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

Optimal. Leaf size=346 \[ -\frac{(b c-a d) (9 b c-a d) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{8 \sqrt{2} c^{3/4} d^{13/4}}+\frac{(b c-a d) (9 b c-a d) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{8 \sqrt{2} c^{3/4} d^{13/4}}-\frac{(b c-a d) (9 b c-a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{4 \sqrt{2} c^{3/4} d^{13/4}}+\frac{(b c-a d) (9 b c-a d) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{4 \sqrt{2} c^{3/4} d^{13/4}}-\frac{\sqrt{x} (b c-a d) (9 b c-a d)}{2 c d^3}+\frac{x^{5/2} (b c-a d)^2}{2 c d^2 \left (c+d x^2\right )}+\frac{2 b^2 x^{5/2}}{5 d^2} \]

[Out]

-((b*c - a*d)*(9*b*c - a*d)*Sqrt[x])/(2*c*d^3) + (2*b^2*x^(5/2))/(5*d^2) + ((b*c
- a*d)^2*x^(5/2))/(2*c*d^2*(c + d*x^2)) - ((b*c - a*d)*(9*b*c - a*d)*ArcTan[1 -
(Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(4*Sqrt[2]*c^(3/4)*d^(13/4)) + ((b*c - a*d)
*(9*b*c - a*d)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(4*Sqrt[2]*c^(3/4)
*d^(13/4)) - ((b*c - a*d)*(9*b*c - a*d)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sq
rt[x] + Sqrt[d]*x])/(8*Sqrt[2]*c^(3/4)*d^(13/4)) + ((b*c - a*d)*(9*b*c - a*d)*Lo
g[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(8*Sqrt[2]*c^(3/4)*d^(
13/4))

_______________________________________________________________________________________

Rubi [A]  time = 0.679314, antiderivative size = 346, 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 \[ -\frac{(b c-a d) (9 b c-a d) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{8 \sqrt{2} c^{3/4} d^{13/4}}+\frac{(b c-a d) (9 b c-a d) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{8 \sqrt{2} c^{3/4} d^{13/4}}-\frac{(b c-a d) (9 b c-a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{4 \sqrt{2} c^{3/4} d^{13/4}}+\frac{(b c-a d) (9 b c-a d) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{4 \sqrt{2} c^{3/4} d^{13/4}}-\frac{\sqrt{x} (b c-a d) (9 b c-a d)}{2 c d^3}+\frac{x^{5/2} (b c-a d)^2}{2 c d^2 \left (c+d x^2\right )}+\frac{2 b^2 x^{5/2}}{5 d^2} \]

Antiderivative was successfully verified.

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

[Out]

-((b*c - a*d)*(9*b*c - a*d)*Sqrt[x])/(2*c*d^3) + (2*b^2*x^(5/2))/(5*d^2) + ((b*c
- a*d)^2*x^(5/2))/(2*c*d^2*(c + d*x^2)) - ((b*c - a*d)*(9*b*c - a*d)*ArcTan[1 -
(Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(4*Sqrt[2]*c^(3/4)*d^(13/4)) + ((b*c - a*d)
*(9*b*c - a*d)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(4*Sqrt[2]*c^(3/4)
*d^(13/4)) - ((b*c - a*d)*(9*b*c - a*d)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sq
rt[x] + Sqrt[d]*x])/(8*Sqrt[2]*c^(3/4)*d^(13/4)) + ((b*c - a*d)*(9*b*c - a*d)*Lo
g[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(8*Sqrt[2]*c^(3/4)*d^(
13/4))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 109.982, size = 313, normalized size = 0.9 \[ \frac{2 b^{2} x^{\frac{5}{2}}}{5 d^{2}} + \frac{x^{\frac{5}{2}} \left (a d - b c\right )^{2}}{2 c d^{2} \left (c + d x^{2}\right )} - \frac{\sqrt{x} \left (a d - 9 b c\right ) \left (a d - b c\right )}{2 c d^{3}} - \frac{\sqrt{2} \left (a d - 9 b c\right ) \left (a d - b c\right ) \log{\left (- \sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x} + \sqrt{c} + \sqrt{d} x \right )}}{16 c^{\frac{3}{4}} d^{\frac{13}{4}}} + \frac{\sqrt{2} \left (a d - 9 b c\right ) \left (a d - b c\right ) \log{\left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x} + \sqrt{c} + \sqrt{d} x \right )}}{16 c^{\frac{3}{4}} d^{\frac{13}{4}}} - \frac{\sqrt{2} \left (a d - 9 b c\right ) \left (a d - b c\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}} \right )}}{8 c^{\frac{3}{4}} d^{\frac{13}{4}}} + \frac{\sqrt{2} \left (a d - 9 b c\right ) \left (a d - b c\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}} \right )}}{8 c^{\frac{3}{4}} d^{\frac{13}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*b**2*x**(5/2)/(5*d**2) + x**(5/2)*(a*d - b*c)**2/(2*c*d**2*(c + d*x**2)) - sqr
t(x)*(a*d - 9*b*c)*(a*d - b*c)/(2*c*d**3) - sqrt(2)*(a*d - 9*b*c)*(a*d - b*c)*lo
g(-sqrt(2)*c**(1/4)*d**(1/4)*sqrt(x) + sqrt(c) + sqrt(d)*x)/(16*c**(3/4)*d**(13/
4)) + sqrt(2)*(a*d - 9*b*c)*(a*d - b*c)*log(sqrt(2)*c**(1/4)*d**(1/4)*sqrt(x) +
sqrt(c) + sqrt(d)*x)/(16*c**(3/4)*d**(13/4)) - sqrt(2)*(a*d - 9*b*c)*(a*d - b*c)
*atan(1 - sqrt(2)*d**(1/4)*sqrt(x)/c**(1/4))/(8*c**(3/4)*d**(13/4)) + sqrt(2)*(a
*d - 9*b*c)*(a*d - b*c)*atan(1 + sqrt(2)*d**(1/4)*sqrt(x)/c**(1/4))/(8*c**(3/4)*
d**(13/4))

_______________________________________________________________________________________

Mathematica [A]  time = 0.303945, size = 333, normalized size = 0.96 \[ \frac{-\frac{5 \sqrt{2} \left (a^2 d^2-10 a b c d+9 b^2 c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{c^{3/4}}+\frac{5 \sqrt{2} \left (a^2 d^2-10 a b c d+9 b^2 c^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{c^{3/4}}-\frac{10 \sqrt{2} \left (a^2 d^2-10 a b c d+9 b^2 c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{c^{3/4}}+\frac{10 \sqrt{2} \left (a^2 d^2-10 a b c d+9 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{c^{3/4}}-\frac{40 \sqrt [4]{d} \sqrt{x} (b c-a d)^2}{c+d x^2}-320 b \sqrt [4]{d} \sqrt{x} (b c-a d)+32 b^2 d^{5/4} x^{5/2}}{80 d^{13/4}} \]

Antiderivative was successfully verified.

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

[Out]

(-320*b*d^(1/4)*(b*c - a*d)*Sqrt[x] + 32*b^2*d^(5/4)*x^(5/2) - (40*d^(1/4)*(b*c
- a*d)^2*Sqrt[x])/(c + d*x^2) - (10*Sqrt[2]*(9*b^2*c^2 - 10*a*b*c*d + a^2*d^2)*A
rcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/c^(3/4) + (10*Sqrt[2]*(9*b^2*c^2 -
10*a*b*c*d + a^2*d^2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/c^(3/4) -
(5*Sqrt[2]*(9*b^2*c^2 - 10*a*b*c*d + 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]*(9*b^2*c^2 - 10*a*b*c*d + a^2*d^2
)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/c^(3/4))/(80*d^(13
/4))

_______________________________________________________________________________________

Maple [A]  time = 0.023, size = 523, normalized size = 1.5 \[{\frac{2\,{b}^{2}}{5\,{d}^{2}}{x}^{{\frac{5}{2}}}}+4\,{\frac{ab\sqrt{x}}{{d}^{2}}}-4\,{\frac{{b}^{2}\sqrt{x}c}{{d}^{3}}}-{\frac{{a}^{2}}{2\,d \left ( d{x}^{2}+c \right ) }\sqrt{x}}+{\frac{abc}{{d}^{2} \left ( d{x}^{2}+c \right ) }\sqrt{x}}-{\frac{{b}^{2}{c}^{2}}{2\,{d}^{3} \left ( d{x}^{2}+c \right ) }\sqrt{x}}+{\frac{\sqrt{2}{a}^{2}}{8\,cd}\sqrt [4]{{\frac{c}{d}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}+1 \right ) }-{\frac{5\,\sqrt{2}ab}{4\,{d}^{2}}\sqrt [4]{{\frac{c}{d}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}+1 \right ) }+{\frac{9\,c\sqrt{2}{b}^{2}}{8\,{d}^{3}}\sqrt [4]{{\frac{c}{d}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}+1 \right ) }+{\frac{\sqrt{2}{a}^{2}}{8\,cd}\sqrt [4]{{\frac{c}{d}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}-1 \right ) }-{\frac{5\,\sqrt{2}ab}{4\,{d}^{2}}\sqrt [4]{{\frac{c}{d}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}-1 \right ) }+{\frac{9\,c\sqrt{2}{b}^{2}}{8\,{d}^{3}}\sqrt [4]{{\frac{c}{d}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}-1 \right ) }+{\frac{\sqrt{2}{a}^{2}}{16\,cd}\sqrt [4]{{\frac{c}{d}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{c}{d}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{c}{d}}} \right ) \left ( x-\sqrt [4]{{\frac{c}{d}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{c}{d}}} \right ) ^{-1}} \right ) }-{\frac{5\,\sqrt{2}ab}{8\,{d}^{2}}\sqrt [4]{{\frac{c}{d}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{c}{d}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{c}{d}}} \right ) \left ( x-\sqrt [4]{{\frac{c}{d}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{c}{d}}} \right ) ^{-1}} \right ) }+{\frac{9\,c\sqrt{2}{b}^{2}}{16\,{d}^{3}}\sqrt [4]{{\frac{c}{d}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{c}{d}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{c}{d}}} \right ) \left ( x-\sqrt [4]{{\frac{c}{d}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{c}{d}}} \right ) ^{-1}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/5*b^2*x^(5/2)/d^2+4/d^2*b*a*x^(1/2)-4/d^3*b^2*x^(1/2)*c-1/2/d*x^(1/2)/(d*x^2+c
)*a^2+1/d^2*x^(1/2)/(d*x^2+c)*c*a*b-1/2/d^3*x^(1/2)/(d*x^2+c)*b^2*c^2+1/8/d*(c/d
)^(1/4)/c*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)+1)*a^2-5/4/d^2*(c/d)^(1/4)*
2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)+1)*a*b+9/8/d^3*(c/d)^(1/4)*c*2^(1/2)*
arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)+1)*b^2+1/8/d*(c/d)^(1/4)/c*2^(1/2)*arctan(2^(
1/2)/(c/d)^(1/4)*x^(1/2)-1)*a^2-5/4/d^2*(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)
^(1/4)*x^(1/2)-1)*a*b+9/8/d^3*(c/d)^(1/4)*c*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x
^(1/2)-1)*b^2+1/16/d*(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^2-5/8/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)))*a*b+9/16/d^3*(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

_______________________________________________________________________________________

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

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.269553, size = 1482, normalized size = 4.28 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/40*(20*(d^4*x^2 + c*d^3)*(-(6561*b^8*c^8 - 29160*a*b^7*c^7*d + 51516*a^2*b^6*
c^6*d^2 - 45720*a^3*b^5*c^5*d^3 + 21286*a^4*b^4*c^4*d^4 - 5080*a^5*b^3*c^3*d^5 +
636*a^6*b^2*c^2*d^6 - 40*a^7*b*c*d^7 + a^8*d^8)/(c^3*d^13))^(1/4)*arctan(c*d^3*
(-(6561*b^8*c^8 - 29160*a*b^7*c^7*d + 51516*a^2*b^6*c^6*d^2 - 45720*a^3*b^5*c^5*
d^3 + 21286*a^4*b^4*c^4*d^4 - 5080*a^5*b^3*c^3*d^5 + 636*a^6*b^2*c^2*d^6 - 40*a^
7*b*c*d^7 + a^8*d^8)/(c^3*d^13))^(1/4)/((9*b^2*c^2 - 10*a*b*c*d + a^2*d^2)*sqrt(
x) + sqrt(c^2*d^6*sqrt(-(6561*b^8*c^8 - 29160*a*b^7*c^7*d + 51516*a^2*b^6*c^6*d^
2 - 45720*a^3*b^5*c^5*d^3 + 21286*a^4*b^4*c^4*d^4 - 5080*a^5*b^3*c^3*d^5 + 636*a
^6*b^2*c^2*d^6 - 40*a^7*b*c*d^7 + a^8*d^8)/(c^3*d^13)) + (81*b^4*c^4 - 180*a*b^3
*c^3*d + 118*a^2*b^2*c^2*d^2 - 20*a^3*b*c*d^3 + a^4*d^4)*x))) - 5*(d^4*x^2 + c*d
^3)*(-(6561*b^8*c^8 - 29160*a*b^7*c^7*d + 51516*a^2*b^6*c^6*d^2 - 45720*a^3*b^5*
c^5*d^3 + 21286*a^4*b^4*c^4*d^4 - 5080*a^5*b^3*c^3*d^5 + 636*a^6*b^2*c^2*d^6 - 4
0*a^7*b*c*d^7 + a^8*d^8)/(c^3*d^13))^(1/4)*log(c*d^3*(-(6561*b^8*c^8 - 29160*a*b
^7*c^7*d + 51516*a^2*b^6*c^6*d^2 - 45720*a^3*b^5*c^5*d^3 + 21286*a^4*b^4*c^4*d^4
- 5080*a^5*b^3*c^3*d^5 + 636*a^6*b^2*c^2*d^6 - 40*a^7*b*c*d^7 + a^8*d^8)/(c^3*d
^13))^(1/4) + (9*b^2*c^2 - 10*a*b*c*d + a^2*d^2)*sqrt(x)) + 5*(d^4*x^2 + c*d^3)*
(-(6561*b^8*c^8 - 29160*a*b^7*c^7*d + 51516*a^2*b^6*c^6*d^2 - 45720*a^3*b^5*c^5*
d^3 + 21286*a^4*b^4*c^4*d^4 - 5080*a^5*b^3*c^3*d^5 + 636*a^6*b^2*c^2*d^6 - 40*a^
7*b*c*d^7 + a^8*d^8)/(c^3*d^13))^(1/4)*log(-c*d^3*(-(6561*b^8*c^8 - 29160*a*b^7*
c^7*d + 51516*a^2*b^6*c^6*d^2 - 45720*a^3*b^5*c^5*d^3 + 21286*a^4*b^4*c^4*d^4 -
5080*a^5*b^3*c^3*d^5 + 636*a^6*b^2*c^2*d^6 - 40*a^7*b*c*d^7 + a^8*d^8)/(c^3*d^13
))^(1/4) + (9*b^2*c^2 - 10*a*b*c*d + a^2*d^2)*sqrt(x)) - 4*(4*b^2*d^2*x^4 - 45*b
^2*c^2 + 50*a*b*c*d - 5*a^2*d^2 - 4*(9*b^2*c*d - 10*a*b*d^2)*x^2)*sqrt(x))/(d^4*
x^2 + c*d^3)

_______________________________________________________________________________________

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

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.24736, size = 551, normalized size = 1.59 \[ \frac{\sqrt{2}{\left (9 \, \left (c d^{3}\right )^{\frac{1}{4}} b^{2} c^{2} - 10 \, \left (c d^{3}\right )^{\frac{1}{4}} a b c d + \left (c d^{3}\right )^{\frac{1}{4}} a^{2} d^{2}\right )} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{c}{d}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{c}{d}\right )^{\frac{1}{4}}}\right )}{8 \, c d^{4}} + \frac{\sqrt{2}{\left (9 \, \left (c d^{3}\right )^{\frac{1}{4}} b^{2} c^{2} - 10 \, \left (c d^{3}\right )^{\frac{1}{4}} a b c d + \left (c d^{3}\right )^{\frac{1}{4}} a^{2} d^{2}\right )} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{c}{d}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{c}{d}\right )^{\frac{1}{4}}}\right )}{8 \, c d^{4}} + \frac{\sqrt{2}{\left (9 \, \left (c d^{3}\right )^{\frac{1}{4}} b^{2} c^{2} - 10 \, \left (c d^{3}\right )^{\frac{1}{4}} a b c d + \left (c d^{3}\right )^{\frac{1}{4}} a^{2} d^{2}\right )}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{c}{d}\right )^{\frac{1}{4}} + x + \sqrt{\frac{c}{d}}\right )}{16 \, c d^{4}} - \frac{\sqrt{2}{\left (9 \, \left (c d^{3}\right )^{\frac{1}{4}} b^{2} c^{2} - 10 \, \left (c d^{3}\right )^{\frac{1}{4}} a b c d + \left (c d^{3}\right )^{\frac{1}{4}} a^{2} d^{2}\right )}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{c}{d}\right )^{\frac{1}{4}} + x + \sqrt{\frac{c}{d}}\right )}{16 \, c d^{4}} - \frac{b^{2} c^{2} \sqrt{x} - 2 \, a b c d \sqrt{x} + a^{2} d^{2} \sqrt{x}}{2 \,{\left (d x^{2} + c\right )} d^{3}} + \frac{2 \,{\left (b^{2} d^{8} x^{\frac{5}{2}} - 10 \, b^{2} c d^{7} \sqrt{x} + 10 \, a b d^{8} \sqrt{x}\right )}}{5 \, d^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*sqrt(2)*(9*(c*d^3)^(1/4)*b^2*c^2 - 10*(c*d^3)^(1/4)*a*b*c*d + (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^
4) + 1/8*sqrt(2)*(9*(c*d^3)^(1/4)*b^2*c^2 - 10*(c*d^3)^(1/4)*a*b*c*d + (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^4) + 1/16*sqrt(2)*(9*(c*d^3)^(1/4)*b^2*c^2 - 10*(c*d^3)^(1/4)*a*b*c*d + (c
*d^3)^(1/4)*a^2*d^2)*ln(sqrt(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt(c/d))/(c*d^4) - 1
/16*sqrt(2)*(9*(c*d^3)^(1/4)*b^2*c^2 - 10*(c*d^3)^(1/4)*a*b*c*d + (c*d^3)^(1/4)*
a^2*d^2)*ln(-sqrt(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt(c/d))/(c*d^4) - 1/2*(b^2*c^2
*sqrt(x) - 2*a*b*c*d*sqrt(x) + a^2*d^2*sqrt(x))/((d*x^2 + c)*d^3) + 2/5*(b^2*d^8
*x^(5/2) - 10*b^2*c*d^7*sqrt(x) + 10*a*b*d^8*sqrt(x))/d^10