3.458 \(\int \frac{\left (c+d x^2\right )^3}{x^{5/2} \left (a+b x^2\right )^2} \, dx\)

Optimal. Leaf size=367 \[ \frac{(b c-a d)^2 (5 a d+7 b c) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{11/4} b^{9/4}}-\frac{(b c-a d)^2 (5 a d+7 b c) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{11/4} b^{9/4}}+\frac{(b c-a d)^2 (5 a d+7 b c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{11/4} b^{9/4}}-\frac{(b c-a d)^2 (5 a d+7 b c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} a^{11/4} b^{9/4}}-\frac{c^2 (7 b c-3 a d)}{6 a^2 b x^{3/2}}-\frac{d^2 \sqrt{x} (b c-5 a d)}{2 a b^2}+\frac{\left (c+d x^2\right )^2 (b c-a d)}{2 a b x^{3/2} \left (a+b x^2\right )} \]

[Out]

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

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Rubi [A]  time = 0.911418, antiderivative size = 367, 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 \[ \frac{(b c-a d)^2 (5 a d+7 b c) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{11/4} b^{9/4}}-\frac{(b c-a d)^2 (5 a d+7 b c) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{11/4} b^{9/4}}+\frac{(b c-a d)^2 (5 a d+7 b c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{11/4} b^{9/4}}-\frac{(b c-a d)^2 (5 a d+7 b c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} a^{11/4} b^{9/4}}-\frac{c^2 (7 b c-3 a d)}{6 a^2 b x^{3/2}}-\frac{d^2 \sqrt{x} (b c-5 a d)}{2 a b^2}+\frac{\left (c+d x^2\right )^2 (b c-a d)}{2 a b x^{3/2} \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x^2)^3/(x^(5/2)*(a + b*x^2)^2),x]

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{d^{2} \left (5 a d - b c\right ) \int ^{\sqrt{x}} \frac{1}{b}\, dx}{2 a b} - \frac{\left (c + d x^{2}\right )^{2} \left (a d - b c\right )}{2 a b x^{\frac{3}{2}} \left (a + b x^{2}\right )} + \frac{c^{2} \left (3 a d - 7 b c\right )}{6 a^{2} b x^{\frac{3}{2}}} + \frac{\sqrt{2} \left (a d - b c\right )^{2} \left (5 a d + 7 b c\right ) \log{\left (- \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x} + \sqrt{a} + \sqrt{b} x \right )}}{16 a^{\frac{11}{4}} b^{\frac{9}{4}}} - \frac{\sqrt{2} \left (a d - b c\right )^{2} \left (5 a d + 7 b c\right ) \log{\left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x} + \sqrt{a} + \sqrt{b} x \right )}}{16 a^{\frac{11}{4}} b^{\frac{9}{4}}} + \frac{\sqrt{2} \left (a d - b c\right )^{2} \left (5 a d + 7 b c\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}}{8 a^{\frac{11}{4}} b^{\frac{9}{4}}} - \frac{\sqrt{2} \left (a d - b c\right )^{2} \left (5 a d + 7 b c\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}}{8 a^{\frac{11}{4}} b^{\frac{9}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x**2+c)**3/x**(5/2)/(b*x**2+a)**2,x)

[Out]

d**2*(5*a*d - b*c)*Integral(1/b, (x, sqrt(x)))/(2*a*b) - (c + d*x**2)**2*(a*d -
b*c)/(2*a*b*x**(3/2)*(a + b*x**2)) + c**2*(3*a*d - 7*b*c)/(6*a**2*b*x**(3/2)) +
sqrt(2)*(a*d - b*c)**2*(5*a*d + 7*b*c)*log(-sqrt(2)*a**(1/4)*b**(1/4)*sqrt(x) +
sqrt(a) + sqrt(b)*x)/(16*a**(11/4)*b**(9/4)) - sqrt(2)*(a*d - b*c)**2*(5*a*d + 7
*b*c)*log(sqrt(2)*a**(1/4)*b**(1/4)*sqrt(x) + sqrt(a) + sqrt(b)*x)/(16*a**(11/4)
*b**(9/4)) + sqrt(2)*(a*d - b*c)**2*(5*a*d + 7*b*c)*atan(1 - sqrt(2)*b**(1/4)*sq
rt(x)/a**(1/4))/(8*a**(11/4)*b**(9/4)) - sqrt(2)*(a*d - b*c)**2*(5*a*d + 7*b*c)*
atan(1 + sqrt(2)*b**(1/4)*sqrt(x)/a**(1/4))/(8*a**(11/4)*b**(9/4))

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Mathematica [A]  time = 0.388902, size = 327, normalized size = 0.89 \[ \frac{1}{48} \left (\frac{3 \sqrt{2} (b c-a d)^2 (5 a d+7 b c) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{a^{11/4} b^{9/4}}-\frac{3 \sqrt{2} (b c-a d)^2 (5 a d+7 b c) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{a^{11/4} b^{9/4}}+\frac{6 \sqrt{2} (b c-a d)^2 (5 a d+7 b c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{a^{11/4} b^{9/4}}-\frac{6 \sqrt{2} (b c-a d)^2 (5 a d+7 b c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{a^{11/4} b^{9/4}}+\frac{24 \sqrt{x} (a d-b c)^3}{a^2 b^2 \left (a+b x^2\right )}-\frac{32 c^3}{a^2 x^{3/2}}+\frac{96 d^3 \sqrt{x}}{b^2}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x^2)^3/(x^(5/2)*(a + b*x^2)^2),x]

[Out]

((-32*c^3)/(a^2*x^(3/2)) + (96*d^3*Sqrt[x])/b^2 + (24*(-(b*c) + a*d)^3*Sqrt[x])/
(a^2*b^2*(a + b*x^2)) + (6*Sqrt[2]*(b*c - a*d)^2*(7*b*c + 5*a*d)*ArcTan[1 - (Sqr
t[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(a^(11/4)*b^(9/4)) - (6*Sqrt[2]*(b*c - a*d)^2*(7
*b*c + 5*a*d)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(a^(11/4)*b^(9/4))
+ (3*Sqrt[2]*(b*c - a*d)^2*(7*b*c + 5*a*d)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)
*Sqrt[x] + Sqrt[b]*x])/(a^(11/4)*b^(9/4)) - (3*Sqrt[2]*(b*c - a*d)^2*(7*b*c + 5*
a*d)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(a^(11/4)*b^(9/
4)))/48

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Maple [B]  time = 0.029, size = 682, normalized size = 1.9 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x^2+c)^3/x^(5/2)/(b*x^2+a)^2,x)

[Out]

2*d^3*x^(1/2)/b^2-2/3*c^3/a^2/x^(3/2)+1/2*a/b^2*x^(1/2)/(b*x^2+a)*d^3-3/2/b*x^(1
/2)/(b*x^2+a)*c*d^2+3/2/a*x^(1/2)/(b*x^2+a)*c^2*d-1/2/a^2*b*x^(1/2)/(b*x^2+a)*c^
3-5/8/b^2*(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)*d^3+3/8/a/b*
(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)*c*d^2+9/8/a^2*(a/b)^(1
/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)*c^2*d-7/8/a^3*b*(a/b)^(1/4)*2^
(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)*c^3-5/8/b^2*(a/b)^(1/4)*2^(1/2)*arct
an(2^(1/2)/(a/b)^(1/4)*x^(1/2)-1)*d^3+3/8/a/b*(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)
/(a/b)^(1/4)*x^(1/2)-1)*c*d^2+9/8/a^2*(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(
1/4)*x^(1/2)-1)*c^2*d-7/8/a^3*b*(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x
^(1/2)-1)*c^3-5/16/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)))*d^3+3/16/a/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*d^2+9/16/a^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)))*c^2*d-7/16/a^
3*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/((b*x^2 + a)^2*x^(5/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.264506, size = 2183, normalized size = 5.95 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^2 + c)^3/((b*x^2 + a)^2*x^(5/2)),x, algorithm="fricas")

[Out]

1/24*(48*a^2*b*d^3*x^4 - 16*a*b^2*c^3 - 4*(7*b^3*c^3 - 9*a*b^2*c^2*d + 9*a^2*b*c
*d^2 - 15*a^3*d^3)*x^2 + 12*(a^2*b^3*x^3 + a^3*b^2*x)*sqrt(x)*(-(2401*b^12*c^12
- 12348*a*b^11*c^11*d + 19698*a^2*b^10*c^10*d^2 + 2324*a^3*b^9*c^9*d^3 - 37665*a
^4*b^8*c^8*d^4 + 27144*a^5*b^7*c^7*d^5 + 19068*a^6*b^6*c^6*d^6 - 28728*a^7*b^5*c
^5*d^7 + 1071*a^8*b^4*c^4*d^8 + 11060*a^9*b^3*c^3*d^9 - 3150*a^10*b^2*c^2*d^10 -
1500*a^11*b*c*d^11 + 625*a^12*d^12)/(a^11*b^9))^(1/4)*arctan(a^3*b^2*(-(2401*b^
12*c^12 - 12348*a*b^11*c^11*d + 19698*a^2*b^10*c^10*d^2 + 2324*a^3*b^9*c^9*d^3 -
37665*a^4*b^8*c^8*d^4 + 27144*a^5*b^7*c^7*d^5 + 19068*a^6*b^6*c^6*d^6 - 28728*a
^7*b^5*c^5*d^7 + 1071*a^8*b^4*c^4*d^8 + 11060*a^9*b^3*c^3*d^9 - 3150*a^10*b^2*c^
2*d^10 - 1500*a^11*b*c*d^11 + 625*a^12*d^12)/(a^11*b^9))^(1/4)/((7*b^3*c^3 - 9*a
*b^2*c^2*d - 3*a^2*b*c*d^2 + 5*a^3*d^3)*sqrt(x) + sqrt(a^6*b^4*sqrt(-(2401*b^12*
c^12 - 12348*a*b^11*c^11*d + 19698*a^2*b^10*c^10*d^2 + 2324*a^3*b^9*c^9*d^3 - 37
665*a^4*b^8*c^8*d^4 + 27144*a^5*b^7*c^7*d^5 + 19068*a^6*b^6*c^6*d^6 - 28728*a^7*
b^5*c^5*d^7 + 1071*a^8*b^4*c^4*d^8 + 11060*a^9*b^3*c^3*d^9 - 3150*a^10*b^2*c^2*d
^10 - 1500*a^11*b*c*d^11 + 625*a^12*d^12)/(a^11*b^9)) + (49*b^6*c^6 - 126*a*b^5*
c^5*d + 39*a^2*b^4*c^4*d^2 + 124*a^3*b^3*c^3*d^3 - 81*a^4*b^2*c^2*d^4 - 30*a^5*b
*c*d^5 + 25*a^6*d^6)*x))) - 3*(a^2*b^3*x^3 + a^3*b^2*x)*sqrt(x)*(-(2401*b^12*c^1
2 - 12348*a*b^11*c^11*d + 19698*a^2*b^10*c^10*d^2 + 2324*a^3*b^9*c^9*d^3 - 37665
*a^4*b^8*c^8*d^4 + 27144*a^5*b^7*c^7*d^5 + 19068*a^6*b^6*c^6*d^6 - 28728*a^7*b^5
*c^5*d^7 + 1071*a^8*b^4*c^4*d^8 + 11060*a^9*b^3*c^3*d^9 - 3150*a^10*b^2*c^2*d^10
- 1500*a^11*b*c*d^11 + 625*a^12*d^12)/(a^11*b^9))^(1/4)*log(a^3*b^2*(-(2401*b^1
2*c^12 - 12348*a*b^11*c^11*d + 19698*a^2*b^10*c^10*d^2 + 2324*a^3*b^9*c^9*d^3 -
37665*a^4*b^8*c^8*d^4 + 27144*a^5*b^7*c^7*d^5 + 19068*a^6*b^6*c^6*d^6 - 28728*a^
7*b^5*c^5*d^7 + 1071*a^8*b^4*c^4*d^8 + 11060*a^9*b^3*c^3*d^9 - 3150*a^10*b^2*c^2
*d^10 - 1500*a^11*b*c*d^11 + 625*a^12*d^12)/(a^11*b^9))^(1/4) + (7*b^3*c^3 - 9*a
*b^2*c^2*d - 3*a^2*b*c*d^2 + 5*a^3*d^3)*sqrt(x)) + 3*(a^2*b^3*x^3 + a^3*b^2*x)*s
qrt(x)*(-(2401*b^12*c^12 - 12348*a*b^11*c^11*d + 19698*a^2*b^10*c^10*d^2 + 2324*
a^3*b^9*c^9*d^3 - 37665*a^4*b^8*c^8*d^4 + 27144*a^5*b^7*c^7*d^5 + 19068*a^6*b^6*
c^6*d^6 - 28728*a^7*b^5*c^5*d^7 + 1071*a^8*b^4*c^4*d^8 + 11060*a^9*b^3*c^3*d^9 -
3150*a^10*b^2*c^2*d^10 - 1500*a^11*b*c*d^11 + 625*a^12*d^12)/(a^11*b^9))^(1/4)*
log(-a^3*b^2*(-(2401*b^12*c^12 - 12348*a*b^11*c^11*d + 19698*a^2*b^10*c^10*d^2 +
2324*a^3*b^9*c^9*d^3 - 37665*a^4*b^8*c^8*d^4 + 27144*a^5*b^7*c^7*d^5 + 19068*a^
6*b^6*c^6*d^6 - 28728*a^7*b^5*c^5*d^7 + 1071*a^8*b^4*c^4*d^8 + 11060*a^9*b^3*c^3
*d^9 - 3150*a^10*b^2*c^2*d^10 - 1500*a^11*b*c*d^11 + 625*a^12*d^12)/(a^11*b^9))^
(1/4) + (7*b^3*c^3 - 9*a*b^2*c^2*d - 3*a^2*b*c*d^2 + 5*a^3*d^3)*sqrt(x)))/((a^2*
b^3*x^3 + a^3*b^2*x)*sqrt(x))

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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((d*x**2+c)**3/x**(5/2)/(b*x**2+a)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.288542, size = 676, normalized size = 1.84 \[ \frac{2 \, d^{3} \sqrt{x}}{b^{2}} - \frac{2 \, c^{3}}{3 \, a^{2} x^{\frac{3}{2}}} - \frac{\sqrt{2}{\left (7 \, \left (a b^{3}\right )^{\frac{1}{4}} b^{3} c^{3} - 9 \, \left (a b^{3}\right )^{\frac{1}{4}} a b^{2} c^{2} d - 3 \, \left (a b^{3}\right )^{\frac{1}{4}} a^{2} b c d^{2} + 5 \, \left (a b^{3}\right )^{\frac{1}{4}} a^{3} d^{3}\right )} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{8 \, a^{3} b^{3}} - \frac{\sqrt{2}{\left (7 \, \left (a b^{3}\right )^{\frac{1}{4}} b^{3} c^{3} - 9 \, \left (a b^{3}\right )^{\frac{1}{4}} a b^{2} c^{2} d - 3 \, \left (a b^{3}\right )^{\frac{1}{4}} a^{2} b c d^{2} + 5 \, \left (a b^{3}\right )^{\frac{1}{4}} a^{3} d^{3}\right )} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{8 \, a^{3} b^{3}} - \frac{\sqrt{2}{\left (7 \, \left (a b^{3}\right )^{\frac{1}{4}} b^{3} c^{3} - 9 \, \left (a b^{3}\right )^{\frac{1}{4}} a b^{2} c^{2} d - 3 \, \left (a b^{3}\right )^{\frac{1}{4}} a^{2} b c d^{2} + 5 \, \left (a b^{3}\right )^{\frac{1}{4}} a^{3} d^{3}\right )}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{b}}\right )}{16 \, a^{3} b^{3}} + \frac{\sqrt{2}{\left (7 \, \left (a b^{3}\right )^{\frac{1}{4}} b^{3} c^{3} - 9 \, \left (a b^{3}\right )^{\frac{1}{4}} a b^{2} c^{2} d - 3 \, \left (a b^{3}\right )^{\frac{1}{4}} a^{2} b c d^{2} + 5 \, \left (a b^{3}\right )^{\frac{1}{4}} a^{3} d^{3}\right )}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{b}}\right )}{16 \, a^{3} b^{3}} - \frac{b^{3} c^{3} \sqrt{x} - 3 \, a b^{2} c^{2} d \sqrt{x} + 3 \, a^{2} b c d^{2} \sqrt{x} - a^{3} d^{3} \sqrt{x}}{2 \,{\left (b x^{2} + a\right )} a^{2} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^2 + c)^3/((b*x^2 + a)^2*x^(5/2)),x, algorithm="giac")

[Out]

2*d^3*sqrt(x)/b^2 - 2/3*c^3/(a^2*x^(3/2)) - 1/8*sqrt(2)*(7*(a*b^3)^(1/4)*b^3*c^3
- 9*(a*b^3)^(1/4)*a*b^2*c^2*d - 3*(a*b^3)^(1/4)*a^2*b*c*d^2 + 5*(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))/(a^3*b
^3) - 1/8*sqrt(2)*(7*(a*b^3)^(1/4)*b^3*c^3 - 9*(a*b^3)^(1/4)*a*b^2*c^2*d - 3*(a*
b^3)^(1/4)*a^2*b*c*d^2 + 5*(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))/(a^3*b^3) - 1/16*sqrt(2)*(7*(a*b^3)^(1/4)*b
^3*c^3 - 9*(a*b^3)^(1/4)*a*b^2*c^2*d - 3*(a*b^3)^(1/4)*a^2*b*c*d^2 + 5*(a*b^3)^(
1/4)*a^3*d^3)*ln(sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(a^3*b^3) + 1/16*s
qrt(2)*(7*(a*b^3)^(1/4)*b^3*c^3 - 9*(a*b^3)^(1/4)*a*b^2*c^2*d - 3*(a*b^3)^(1/4)*
a^2*b*c*d^2 + 5*(a*b^3)^(1/4)*a^3*d^3)*ln(-sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqr
t(a/b))/(a^3*b^3) - 1/2*(b^3*c^3*sqrt(x) - 3*a*b^2*c^2*d*sqrt(x) + 3*a^2*b*c*d^2
*sqrt(x) - a^3*d^3*sqrt(x))/((b*x^2 + a)*a^2*b^2)