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

Optimal. Leaf size=399 \[ -\frac{x^{3/2} \left (9 a^2 d^2-2 a b c d+b^2 c^2\right )}{4 c^2 d \left (c+d x^2\right )^2}-\frac{2 a^2}{c \sqrt{x} \left (c+d x^2\right )^2}+\frac{\left (5 a d (2 b c-9 a d)+3 b^2 c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{13/4} d^{7/4}}-\frac{\left (5 a d (2 b c-9 a d)+3 b^2 c^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{13/4} d^{7/4}}-\frac{\left (5 a d (2 b c-9 a d)+3 b^2 c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{32 \sqrt{2} c^{13/4} d^{7/4}}+\frac{\left (5 a d (2 b c-9 a d)+3 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{32 \sqrt{2} c^{13/4} d^{7/4}}+\frac{x^{3/2} \left (5 a d (2 b c-9 a d)+3 b^2 c^2\right )}{16 c^3 d \left (c+d x^2\right )} \]

[Out]

(-2*a^2)/(c*Sqrt[x]*(c + d*x^2)^2) - ((b^2*c^2 - 2*a*b*c*d + 9*a^2*d^2)*x^(3/2))
/(4*c^2*d*(c + d*x^2)^2) + ((3*b^2*c^2 + 5*a*d*(2*b*c - 9*a*d))*x^(3/2))/(16*c^3
*d*(c + d*x^2)) - ((3*b^2*c^2 + 5*a*d*(2*b*c - 9*a*d))*ArcTan[1 - (Sqrt[2]*d^(1/
4)*Sqrt[x])/c^(1/4)])/(32*Sqrt[2]*c^(13/4)*d^(7/4)) + ((3*b^2*c^2 + 5*a*d*(2*b*c
- 9*a*d))*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(32*Sqrt[2]*c^(13/4)*d
^(7/4)) + ((3*b^2*c^2 + 5*a*d*(2*b*c - 9*a*d))*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(
1/4)*Sqrt[x] + Sqrt[d]*x])/(64*Sqrt[2]*c^(13/4)*d^(7/4)) - ((3*b^2*c^2 + 5*a*d*(
2*b*c - 9*a*d))*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(64*
Sqrt[2]*c^(13/4)*d^(7/4))

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Rubi [A]  time = 0.830139, antiderivative size = 398, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 10, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417 \[ -\frac{x^{3/2} \left (9 a^2 d^2-2 a b c d+b^2 c^2\right )}{4 c^2 d \left (c+d x^2\right )^2}-\frac{2 a^2}{c \sqrt{x} \left (c+d x^2\right )^2}+\frac{x^{3/2} \left (\frac{5 a (2 b c-9 a d)}{c^2}+\frac{3 b^2}{d}\right )}{16 c \left (c+d x^2\right )}+\frac{\left (5 a d (2 b c-9 a d)+3 b^2 c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{13/4} d^{7/4}}-\frac{\left (5 a d (2 b c-9 a d)+3 b^2 c^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{13/4} d^{7/4}}-\frac{\left (5 a d (2 b c-9 a d)+3 b^2 c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{32 \sqrt{2} c^{13/4} d^{7/4}}+\frac{\left (5 a d (2 b c-9 a d)+3 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{32 \sqrt{2} c^{13/4} d^{7/4}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*a^2)/(c*Sqrt[x]*(c + d*x^2)^2) - ((b^2*c^2 - 2*a*b*c*d + 9*a^2*d^2)*x^(3/2))
/(4*c^2*d*(c + d*x^2)^2) + (((3*b^2)/d + (5*a*(2*b*c - 9*a*d))/c^2)*x^(3/2))/(16
*c*(c + d*x^2)) - ((3*b^2*c^2 + 5*a*d*(2*b*c - 9*a*d))*ArcTan[1 - (Sqrt[2]*d^(1/
4)*Sqrt[x])/c^(1/4)])/(32*Sqrt[2]*c^(13/4)*d^(7/4)) + ((3*b^2*c^2 + 5*a*d*(2*b*c
- 9*a*d))*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(32*Sqrt[2]*c^(13/4)*d
^(7/4)) + ((3*b^2*c^2 + 5*a*d*(2*b*c - 9*a*d))*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(
1/4)*Sqrt[x] + Sqrt[d]*x])/(64*Sqrt[2]*c^(13/4)*d^(7/4)) - ((3*b^2*c^2 + 5*a*d*(
2*b*c - 9*a*d))*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(64*
Sqrt[2]*c^(13/4)*d^(7/4))

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Rubi in Sympy [A]  time = 110.094, size = 381, normalized size = 0.95 \[ - \frac{2 a^{2}}{c \sqrt{x} \left (c + d x^{2}\right )^{2}} - \frac{x^{\frac{3}{2}} \left (a d \left (9 a d - 2 b c\right ) + b^{2} c^{2}\right )}{4 c^{2} d \left (c + d x^{2}\right )^{2}} + \frac{x^{\frac{3}{2}} \left (- 5 a d \left (9 a d - 2 b c\right ) + 3 b^{2} c^{2}\right )}{16 c^{3} d \left (c + d x^{2}\right )} + \frac{\sqrt{2} \left (- 5 a d \left (9 a d - 2 b c\right ) + 3 b^{2} c^{2}\right ) \log{\left (- \sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x} + \sqrt{c} + \sqrt{d} x \right )}}{128 c^{\frac{13}{4}} d^{\frac{7}{4}}} - \frac{\sqrt{2} \left (- 5 a d \left (9 a d - 2 b c\right ) + 3 b^{2} c^{2}\right ) \log{\left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x} + \sqrt{c} + \sqrt{d} x \right )}}{128 c^{\frac{13}{4}} d^{\frac{7}{4}}} - \frac{\sqrt{2} \left (- 5 a d \left (9 a d - 2 b c\right ) + 3 b^{2} c^{2}\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}} \right )}}{64 c^{\frac{13}{4}} d^{\frac{7}{4}}} + \frac{\sqrt{2} \left (- 5 a d \left (9 a d - 2 b c\right ) + 3 b^{2} c^{2}\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}} \right )}}{64 c^{\frac{13}{4}} d^{\frac{7}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*a**2/(c*sqrt(x)*(c + d*x**2)**2) - x**(3/2)*(a*d*(9*a*d - 2*b*c) + b**2*c**2)
/(4*c**2*d*(c + d*x**2)**2) + x**(3/2)*(-5*a*d*(9*a*d - 2*b*c) + 3*b**2*c**2)/(1
6*c**3*d*(c + d*x**2)) + sqrt(2)*(-5*a*d*(9*a*d - 2*b*c) + 3*b**2*c**2)*log(-sqr
t(2)*c**(1/4)*d**(1/4)*sqrt(x) + sqrt(c) + sqrt(d)*x)/(128*c**(13/4)*d**(7/4)) -
sqrt(2)*(-5*a*d*(9*a*d - 2*b*c) + 3*b**2*c**2)*log(sqrt(2)*c**(1/4)*d**(1/4)*sq
rt(x) + sqrt(c) + sqrt(d)*x)/(128*c**(13/4)*d**(7/4)) - sqrt(2)*(-5*a*d*(9*a*d -
2*b*c) + 3*b**2*c**2)*atan(1 - sqrt(2)*d**(1/4)*sqrt(x)/c**(1/4))/(64*c**(13/4)
*d**(7/4)) + sqrt(2)*(-5*a*d*(9*a*d - 2*b*c) + 3*b**2*c**2)*atan(1 + sqrt(2)*d**
(1/4)*sqrt(x)/c**(1/4))/(64*c**(13/4)*d**(7/4))

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Mathematica [A]  time = 0.744102, size = 364, normalized size = 0.91 \[ \frac{\frac{8 \sqrt [4]{c} x^{3/2} \left (-13 a^2 d^2+10 a b c d+3 b^2 c^2\right )}{d \left (c+d x^2\right )}+\frac{\sqrt{2} \left (-45 a^2 d^2+10 a b c d+3 b^2 c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{d^{7/4}}+\frac{\sqrt{2} \left (45 a^2 d^2-10 a b c d-3 b^2 c^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{d^{7/4}}+\frac{2 \sqrt{2} \left (45 a^2 d^2-10 a b c d-3 b^2 c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{d^{7/4}}+\frac{2 \sqrt{2} \left (-45 a^2 d^2+10 a b c d+3 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{d^{7/4}}-\frac{256 a^2 \sqrt [4]{c}}{\sqrt{x}}-\frac{32 c^{5/4} x^{3/2} (b c-a d)^2}{d \left (c+d x^2\right )^2}}{128 c^{13/4}} \]

Antiderivative was successfully verified.

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

[Out]

((-256*a^2*c^(1/4))/Sqrt[x] - (32*c^(5/4)*(b*c - a*d)^2*x^(3/2))/(d*(c + d*x^2)^
2) + (8*c^(1/4)*(3*b^2*c^2 + 10*a*b*c*d - 13*a^2*d^2)*x^(3/2))/(d*(c + d*x^2)) +
(2*Sqrt[2]*(-3*b^2*c^2 - 10*a*b*c*d + 45*a^2*d^2)*ArcTan[1 - (Sqrt[2]*d^(1/4)*S
qrt[x])/c^(1/4)])/d^(7/4) + (2*Sqrt[2]*(3*b^2*c^2 + 10*a*b*c*d - 45*a^2*d^2)*Arc
Tan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/d^(7/4) + (Sqrt[2]*(3*b^2*c^2 + 10*a
*b*c*d - 45*a^2*d^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])
/d^(7/4) + (Sqrt[2]*(-3*b^2*c^2 - 10*a*b*c*d + 45*a^2*d^2)*Log[Sqrt[c] + Sqrt[2]
*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/d^(7/4))/(128*c^(13/4))

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Maple [A]  time = 0.031, size = 568, normalized size = 1.4 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-13/16/c^3/(d*x^2+c)^2*x^(7/2)*a^2*d^2+5/8/c^2/(d*x^2+c)^2*x^(7/2)*a*b*d+3/16/c/
(d*x^2+c)^2*x^(7/2)*b^2-17/16/c^2/(d*x^2+c)^2*d*x^(3/2)*a^2+9/8/c/(d*x^2+c)^2*x^
(3/2)*a*b-1/16/(d*x^2+c)^2/d*x^(3/2)*b^2-45/64/c^3/(c/d)^(1/4)*2^(1/2)*arctan(2^
(1/2)/(c/d)^(1/4)*x^(1/2)-1)*a^2+5/32/c^2/d/(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(
c/d)^(1/4)*x^(1/2)-1)*a*b+3/64/c/d^2/(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1
/4)*x^(1/2)-1)*b^2-45/128/c^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^2+5/64/c^2/d/(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+3/128/c/d^2/(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)))*b^2
-45/64/c^3/(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)+1)*a^2+5/32/c^
2/d/(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)+1)*a*b+3/64/c/d^2/(c/
d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)+1)*b^2-2*a^2/c^3/x^(1/2)

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.279397, size = 2147, normalized size = 5.38 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/64*(128*a^2*c^2*d - 4*(3*b^2*c^2*d + 10*a*b*c*d^2 - 45*a^2*d^3)*x^4 + 4*(b^2*
c^3 - 18*a*b*c^2*d + 81*a^2*c*d^2)*x^2 + 4*(c^3*d^3*x^4 + 2*c^4*d^2*x^2 + c^5*d)
*sqrt(x)*(-(81*b^8*c^8 + 1080*a*b^7*c^7*d + 540*a^2*b^6*c^6*d^2 - 36600*a^3*b^5*
c^5*d^3 - 42650*a^4*b^4*c^4*d^4 + 549000*a^5*b^3*c^3*d^5 + 121500*a^6*b^2*c^2*d^
6 - 3645000*a^7*b*c*d^7 + 4100625*a^8*d^8)/(c^13*d^7))^(1/4)*arctan(-c^10*d^5*(-
(81*b^8*c^8 + 1080*a*b^7*c^7*d + 540*a^2*b^6*c^6*d^2 - 36600*a^3*b^5*c^5*d^3 - 4
2650*a^4*b^4*c^4*d^4 + 549000*a^5*b^3*c^3*d^5 + 121500*a^6*b^2*c^2*d^6 - 3645000
*a^7*b*c*d^7 + 4100625*a^8*d^8)/(c^13*d^7))^(3/4)/((27*b^6*c^6 + 270*a*b^5*c^5*d
- 315*a^2*b^4*c^4*d^2 - 7100*a^3*b^3*c^3*d^3 + 4725*a^4*b^2*c^2*d^4 + 60750*a^5
*b*c*d^5 - 91125*a^6*d^6)*sqrt(x) - sqrt((729*b^12*c^12 + 14580*a*b^11*c^11*d +
55890*a^2*b^10*c^10*d^2 - 553500*a^3*b^9*c^9*d^3 - 3479625*a^4*b^8*c^8*d^4 + 103
05000*a^5*b^7*c^7*d^5 + 75317500*a^6*b^6*c^6*d^6 - 154575000*a^7*b^5*c^5*d^7 - 7
82915625*a^8*b^4*c^4*d^8 + 1868062500*a^9*b^3*c^3*d^9 + 2829431250*a^10*b^2*c^2*
d^10 - 11071687500*a^11*b*c*d^11 + 8303765625*a^12*d^12)*x - (81*b^8*c^15*d^3 +
1080*a*b^7*c^14*d^4 + 540*a^2*b^6*c^13*d^5 - 36600*a^3*b^5*c^12*d^6 - 42650*a^4*
b^4*c^11*d^7 + 549000*a^5*b^3*c^10*d^8 + 121500*a^6*b^2*c^9*d^9 - 3645000*a^7*b*
c^8*d^10 + 4100625*a^8*c^7*d^11)*sqrt(-(81*b^8*c^8 + 1080*a*b^7*c^7*d + 540*a^2*
b^6*c^6*d^2 - 36600*a^3*b^5*c^5*d^3 - 42650*a^4*b^4*c^4*d^4 + 549000*a^5*b^3*c^3
*d^5 + 121500*a^6*b^2*c^2*d^6 - 3645000*a^7*b*c*d^7 + 4100625*a^8*d^8)/(c^13*d^7
))))) + (c^3*d^3*x^4 + 2*c^4*d^2*x^2 + c^5*d)*sqrt(x)*(-(81*b^8*c^8 + 1080*a*b^7
*c^7*d + 540*a^2*b^6*c^6*d^2 - 36600*a^3*b^5*c^5*d^3 - 42650*a^4*b^4*c^4*d^4 + 5
49000*a^5*b^3*c^3*d^5 + 121500*a^6*b^2*c^2*d^6 - 3645000*a^7*b*c*d^7 + 4100625*a
^8*d^8)/(c^13*d^7))^(1/4)*log(c^10*d^5*(-(81*b^8*c^8 + 1080*a*b^7*c^7*d + 540*a^
2*b^6*c^6*d^2 - 36600*a^3*b^5*c^5*d^3 - 42650*a^4*b^4*c^4*d^4 + 549000*a^5*b^3*c
^3*d^5 + 121500*a^6*b^2*c^2*d^6 - 3645000*a^7*b*c*d^7 + 4100625*a^8*d^8)/(c^13*d
^7))^(3/4) - (27*b^6*c^6 + 270*a*b^5*c^5*d - 315*a^2*b^4*c^4*d^2 - 7100*a^3*b^3*
c^3*d^3 + 4725*a^4*b^2*c^2*d^4 + 60750*a^5*b*c*d^5 - 91125*a^6*d^6)*sqrt(x)) - (
c^3*d^3*x^4 + 2*c^4*d^2*x^2 + c^5*d)*sqrt(x)*(-(81*b^8*c^8 + 1080*a*b^7*c^7*d +
540*a^2*b^6*c^6*d^2 - 36600*a^3*b^5*c^5*d^3 - 42650*a^4*b^4*c^4*d^4 + 549000*a^5
*b^3*c^3*d^5 + 121500*a^6*b^2*c^2*d^6 - 3645000*a^7*b*c*d^7 + 4100625*a^8*d^8)/(
c^13*d^7))^(1/4)*log(-c^10*d^5*(-(81*b^8*c^8 + 1080*a*b^7*c^7*d + 540*a^2*b^6*c^
6*d^2 - 36600*a^3*b^5*c^5*d^3 - 42650*a^4*b^4*c^4*d^4 + 549000*a^5*b^3*c^3*d^5 +
121500*a^6*b^2*c^2*d^6 - 3645000*a^7*b*c*d^7 + 4100625*a^8*d^8)/(c^13*d^7))^(3/
4) - (27*b^6*c^6 + 270*a*b^5*c^5*d - 315*a^2*b^4*c^4*d^2 - 7100*a^3*b^3*c^3*d^3
+ 4725*a^4*b^2*c^2*d^4 + 60750*a^5*b*c*d^5 - 91125*a^6*d^6)*sqrt(x)))/((c^3*d^3*
x^4 + 2*c^4*d^2*x^2 + c^5*d)*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((b*x**2+a)**2/x**(3/2)/(d*x**2+c)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.336035, size = 576, normalized size = 1.44 \[ -\frac{2 \, a^{2}}{c^{3} \sqrt{x}} + \frac{3 \, b^{2} c^{2} d x^{\frac{7}{2}} + 10 \, a b c d^{2} x^{\frac{7}{2}} - 13 \, a^{2} d^{3} x^{\frac{7}{2}} - b^{2} c^{3} x^{\frac{3}{2}} + 18 \, a b c^{2} d x^{\frac{3}{2}} - 17 \, a^{2} c d^{2} x^{\frac{3}{2}}}{16 \,{\left (d x^{2} + c\right )}^{2} c^{3} d} + \frac{\sqrt{2}{\left (3 \, \left (c d^{3}\right )^{\frac{3}{4}} b^{2} c^{2} + 10 \, \left (c d^{3}\right )^{\frac{3}{4}} a b c d - 45 \, \left (c d^{3}\right )^{\frac{3}{4}} a^{2} d^{2}\right )} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{c}{d}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{c}{d}\right )^{\frac{1}{4}}}\right )}{64 \, c^{4} d^{4}} + \frac{\sqrt{2}{\left (3 \, \left (c d^{3}\right )^{\frac{3}{4}} b^{2} c^{2} + 10 \, \left (c d^{3}\right )^{\frac{3}{4}} a b c d - 45 \, \left (c d^{3}\right )^{\frac{3}{4}} a^{2} d^{2}\right )} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{c}{d}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{c}{d}\right )^{\frac{1}{4}}}\right )}{64 \, c^{4} d^{4}} - \frac{\sqrt{2}{\left (3 \, \left (c d^{3}\right )^{\frac{3}{4}} b^{2} c^{2} + 10 \, \left (c d^{3}\right )^{\frac{3}{4}} a b c d - 45 \, \left (c d^{3}\right )^{\frac{3}{4}} a^{2} d^{2}\right )}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{c}{d}\right )^{\frac{1}{4}} + x + \sqrt{\frac{c}{d}}\right )}{128 \, c^{4} d^{4}} + \frac{\sqrt{2}{\left (3 \, \left (c d^{3}\right )^{\frac{3}{4}} b^{2} c^{2} + 10 \, \left (c d^{3}\right )^{\frac{3}{4}} a b c d - 45 \, \left (c d^{3}\right )^{\frac{3}{4}} a^{2} d^{2}\right )}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{c}{d}\right )^{\frac{1}{4}} + x + \sqrt{\frac{c}{d}}\right )}{128 \, c^{4} d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*a^2/(c^3*sqrt(x)) + 1/16*(3*b^2*c^2*d*x^(7/2) + 10*a*b*c*d^2*x^(7/2) - 13*a^2
*d^3*x^(7/2) - b^2*c^3*x^(3/2) + 18*a*b*c^2*d*x^(3/2) - 17*a^2*c*d^2*x^(3/2))/((
d*x^2 + c)^2*c^3*d) + 1/64*sqrt(2)*(3*(c*d^3)^(3/4)*b^2*c^2 + 10*(c*d^3)^(3/4)*a
*b*c*d - 45*(c*d^3)^(3/4)*a^2*d^2)*arctan(1/2*sqrt(2)*(sqrt(2)*(c/d)^(1/4) + 2*s
qrt(x))/(c/d)^(1/4))/(c^4*d^4) + 1/64*sqrt(2)*(3*(c*d^3)^(3/4)*b^2*c^2 + 10*(c*d
^3)^(3/4)*a*b*c*d - 45*(c*d^3)^(3/4)*a^2*d^2)*arctan(-1/2*sqrt(2)*(sqrt(2)*(c/d)
^(1/4) - 2*sqrt(x))/(c/d)^(1/4))/(c^4*d^4) - 1/128*sqrt(2)*(3*(c*d^3)^(3/4)*b^2*
c^2 + 10*(c*d^3)^(3/4)*a*b*c*d - 45*(c*d^3)^(3/4)*a^2*d^2)*ln(sqrt(2)*sqrt(x)*(c
/d)^(1/4) + x + sqrt(c/d))/(c^4*d^4) + 1/128*sqrt(2)*(3*(c*d^3)^(3/4)*b^2*c^2 +
10*(c*d^3)^(3/4)*a*b*c*d - 45*(c*d^3)^(3/4)*a^2*d^2)*ln(-sqrt(2)*sqrt(x)*(c/d)^(
1/4) + x + sqrt(c/d))/(c^4*d^4)