.015, size = 712, normalized size = 2.2 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/13*d^3*x^(13/2)/b-2/9/b^2*x^(9/2)*a*d^3+2/3/b*x^(9/2)*c*d^2+2/5/b^3*x^(5/2)*a^
2*d^3-6/5/b^2*x^(5/2)*a*c*d^2+6/5/b*x^(5/2)*c^2*d-2/b^4*a^3*d^3*x^(1/2)+6/b^3*a^
2*c*d^2*x^(1/2)-6/b^2*a*c^2*d*x^(1/2)+2/b*c^3*x^(1/2)+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)*arct
an(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)*x^(1/2)*2^(1/2)+(a/b)^(1/2))/(x-(a/b
)^(1/4)*x^(1/2)*2^(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)*x^(1/2)*2^(1/2)+(a/b)^(1/2))/(x-(a/b)^(1/4)*x^(1/2)*2^(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)*x^(1/2)*2^(1/2)+(a
/b)^(1/2))/(x-(a/b)^(1/4)*x^(1/2)*2^(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)*x^(1/2)*2^(1/2)+(a/b)^(1/2))/(x-(a/b)^(1/4)*x^(1/2)*
2^(1/2)+(a/b)^(1/2)))*c^3

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.264048, size = 2106, normalized size = 6.46 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/1170*(2340*b^4*(-(a*b^12*c^12 - 12*a^2*b^11*c^11*d + 66*a^3*b^10*c^10*d^2 - 2
20*a^4*b^9*c^9*d^3 + 495*a^5*b^8*c^8*d^4 - 792*a^6*b^7*c^7*d^5 + 924*a^7*b^6*c^6
*d^6 - 792*a^8*b^5*c^5*d^7 + 495*a^9*b^4*c^4*d^8 - 220*a^10*b^3*c^3*d^9 + 66*a^1
1*b^2*c^2*d^10 - 12*a^12*b*c*d^11 + a^13*d^12)/b^17)^(1/4)*arctan(-b^4*(-(a*b^12
*c^12 - 12*a^2*b^11*c^11*d + 66*a^3*b^10*c^10*d^2 - 220*a^4*b^9*c^9*d^3 + 495*a^
5*b^8*c^8*d^4 - 792*a^6*b^7*c^7*d^5 + 924*a^7*b^6*c^6*d^6 - 792*a^8*b^5*c^5*d^7
+ 495*a^9*b^4*c^4*d^8 - 220*a^10*b^3*c^3*d^9 + 66*a^11*b^2*c^2*d^10 - 12*a^12*b*
c*d^11 + a^13*d^12)/b^17)^(1/4)/((b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*
d^3)*sqrt(x) - sqrt(b^8*sqrt(-(a*b^12*c^12 - 12*a^2*b^11*c^11*d + 66*a^3*b^10*c^
10*d^2 - 220*a^4*b^9*c^9*d^3 + 495*a^5*b^8*c^8*d^4 - 792*a^6*b^7*c^7*d^5 + 924*a
^7*b^6*c^6*d^6 - 792*a^8*b^5*c^5*d^7 + 495*a^9*b^4*c^4*d^8 - 220*a^10*b^3*c^3*d^
9 + 66*a^11*b^2*c^2*d^10 - 12*a^12*b*c*d^11 + a^13*d^12)/b^17) + (b^6*c^6 - 6*a*
b^5*c^5*d + 15*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a^5
*b*c*d^5 + a^6*d^6)*x))) - 585*b^4*(-(a*b^12*c^12 - 12*a^2*b^11*c^11*d + 66*a^3*
b^10*c^10*d^2 - 220*a^4*b^9*c^9*d^3 + 495*a^5*b^8*c^8*d^4 - 792*a^6*b^7*c^7*d^5
+ 924*a^7*b^6*c^6*d^6 - 792*a^8*b^5*c^5*d^7 + 495*a^9*b^4*c^4*d^8 - 220*a^10*b^3
*c^3*d^9 + 66*a^11*b^2*c^2*d^10 - 12*a^12*b*c*d^11 + a^13*d^12)/b^17)^(1/4)*log(
b^4*(-(a*b^12*c^12 - 12*a^2*b^11*c^11*d + 66*a^3*b^10*c^10*d^2 - 220*a^4*b^9*c^9
*d^3 + 495*a^5*b^8*c^8*d^4 - 792*a^6*b^7*c^7*d^5 + 924*a^7*b^6*c^6*d^6 - 792*a^8
*b^5*c^5*d^7 + 495*a^9*b^4*c^4*d^8 - 220*a^10*b^3*c^3*d^9 + 66*a^11*b^2*c^2*d^10
- 12*a^12*b*c*d^11 + a^13*d^12)/b^17)^(1/4) - (b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*
b*c*d^2 - a^3*d^3)*sqrt(x)) + 585*b^4*(-(a*b^12*c^12 - 12*a^2*b^11*c^11*d + 66*a
^3*b^10*c^10*d^2 - 220*a^4*b^9*c^9*d^3 + 495*a^5*b^8*c^8*d^4 - 792*a^6*b^7*c^7*d
^5 + 924*a^7*b^6*c^6*d^6 - 792*a^8*b^5*c^5*d^7 + 495*a^9*b^4*c^4*d^8 - 220*a^10*
b^3*c^3*d^9 + 66*a^11*b^2*c^2*d^10 - 12*a^12*b*c*d^11 + a^13*d^12)/b^17)^(1/4)*l
og(-b^4*(-(a*b^12*c^12 - 12*a^2*b^11*c^11*d + 66*a^3*b^10*c^10*d^2 - 220*a^4*b^9
*c^9*d^3 + 495*a^5*b^8*c^8*d^4 - 792*a^6*b^7*c^7*d^5 + 924*a^7*b^6*c^6*d^6 - 792
*a^8*b^5*c^5*d^7 + 495*a^9*b^4*c^4*d^8 - 220*a^10*b^3*c^3*d^9 + 66*a^11*b^2*c^2*
d^10 - 12*a^12*b*c*d^11 + a^13*d^12)/b^17)^(1/4) - (b^3*c^3 - 3*a*b^2*c^2*d + 3*
a^2*b*c*d^2 - a^3*d^3)*sqrt(x)) - 4*(45*b^3*d^3*x^6 + 585*b^3*c^3 - 1755*a*b^2*c
^2*d + 1755*a^2*b*c*d^2 - 585*a^3*d^3 + 65*(3*b^3*c*d^2 - a*b^2*d^3)*x^4 + 117*(
3*b^3*c^2*d - 3*a*b^2*c*d^2 + a^2*b*d^3)*x^2)*sqrt(x))/b^4

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.317018, size = 717, normalized size = 2.2 \[ -\frac{\sqrt{2}{\left (\left (a b^{3}\right )^{\frac{1}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac{1}{4}} a b^{2} c^{2} d + 3 \, \left (a b^{3}\right )^{\frac{1}{4}} a^{2} b c d^{2} - \left (a b^{3}\right )^{\frac{1}{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 \, b^{5}} - \frac{\sqrt{2}{\left (\left (a b^{3}\right )^{\frac{1}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac{1}{4}} a b^{2} c^{2} d + 3 \, \left (a b^{3}\right )^{\frac{1}{4}} a^{2} b c d^{2} - \left (a b^{3}\right )^{\frac{1}{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 \, b^{5}} - \frac{\sqrt{2}{\left (\left (a b^{3}\right )^{\frac{1}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac{1}{4}} a b^{2} c^{2} d + 3 \, \left (a b^{3}\right )^{\frac{1}{4}} a^{2} b c d^{2} - \left (a b^{3}\right )^{\frac{1}{4}} a^{3} d^{3}\right )}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{b}}\right )}{4 \, b^{5}} + \frac{\sqrt{2}{\left (\left (a b^{3}\right )^{\frac{1}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac{1}{4}} a b^{2} c^{2} d + 3 \, \left (a b^{3}\right )^{\frac{1}{4}} a^{2} b c d^{2} - \left (a b^{3}\right )^{\frac{1}{4}} a^{3} d^{3}\right )}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{b}}\right )}{4 \, b^{5}} + \frac{2 \,{\left (45 \, b^{12} d^{3} x^{\frac{13}{2}} + 195 \, b^{12} c d^{2} x^{\frac{9}{2}} - 65 \, a b^{11} d^{3} x^{\frac{9}{2}} + 351 \, b^{12} c^{2} d x^{\frac{5}{2}} - 351 \, a b^{11} c d^{2} x^{\frac{5}{2}} + 117 \, a^{2} b^{10} d^{3} x^{\frac{5}{2}} + 585 \, b^{12} c^{3} \sqrt{x} - 1755 \, a b^{11} c^{2} d \sqrt{x} + 1755 \, a^{2} b^{10} c d^{2} \sqrt{x} - 585 \, a^{3} b^{9} d^{3} \sqrt{x}\right )}}{585 \, b^{13}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*sqrt(2)*((a*b^3)^(1/4)*b^3*c^3 - 3*(a*b^3)^(1/4)*a*b^2*c^2*d + 3*(a*b^3)^(1
/4)*a^2*b*c*d^2 - (a*b^3)^(1/4)*a^3*d^3)*arctan(1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4)
+ 2*sqrt(x))/(a/b)^(1/4))/b^5 - 1/2*sqrt(2)*((a*b^3)^(1/4)*b^3*c^3 - 3*(a*b^3)^
(1/4)*a*b^2*c^2*d + 3*(a*b^3)^(1/4)*a^2*b*c*d^2 - (a*b^3)^(1/4)*a^3*d^3)*arctan(
-1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) - 2*sqrt(x))/(a/b)^(1/4))/b^5 - 1/4*sqrt(2)*((
a*b^3)^(1/4)*b^3*c^3 - 3*(a*b^3)^(1/4)*a*b^2*c^2*d + 3*(a*b^3)^(1/4)*a^2*b*c*d^2
- (a*b^3)^(1/4)*a^3*d^3)*ln(sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/b^5 +
1/4*sqrt(2)*((a*b^3)^(1/4)*b^3*c^3 - 3*(a*b^3)^(1/4)*a*b^2*c^2*d + 3*(a*b^3)^(1/
4)*a^2*b*c*d^2 - (a*b^3)^(1/4)*a^3*d^3)*ln(-sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sq
rt(a/b))/b^5 + 2/585*(45*b^12*d^3*x^(13/2) + 195*b^12*c*d^2*x^(9/2) - 65*a*b^11*
d^3*x^(9/2) + 351*b^12*c^2*d*x^(5/2) - 351*a*b^11*c*d^2*x^(5/2) + 117*a^2*b^10*d
^3*x^(5/2) + 585*b^12*c^3*sqrt(x) - 1755*a*b^11*c^2*d*sqrt(x) + 1755*a^2*b^10*c*
d^2*sqrt(x) - 585*a^3*b^9*d^3*sqrt(x))/b^13