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

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

[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 - 9
0*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)])/(32*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)*Sqr
t[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))

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Rubi [A]  time = 0.824478, antiderivative size = 440, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 10, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417 \[ -\frac{\sqrt{x} \left (5 a^2 d^2-90 a b c d+117 b^2 c^2\right )}{16 c d^4}+\frac{x^{5/2} \left (5 a^2 d^2-90 a b c d+117 b^2 c^2\right )}{80 c^2 d^3}-\frac{\left (5 a^2 d^2-90 a b c d+117 b^2 c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{3/4} d^{17/4}}+\frac{\left (5 a^2 d^2-90 a b c d+117 b^2 c^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{3/4} d^{17/4}}-\frac{\left (5 a^2 d^2-90 a b c d+117 b^2 c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{32 \sqrt{2} c^{3/4} d^{17/4}}+\frac{\left (5 a^2 d^2-90 a b c d+117 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{32 \sqrt{2} c^{3/4} d^{17/4}}-\frac{x^{9/2} (b c-a d) (17 b c-a d)}{16 c^2 d^2 \left (c+d x^2\right )}+\frac{x^{9/2} (b c-a d)^2}{4 c d^2 \left (c+d x^2\right )^2} \]

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 - 9
0*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)])/(32*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)*Sqr
t[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))

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Rubi in Sympy [A]  time = 124.299, size = 425, normalized size = 0.97 \[ \frac{x^{\frac{9}{2}} \left (a d - b c\right )^{2}}{4 c d^{2} \left (c + d x^{2}\right )^{2}} - \frac{\sqrt{x} \left (5 a^{2} d^{2} - 90 a b c d + 117 b^{2} c^{2}\right )}{16 c d^{4}} - \frac{x^{\frac{9}{2}} \left (a d - 17 b c\right ) \left (a d - b c\right )}{16 c^{2} d^{2} \left (c + d x^{2}\right )} + \frac{x^{\frac{5}{2}} \left (5 a^{2} d^{2} - 90 a b c d + 117 b^{2} c^{2}\right )}{80 c^{2} d^{3}} - \frac{\sqrt{2} \left (5 a^{2} d^{2} - 90 a b c d + 117 b^{2} c^{2}\right ) \log{\left (- \sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x} + \sqrt{c} + \sqrt{d} x \right )}}{128 c^{\frac{3}{4}} d^{\frac{17}{4}}} + \frac{\sqrt{2} \left (5 a^{2} d^{2} - 90 a b c d + 117 b^{2} c^{2}\right ) \log{\left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x} + \sqrt{c} + \sqrt{d} x \right )}}{128 c^{\frac{3}{4}} d^{\frac{17}{4}}} - \frac{\sqrt{2} \left (5 a^{2} d^{2} - 90 a b c d + 117 b^{2} c^{2}\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}} \right )}}{64 c^{\frac{3}{4}} d^{\frac{17}{4}}} + \frac{\sqrt{2} \left (5 a^{2} d^{2} - 90 a b c d + 117 b^{2} c^{2}\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}} \right )}}{64 c^{\frac{3}{4}} d^{\frac{17}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**(9/2)*(a*d - b*c)**2/(4*c*d**2*(c + d*x**2)**2) - sqrt(x)*(5*a**2*d**2 - 90*a
*b*c*d + 117*b**2*c**2)/(16*c*d**4) - x**(9/2)*(a*d - 17*b*c)*(a*d - b*c)/(16*c*
*2*d**2*(c + d*x**2)) + x**(5/2)*(5*a**2*d**2 - 90*a*b*c*d + 117*b**2*c**2)/(80*
c**2*d**3) - sqrt(2)*(5*a**2*d**2 - 90*a*b*c*d + 117*b**2*c**2)*log(-sqrt(2)*c**
(1/4)*d**(1/4)*sqrt(x) + sqrt(c) + sqrt(d)*x)/(128*c**(3/4)*d**(17/4)) + sqrt(2)
*(5*a**2*d**2 - 90*a*b*c*d + 117*b**2*c**2)*log(sqrt(2)*c**(1/4)*d**(1/4)*sqrt(x
) + sqrt(c) + sqrt(d)*x)/(128*c**(3/4)*d**(17/4)) - sqrt(2)*(5*a**2*d**2 - 90*a*
b*c*d + 117*b**2*c**2)*atan(1 - sqrt(2)*d**(1/4)*sqrt(x)/c**(1/4))/(64*c**(3/4)*
d**(17/4)) + sqrt(2)*(5*a**2*d**2 - 90*a*b*c*d + 117*b**2*c**2)*atan(1 + sqrt(2)
*d**(1/4)*sqrt(x)/c**(1/4))/(64*c**(3/4)*d**(17/4))

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Mathematica [A]  time = 0.891102, size = 383, normalized size = 0.87 \[ \frac{-\frac{40 \sqrt [4]{d} \sqrt{x} \left (9 a^2 d^2-34 a b c d+25 b^2 c^2\right )}{c+d x^2}-\frac{5 \sqrt{2} \left (5 a^2 d^2-90 a b c d+117 b^2 c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{c^{3/4}}+\frac{5 \sqrt{2} \left (5 a^2 d^2-90 a b c d+117 b^2 c^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{c^{3/4}}-\frac{10 \sqrt{2} \left (5 a^2 d^2-90 a b c d+117 b^2 c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{c^{3/4}}+\frac{10 \sqrt{2} \left (5 a^2 d^2-90 a b c d+117 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{c^{3/4}}+\frac{160 c \sqrt [4]{d} \sqrt{x} (b c-a d)^2}{\left (c+d x^2\right )^2}-1280 b \sqrt [4]{d} \sqrt{x} (3 b c-2 a d)+256 b^2 d^{5/4} x^{5/2}}{640 d^{17/4}} \]

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^(17/4))

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Maple [A]  time = 0.028, size = 590, normalized size = 1.3 \[ \text{result too large to display} \]

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/d^3*b^2*x^(5/2)+4/d^3*b*a*x^(1/2)-6/d^4*b^2*x^(1/2)*c-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+5/64/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)*x^(1/2)*2^(1/2)+(c/d)^(1/2))/(x-(c/d)^
(1/4)*x^(1/2)*2^(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)*x^(1/2)*2^(1/2)+(c/d)^(1/2))/(x-(c/d)^(1/4)*x^(1/2)*2^(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)*x^(1/2)*2^(1/2)+(c/d)^
(1/2))/(x-(c/d)^(1/4)*x^(1/2)*2^(1/2)+(c/d)^(1/2)))*b^2

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.280408, size = 1605, normalized size = 3.65 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2*x^(7/2)/(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(c*d^4*(-(187388721*b^8*c^8 - 5765
80680*a*b^7*c^7*d + 697317660*a^2*b^6*c^6*d^2 - 415092600*a^3*b^5*c^5*d^3 + 1245
25350*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 - 450
00*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) + sqrt(c^2*d^8*sqrt(-(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 + 6
25*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))) - 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 - 415092
600*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 + 127
3500*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 - 4
15092600*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 + 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)) - 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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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**(7/2)*(b*x**2+a)**2/(d*x**2+c)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.247895, size = 609, normalized size = 1.38 \[ \frac{\sqrt{2}{\left (117 \, \left (c d^{3}\right )^{\frac{1}{4}} b^{2} c^{2} - 90 \, \left (c d^{3}\right )^{\frac{1}{4}} a b c d + 5 \, \left (c d^{3}\right )^{\frac{1}{4}} a^{2} d^{2}\right )} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{c}{d}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{c}{d}\right )^{\frac{1}{4}}}\right )}{64 \, c d^{5}} + \frac{\sqrt{2}{\left (117 \, \left (c d^{3}\right )^{\frac{1}{4}} b^{2} c^{2} - 90 \, \left (c d^{3}\right )^{\frac{1}{4}} a b c d + 5 \, \left (c d^{3}\right )^{\frac{1}{4}} a^{2} d^{2}\right )} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{c}{d}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{c}{d}\right )^{\frac{1}{4}}}\right )}{64 \, c d^{5}} + \frac{\sqrt{2}{\left (117 \, \left (c d^{3}\right )^{\frac{1}{4}} b^{2} c^{2} - 90 \, \left (c d^{3}\right )^{\frac{1}{4}} a b c d + 5 \, \left (c d^{3}\right )^{\frac{1}{4}} a^{2} d^{2}\right )}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{c}{d}\right )^{\frac{1}{4}} + x + \sqrt{\frac{c}{d}}\right )}{128 \, c d^{5}} - \frac{\sqrt{2}{\left (117 \, \left (c d^{3}\right )^{\frac{1}{4}} b^{2} c^{2} - 90 \, \left (c d^{3}\right )^{\frac{1}{4}} a b c d + 5 \, \left (c d^{3}\right )^{\frac{1}{4}} a^{2} d^{2}\right )}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{c}{d}\right )^{\frac{1}{4}} + x + \sqrt{\frac{c}{d}}\right )}{128 \, c d^{5}} - \frac{25 \, b^{2} c^{2} d x^{\frac{5}{2}} - 34 \, a b c d^{2} x^{\frac{5}{2}} + 9 \, a^{2} d^{3} x^{\frac{5}{2}} + 21 \, b^{2} c^{3} \sqrt{x} - 26 \, a b c^{2} d \sqrt{x} + 5 \, a^{2} c d^{2} \sqrt{x}}{16 \,{\left (d x^{2} + c\right )}^{2} d^{4}} + \frac{2 \,{\left (b^{2} d^{12} x^{\frac{5}{2}} - 15 \, b^{2} c d^{11} \sqrt{x} + 10 \, a b d^{12} \sqrt{x}\right )}}{5 \, d^{15}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2*x^(7/2)/(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)*ln(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)*ln(-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