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

Optimal. Leaf size=402 \[ \frac{\left (-3 a^2 d^2-10 a b c d+45 b^2 c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{7/4} d^{13/4}}-\frac{\left (-3 a^2 d^2-10 a b c d+45 b^2 c^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{7/4} d^{13/4}}+\frac{\left (-3 a^2 d^2-10 a b c d+45 b^2 c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{32 \sqrt{2} c^{7/4} d^{13/4}}-\frac{\left (-3 a^2 d^2-10 a b c d+45 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{32 \sqrt{2} c^{7/4} d^{13/4}}-\frac{\sqrt{x} \left (\frac{3 a^2 d}{c}+10 a b-\frac{45 b^2 c}{d}\right )}{16 c d^2}-\frac{x^{5/2} (b c-a d) (3 a d+13 b c)}{16 c^2 d^2 \left (c+d x^2\right )}+\frac{x^{5/2} (b c-a d)^2}{4 c d^2 \left (c+d x^2\right )^2} \]

[Out]

-((10*a*b - (45*b^2*c)/d + (3*a^2*d)/c)*Sqrt[x])/(16*c*d^2) + ((b*c - a*d)^2*x^(
5/2))/(4*c*d^2*(c + d*x^2)^2) - ((b*c - a*d)*(13*b*c + 3*a*d)*x^(5/2))/(16*c^2*d
^2*(c + d*x^2)) + ((45*b^2*c^2 - 10*a*b*c*d - 3*a^2*d^2)*ArcTan[1 - (Sqrt[2]*d^(
1/4)*Sqrt[x])/c^(1/4)])/(32*Sqrt[2]*c^(7/4)*d^(13/4)) - ((45*b^2*c^2 - 10*a*b*c*
d - 3*a^2*d^2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(32*Sqrt[2]*c^(7/4
)*d^(13/4)) + ((45*b^2*c^2 - 10*a*b*c*d - 3*a^2*d^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/
4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(64*Sqrt[2]*c^(7/4)*d^(13/4)) - ((45*b^2*c^2 -
10*a*b*c*d - 3*a^2*d^2)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*
x])/(64*Sqrt[2]*c^(7/4)*d^(13/4))

_______________________________________________________________________________________

Rubi [A]  time = 0.726052, antiderivative size = 402, 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 \[ \frac{\left (-3 a^2 d^2-10 a b c d+45 b^2 c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{7/4} d^{13/4}}-\frac{\left (-3 a^2 d^2-10 a b c d+45 b^2 c^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{7/4} d^{13/4}}+\frac{\left (-3 a^2 d^2-10 a b c d+45 b^2 c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{32 \sqrt{2} c^{7/4} d^{13/4}}-\frac{\left (-3 a^2 d^2-10 a b c d+45 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{32 \sqrt{2} c^{7/4} d^{13/4}}-\frac{\sqrt{x} \left (\frac{3 a^2 d}{c}+10 a b-\frac{45 b^2 c}{d}\right )}{16 c d^2}-\frac{x^{5/2} (b c-a d) (3 a d+13 b c)}{16 c^2 d^2 \left (c+d x^2\right )}+\frac{x^{5/2} (b c-a d)^2}{4 c d^2 \left (c+d x^2\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

-((10*a*b - (45*b^2*c)/d + (3*a^2*d)/c)*Sqrt[x])/(16*c*d^2) + ((b*c - a*d)^2*x^(
5/2))/(4*c*d^2*(c + d*x^2)^2) - ((b*c - a*d)*(13*b*c + 3*a*d)*x^(5/2))/(16*c^2*d
^2*(c + d*x^2)) + ((45*b^2*c^2 - 10*a*b*c*d - 3*a^2*d^2)*ArcTan[1 - (Sqrt[2]*d^(
1/4)*Sqrt[x])/c^(1/4)])/(32*Sqrt[2]*c^(7/4)*d^(13/4)) - ((45*b^2*c^2 - 10*a*b*c*
d - 3*a^2*d^2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(32*Sqrt[2]*c^(7/4
)*d^(13/4)) + ((45*b^2*c^2 - 10*a*b*c*d - 3*a^2*d^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/
4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(64*Sqrt[2]*c^(7/4)*d^(13/4)) - ((45*b^2*c^2 -
10*a*b*c*d - 3*a^2*d^2)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*
x])/(64*Sqrt[2]*c^(7/4)*d^(13/4))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 115.505, size = 389, normalized size = 0.97 \[ \frac{x^{\frac{5}{2}} \left (a d - b c\right )^{2}}{4 c d^{2} \left (c + d x^{2}\right )^{2}} + \frac{x^{\frac{5}{2}} \left (a d - b c\right ) \left (3 a d + 13 b c\right )}{16 c^{2} d^{2} \left (c + d x^{2}\right )} - \frac{\sqrt{x} \left (3 a^{2} d^{2} + 10 a b c d - 45 b^{2} c^{2}\right )}{16 c^{2} d^{3}} - \frac{\sqrt{2} \left (3 a^{2} d^{2} + 10 a b c d - 45 b^{2} c^{2}\right ) \log{\left (- \sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x} + \sqrt{c} + \sqrt{d} x \right )}}{128 c^{\frac{7}{4}} d^{\frac{13}{4}}} + \frac{\sqrt{2} \left (3 a^{2} d^{2} + 10 a b c d - 45 b^{2} c^{2}\right ) \log{\left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x} + \sqrt{c} + \sqrt{d} x \right )}}{128 c^{\frac{7}{4}} d^{\frac{13}{4}}} - \frac{\sqrt{2} \left (3 a^{2} d^{2} + 10 a b c d - 45 b^{2} c^{2}\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}} \right )}}{64 c^{\frac{7}{4}} d^{\frac{13}{4}}} + \frac{\sqrt{2} \left (3 a^{2} d^{2} + 10 a b c d - 45 b^{2} c^{2}\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}} \right )}}{64 c^{\frac{7}{4}} d^{\frac{13}{4}}} \]

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)**3,x)

[Out]

x**(5/2)*(a*d - b*c)**2/(4*c*d**2*(c + d*x**2)**2) + x**(5/2)*(a*d - b*c)*(3*a*d
+ 13*b*c)/(16*c**2*d**2*(c + d*x**2)) - sqrt(x)*(3*a**2*d**2 + 10*a*b*c*d - 45*
b**2*c**2)/(16*c**2*d**3) - sqrt(2)*(3*a**2*d**2 + 10*a*b*c*d - 45*b**2*c**2)*lo
g(-sqrt(2)*c**(1/4)*d**(1/4)*sqrt(x) + sqrt(c) + sqrt(d)*x)/(128*c**(7/4)*d**(13
/4)) + sqrt(2)*(3*a**2*d**2 + 10*a*b*c*d - 45*b**2*c**2)*log(sqrt(2)*c**(1/4)*d*
*(1/4)*sqrt(x) + sqrt(c) + sqrt(d)*x)/(128*c**(7/4)*d**(13/4)) - sqrt(2)*(3*a**2
*d**2 + 10*a*b*c*d - 45*b**2*c**2)*atan(1 - sqrt(2)*d**(1/4)*sqrt(x)/c**(1/4))/(
64*c**(7/4)*d**(13/4)) + sqrt(2)*(3*a**2*d**2 + 10*a*b*c*d - 45*b**2*c**2)*atan(
1 + sqrt(2)*d**(1/4)*sqrt(x)/c**(1/4))/(64*c**(7/4)*d**(13/4))

_______________________________________________________________________________________

Mathematica [A]  time = 0.792802, size = 361, normalized size = 0.9 \[ \frac{\frac{8 \sqrt [4]{d} \sqrt{x} \left (a^2 d^2-18 a b c d+17 b^2 c^2\right )}{c \left (c+d x^2\right )}+\frac{\sqrt{2} \left (-3 a^2 d^2-10 a b c d+45 b^2 c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{c^{7/4}}-\frac{\sqrt{2} \left (-3 a^2 d^2-10 a b c d+45 b^2 c^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{c^{7/4}}+\frac{2 \sqrt{2} \left (-3 a^2 d^2-10 a b c d+45 b^2 c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{c^{7/4}}-\frac{2 \sqrt{2} \left (-3 a^2 d^2-10 a b c d+45 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{c^{7/4}}-\frac{32 \sqrt [4]{d} \sqrt{x} (b c-a d)^2}{\left (c+d x^2\right )^2}+256 b^2 \sqrt [4]{d} \sqrt{x}}{128 d^{13/4}} \]

Antiderivative was successfully verified.

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

[Out]

(256*b^2*d^(1/4)*Sqrt[x] - (32*d^(1/4)*(b*c - a*d)^2*Sqrt[x])/(c + d*x^2)^2 + (8
*d^(1/4)*(17*b^2*c^2 - 18*a*b*c*d + a^2*d^2)*Sqrt[x])/(c*(c + d*x^2)) + (2*Sqrt[
2]*(45*b^2*c^2 - 10*a*b*c*d - 3*a^2*d^2)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^
(1/4)])/c^(7/4) - (2*Sqrt[2]*(45*b^2*c^2 - 10*a*b*c*d - 3*a^2*d^2)*ArcTan[1 + (S
qrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/c^(7/4) + (Sqrt[2]*(45*b^2*c^2 - 10*a*b*c*d -
3*a^2*d^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/c^(7/4) -
(Sqrt[2]*(45*b^2*c^2 - 10*a*b*c*d - 3*a^2*d^2)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^
(1/4)*Sqrt[x] + Sqrt[d]*x])/c^(7/4))/(128*d^(13/4))

_______________________________________________________________________________________

Maple [A]  time = 0.027, size = 568, normalized size = 1.4 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.270326, size = 1604, normalized size = 3.99 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/64*(4*(c*d^5*x^4 + 2*c^2*d^4*x^2 + c^3*d^3)*(-(4100625*b^8*c^8 - 3645000*a*b^
7*c^7*d + 121500*a^2*b^6*c^6*d^2 + 549000*a^3*b^5*c^5*d^3 - 42650*a^4*b^4*c^4*d^
4 - 36600*a^5*b^3*c^3*d^5 + 540*a^6*b^2*c^2*d^6 + 1080*a^7*b*c*d^7 + 81*a^8*d^8)
/(c^7*d^13))^(1/4)*arctan(-c^2*d^3*(-(4100625*b^8*c^8 - 3645000*a*b^7*c^7*d + 12
1500*a^2*b^6*c^6*d^2 + 549000*a^3*b^5*c^5*d^3 - 42650*a^4*b^4*c^4*d^4 - 36600*a^
5*b^3*c^3*d^5 + 540*a^6*b^2*c^2*d^6 + 1080*a^7*b*c*d^7 + 81*a^8*d^8)/(c^7*d^13))
^(1/4)/((45*b^2*c^2 - 10*a*b*c*d - 3*a^2*d^2)*sqrt(x) - sqrt(c^4*d^6*sqrt(-(4100
625*b^8*c^8 - 3645000*a*b^7*c^7*d + 121500*a^2*b^6*c^6*d^2 + 549000*a^3*b^5*c^5*
d^3 - 42650*a^4*b^4*c^4*d^4 - 36600*a^5*b^3*c^3*d^5 + 540*a^6*b^2*c^2*d^6 + 1080
*a^7*b*c*d^7 + 81*a^8*d^8)/(c^7*d^13)) + (2025*b^4*c^4 - 900*a*b^3*c^3*d - 170*a
^2*b^2*c^2*d^2 + 60*a^3*b*c*d^3 + 9*a^4*d^4)*x))) - (c*d^5*x^4 + 2*c^2*d^4*x^2 +
c^3*d^3)*(-(4100625*b^8*c^8 - 3645000*a*b^7*c^7*d + 121500*a^2*b^6*c^6*d^2 + 54
9000*a^3*b^5*c^5*d^3 - 42650*a^4*b^4*c^4*d^4 - 36600*a^5*b^3*c^3*d^5 + 540*a^6*b
^2*c^2*d^6 + 1080*a^7*b*c*d^7 + 81*a^8*d^8)/(c^7*d^13))^(1/4)*log(c^2*d^3*(-(410
0625*b^8*c^8 - 3645000*a*b^7*c^7*d + 121500*a^2*b^6*c^6*d^2 + 549000*a^3*b^5*c^5
*d^3 - 42650*a^4*b^4*c^4*d^4 - 36600*a^5*b^3*c^3*d^5 + 540*a^6*b^2*c^2*d^6 + 108
0*a^7*b*c*d^7 + 81*a^8*d^8)/(c^7*d^13))^(1/4) - (45*b^2*c^2 - 10*a*b*c*d - 3*a^2
*d^2)*sqrt(x)) + (c*d^5*x^4 + 2*c^2*d^4*x^2 + c^3*d^3)*(-(4100625*b^8*c^8 - 3645
000*a*b^7*c^7*d + 121500*a^2*b^6*c^6*d^2 + 549000*a^3*b^5*c^5*d^3 - 42650*a^4*b^
4*c^4*d^4 - 36600*a^5*b^3*c^3*d^5 + 540*a^6*b^2*c^2*d^6 + 1080*a^7*b*c*d^7 + 81*
a^8*d^8)/(c^7*d^13))^(1/4)*log(-c^2*d^3*(-(4100625*b^8*c^8 - 3645000*a*b^7*c^7*d
+ 121500*a^2*b^6*c^6*d^2 + 549000*a^3*b^5*c^5*d^3 - 42650*a^4*b^4*c^4*d^4 - 366
00*a^5*b^3*c^3*d^5 + 540*a^6*b^2*c^2*d^6 + 1080*a^7*b*c*d^7 + 81*a^8*d^8)/(c^7*d
^13))^(1/4) - (45*b^2*c^2 - 10*a*b*c*d - 3*a^2*d^2)*sqrt(x)) - 4*(32*b^2*c*d^2*x
^4 + 45*b^2*c^3 - 10*a*b*c^2*d - 3*a^2*c*d^2 + (81*b^2*c^2*d - 18*a*b*c*d^2 + a^
2*d^3)*x^2)*sqrt(x))/(c*d^5*x^4 + 2*c^2*d^4*x^2 + c^3*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)**3,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.257104, size = 575, normalized size = 1.43 \[ \frac{2 \, b^{2} \sqrt{x}}{d^{3}} - \frac{\sqrt{2}{\left (45 \, \left (c d^{3}\right )^{\frac{1}{4}} b^{2} c^{2} - 10 \, \left (c d^{3}\right )^{\frac{1}{4}} a b c d - 3 \, \left (c d^{3}\right )^{\frac{1}{4}} a^{2} d^{2}\right )} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{c}{d}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{c}{d}\right )^{\frac{1}{4}}}\right )}{64 \, c^{2} d^{4}} - \frac{\sqrt{2}{\left (45 \, \left (c d^{3}\right )^{\frac{1}{4}} b^{2} c^{2} - 10 \, \left (c d^{3}\right )^{\frac{1}{4}} a b c d - 3 \, \left (c d^{3}\right )^{\frac{1}{4}} a^{2} d^{2}\right )} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{c}{d}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{c}{d}\right )^{\frac{1}{4}}}\right )}{64 \, c^{2} d^{4}} - \frac{\sqrt{2}{\left (45 \, \left (c d^{3}\right )^{\frac{1}{4}} b^{2} c^{2} - 10 \, \left (c d^{3}\right )^{\frac{1}{4}} a b c d - 3 \, \left (c d^{3}\right )^{\frac{1}{4}} a^{2} d^{2}\right )}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{c}{d}\right )^{\frac{1}{4}} + x + \sqrt{\frac{c}{d}}\right )}{128 \, c^{2} d^{4}} + \frac{\sqrt{2}{\left (45 \, \left (c d^{3}\right )^{\frac{1}{4}} b^{2} c^{2} - 10 \, \left (c d^{3}\right )^{\frac{1}{4}} a b c d - 3 \, \left (c d^{3}\right )^{\frac{1}{4}} a^{2} d^{2}\right )}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{c}{d}\right )^{\frac{1}{4}} + x + \sqrt{\frac{c}{d}}\right )}{128 \, c^{2} d^{4}} + \frac{17 \, b^{2} c^{2} d x^{\frac{5}{2}} - 18 \, a b c d^{2} x^{\frac{5}{2}} + a^{2} d^{3} x^{\frac{5}{2}} + 13 \, b^{2} c^{3} \sqrt{x} - 10 \, a b c^{2} d \sqrt{x} - 3 \, a^{2} c d^{2} \sqrt{x}}{16 \,{\left (d x^{2} + c\right )}^{2} c d^{3}} \]

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, algorithm="giac")

[Out]

2*b^2*sqrt(x)/d^3 - 1/64*sqrt(2)*(45*(c*d^3)^(1/4)*b^2*c^2 - 10*(c*d^3)^(1/4)*a*
b*c*d - 3*(c*d^3)^(1/4)*a^2*d^2)*arctan(1/2*sqrt(2)*(sqrt(2)*(c/d)^(1/4) + 2*sqr
t(x))/(c/d)^(1/4))/(c^2*d^4) - 1/64*sqrt(2)*(45*(c*d^3)^(1/4)*b^2*c^2 - 10*(c*d^
3)^(1/4)*a*b*c*d - 3*(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^2*d^4) - 1/128*sqrt(2)*(45*(c*d^3)^(1/4)*b^2*c
^2 - 10*(c*d^3)^(1/4)*a*b*c*d - 3*(c*d^3)^(1/4)*a^2*d^2)*ln(sqrt(2)*sqrt(x)*(c/d
)^(1/4) + x + sqrt(c/d))/(c^2*d^4) + 1/128*sqrt(2)*(45*(c*d^3)^(1/4)*b^2*c^2 - 1
0*(c*d^3)^(1/4)*a*b*c*d - 3*(c*d^3)^(1/4)*a^2*d^2)*ln(-sqrt(2)*sqrt(x)*(c/d)^(1/
4) + x + sqrt(c/d))/(c^2*d^4) + 1/16*(17*b^2*c^2*d*x^(5/2) - 18*a*b*c*d^2*x^(5/2
) + a^2*d^3*x^(5/2) + 13*b^2*c^3*sqrt(x) - 10*a*b*c^2*d*sqrt(x) - 3*a^2*c*d^2*sq
rt(x))/((d*x^2 + c)^2*c*d^3)