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

Optimal. Leaf size=364 \[ \frac{\left (5 a^2 d^2+6 a b c d+21 b^2 c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{9/4} d^{11/4}}-\frac{\left (5 a^2 d^2+6 a b c d+21 b^2 c^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{9/4} d^{11/4}}-\frac{\left (5 a^2 d^2+6 a b c d+21 b^2 c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{32 \sqrt{2} c^{9/4} d^{11/4}}+\frac{\left (5 a^2 d^2+6 a b c d+21 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{32 \sqrt{2} c^{9/4} d^{11/4}}-\frac{x^{3/2} (5 a d+11 b c) (b c-a d)}{16 c^2 d^2 \left (c+d x^2\right )}+\frac{x^{3/2} (b c-a d)^2}{4 c d^2 \left (c+d x^2\right )^2} \]

[Out]

((b*c - a*d)^2*x^(3/2))/(4*c*d^2*(c + d*x^2)^2) - ((b*c - a*d)*(11*b*c + 5*a*d)*
x^(3/2))/(16*c^2*d^2*(c + d*x^2)) - ((21*b^2*c^2 + 6*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^(9/4)*d^(11/4)) + ((21*b^
2*c^2 + 6*a*b*c*d + 5*a^2*d^2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(3
2*Sqrt[2]*c^(9/4)*d^(11/4)) + ((21*b^2*c^2 + 6*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^(9/4)*d^(11/4)) -
((21*b^2*c^2 + 6*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^(9/4)*d^(11/4))

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Rubi [A]  time = 0.644137, antiderivative size = 364, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375 \[ \frac{\left (5 a^2 d^2+6 a b c d+21 b^2 c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{9/4} d^{11/4}}-\frac{\left (5 a^2 d^2+6 a b c d+21 b^2 c^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{9/4} d^{11/4}}-\frac{\left (5 a^2 d^2+6 a b c d+21 b^2 c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{32 \sqrt{2} c^{9/4} d^{11/4}}+\frac{\left (5 a^2 d^2+6 a b c d+21 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{32 \sqrt{2} c^{9/4} d^{11/4}}-\frac{x^{3/2} (5 a d+11 b c) (b c-a d)}{16 c^2 d^2 \left (c+d x^2\right )}+\frac{x^{3/2} (b c-a d)^2}{4 c d^2 \left (c+d x^2\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

((b*c - a*d)^2*x^(3/2))/(4*c*d^2*(c + d*x^2)^2) - ((b*c - a*d)*(11*b*c + 5*a*d)*
x^(3/2))/(16*c^2*d^2*(c + d*x^2)) - ((21*b^2*c^2 + 6*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^(9/4)*d^(11/4)) + ((21*b^
2*c^2 + 6*a*b*c*d + 5*a^2*d^2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(3
2*Sqrt[2]*c^(9/4)*d^(11/4)) + ((21*b^2*c^2 + 6*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^(9/4)*d^(11/4)) -
((21*b^2*c^2 + 6*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^(9/4)*d^(11/4))

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Rubi in Sympy [A]  time = 107.693, size = 350, normalized size = 0.96 \[ \frac{x^{\frac{3}{2}} \left (a d - b c\right )^{2}}{4 c d^{2} \left (c + d x^{2}\right )^{2}} + \frac{x^{\frac{3}{2}} \left (a d - b c\right ) \left (5 a d + 11 b c\right )}{16 c^{2} d^{2} \left (c + d x^{2}\right )} + \frac{\sqrt{2} \left (5 a^{2} d^{2} + 6 a b c d + 21 b^{2} c^{2}\right ) \log{\left (- \sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x} + \sqrt{c} + \sqrt{d} x \right )}}{128 c^{\frac{9}{4}} d^{\frac{11}{4}}} - \frac{\sqrt{2} \left (5 a^{2} d^{2} + 6 a b c d + 21 b^{2} c^{2}\right ) \log{\left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x} + \sqrt{c} + \sqrt{d} x \right )}}{128 c^{\frac{9}{4}} d^{\frac{11}{4}}} - \frac{\sqrt{2} \left (5 a^{2} d^{2} + 6 a b c d + 21 b^{2} c^{2}\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}} \right )}}{64 c^{\frac{9}{4}} d^{\frac{11}{4}}} + \frac{\sqrt{2} \left (5 a^{2} d^{2} + 6 a b c d + 21 b^{2} c^{2}\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}} \right )}}{64 c^{\frac{9}{4}} d^{\frac{11}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**2*x**(1/2)/(d*x**2+c)**3,x)

[Out]

x**(3/2)*(a*d - b*c)**2/(4*c*d**2*(c + d*x**2)**2) + x**(3/2)*(a*d - b*c)*(5*a*d
+ 11*b*c)/(16*c**2*d**2*(c + d*x**2)) + sqrt(2)*(5*a**2*d**2 + 6*a*b*c*d + 21*b
**2*c**2)*log(-sqrt(2)*c**(1/4)*d**(1/4)*sqrt(x) + sqrt(c) + sqrt(d)*x)/(128*c**
(9/4)*d**(11/4)) - sqrt(2)*(5*a**2*d**2 + 6*a*b*c*d + 21*b**2*c**2)*log(sqrt(2)*
c**(1/4)*d**(1/4)*sqrt(x) + sqrt(c) + sqrt(d)*x)/(128*c**(9/4)*d**(11/4)) - sqrt
(2)*(5*a**2*d**2 + 6*a*b*c*d + 21*b**2*c**2)*atan(1 - sqrt(2)*d**(1/4)*sqrt(x)/c
**(1/4))/(64*c**(9/4)*d**(11/4)) + sqrt(2)*(5*a**2*d**2 + 6*a*b*c*d + 21*b**2*c*
*2)*atan(1 + sqrt(2)*d**(1/4)*sqrt(x)/c**(1/4))/(64*c**(9/4)*d**(11/4))

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Mathematica [A]  time = 0.335137, size = 339, normalized size = 0.93 \[ \frac{\sqrt{2} \left (5 a^2 d^2+6 a b c d+21 b^2 c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )-\sqrt{2} \left (5 a^2 d^2+6 a b c d+21 b^2 c^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )-2 \sqrt{2} \left (5 a^2 d^2+6 a b c d+21 b^2 c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )+2 \sqrt{2} \left (5 a^2 d^2+6 a b c d+21 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )-\frac{8 \sqrt [4]{c} d^{3/4} x^{3/2} \left (-5 a^2 d^2-6 a b c d+11 b^2 c^2\right )}{c+d x^2}+\frac{32 c^{5/4} d^{3/4} x^{3/2} (b c-a d)^2}{\left (c+d x^2\right )^2}}{128 c^{9/4} d^{11/4}} \]

Antiderivative was successfully verified.

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

[Out]

((32*c^(5/4)*d^(3/4)*(b*c - a*d)^2*x^(3/2))/(c + d*x^2)^2 - (8*c^(1/4)*d^(3/4)*(
11*b^2*c^2 - 6*a*b*c*d - 5*a^2*d^2)*x^(3/2))/(c + d*x^2) - 2*Sqrt[2]*(21*b^2*c^2
+ 6*a*b*c*d + 5*a^2*d^2)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)] + 2*Sqrt
[2]*(21*b^2*c^2 + 6*a*b*c*d + 5*a^2*d^2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^
(1/4)] + Sqrt[2]*(21*b^2*c^2 + 6*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] - Sqrt[2]*(21*b^2*c^2 + 6*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])/(128*c^(9/4)*d^(11/4
))

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Maple [A]  time = 0.026, size = 514, 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^(1/2)/(d*x^2+c)^3,x)

[Out]

2*(1/32*(5*a^2*d^2+6*a*b*c*d-11*b^2*c^2)/c^2/d*x^(7/2)+1/32*(9*a^2*d^2-2*a*b*c*d
-7*b^2*c^2)/c/d^2*x^(3/2))/(d*x^2+c)^2+5/64/d/c^2/(c/d)^(1/4)*2^(1/2)*arctan(2^(
1/2)/(c/d)^(1/4)*x^(1/2)+1)*a^2+3/32/d^2/c/(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c
/d)^(1/4)*x^(1/2)+1)*a*b+21/64/d^3/(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4
)*x^(1/2)+1)*b^2+5/64/d/c^2/(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/
2)-1)*a^2+3/32/d^2/c/(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1)*a
*b+21/64/d^3/(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1)*b^2+5/128
/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+3/64/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+21/128/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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.276313, size = 2126, normalized size = 5.84 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/64*(4*(c^2*d^4*x^4 + 2*c^3*d^3*x^2 + c^4*d^2)*(-(194481*b^8*c^8 + 222264*a*b^7
*c^7*d + 280476*a^2*b^6*c^6*d^2 + 176904*a^3*b^5*c^5*d^3 + 112806*a^4*b^4*c^4*d^
4 + 42120*a^5*b^3*c^3*d^5 + 15900*a^6*b^2*c^2*d^6 + 3000*a^7*b*c*d^7 + 625*a^8*d
^8)/(c^9*d^11))^(1/4)*arctan(c^7*d^8*(-(194481*b^8*c^8 + 222264*a*b^7*c^7*d + 28
0476*a^2*b^6*c^6*d^2 + 176904*a^3*b^5*c^5*d^3 + 112806*a^4*b^4*c^4*d^4 + 42120*a
^5*b^3*c^3*d^5 + 15900*a^6*b^2*c^2*d^6 + 3000*a^7*b*c*d^7 + 625*a^8*d^8)/(c^9*d^
11))^(3/4)/((9261*b^6*c^6 + 7938*a*b^5*c^5*d + 8883*a^2*b^4*c^4*d^2 + 3996*a^3*b
^3*c^3*d^3 + 2115*a^4*b^2*c^2*d^4 + 450*a^5*b*c*d^5 + 125*a^6*d^6)*sqrt(x) + sqr
t((85766121*b^12*c^12 + 147027636*a*b^11*c^11*d + 227542770*a^2*b^10*c^10*d^2 +
215040420*a^3*b^9*c^9*d^3 + 181522215*a^4*b^8*c^8*d^4 + 112905576*a^5*b^7*c^7*d^
5 + 63002556*a^6*b^6*c^6*d^6 + 26882280*a^7*b^5*c^5*d^7 + 10290375*a^8*b^4*c^4*d
^8 + 2902500*a^9*b^3*c^3*d^9 + 731250*a^10*b^2*c^2*d^10 + 112500*a^11*b*c*d^11 +
15625*a^12*d^12)*x - (194481*b^8*c^13*d^5 + 222264*a*b^7*c^12*d^6 + 280476*a^2*
b^6*c^11*d^7 + 176904*a^3*b^5*c^10*d^8 + 112806*a^4*b^4*c^9*d^9 + 42120*a^5*b^3*
c^8*d^10 + 15900*a^6*b^2*c^7*d^11 + 3000*a^7*b*c^6*d^12 + 625*a^8*c^5*d^13)*sqrt
(-(194481*b^8*c^8 + 222264*a*b^7*c^7*d + 280476*a^2*b^6*c^6*d^2 + 176904*a^3*b^5
*c^5*d^3 + 112806*a^4*b^4*c^4*d^4 + 42120*a^5*b^3*c^3*d^5 + 15900*a^6*b^2*c^2*d^
6 + 3000*a^7*b*c*d^7 + 625*a^8*d^8)/(c^9*d^11))))) + (c^2*d^4*x^4 + 2*c^3*d^3*x^
2 + c^4*d^2)*(-(194481*b^8*c^8 + 222264*a*b^7*c^7*d + 280476*a^2*b^6*c^6*d^2 + 1
76904*a^3*b^5*c^5*d^3 + 112806*a^4*b^4*c^4*d^4 + 42120*a^5*b^3*c^3*d^5 + 15900*a
^6*b^2*c^2*d^6 + 3000*a^7*b*c*d^7 + 625*a^8*d^8)/(c^9*d^11))^(1/4)*log(c^7*d^8*(
-(194481*b^8*c^8 + 222264*a*b^7*c^7*d + 280476*a^2*b^6*c^6*d^2 + 176904*a^3*b^5*
c^5*d^3 + 112806*a^4*b^4*c^4*d^4 + 42120*a^5*b^3*c^3*d^5 + 15900*a^6*b^2*c^2*d^6
+ 3000*a^7*b*c*d^7 + 625*a^8*d^8)/(c^9*d^11))^(3/4) + (9261*b^6*c^6 + 7938*a*b^
5*c^5*d + 8883*a^2*b^4*c^4*d^2 + 3996*a^3*b^3*c^3*d^3 + 2115*a^4*b^2*c^2*d^4 + 4
50*a^5*b*c*d^5 + 125*a^6*d^6)*sqrt(x)) - (c^2*d^4*x^4 + 2*c^3*d^3*x^2 + c^4*d^2)
*(-(194481*b^8*c^8 + 222264*a*b^7*c^7*d + 280476*a^2*b^6*c^6*d^2 + 176904*a^3*b^
5*c^5*d^3 + 112806*a^4*b^4*c^4*d^4 + 42120*a^5*b^3*c^3*d^5 + 15900*a^6*b^2*c^2*d
^6 + 3000*a^7*b*c*d^7 + 625*a^8*d^8)/(c^9*d^11))^(1/4)*log(-c^7*d^8*(-(194481*b^
8*c^8 + 222264*a*b^7*c^7*d + 280476*a^2*b^6*c^6*d^2 + 176904*a^3*b^5*c^5*d^3 + 1
12806*a^4*b^4*c^4*d^4 + 42120*a^5*b^3*c^3*d^5 + 15900*a^6*b^2*c^2*d^6 + 3000*a^7
*b*c*d^7 + 625*a^8*d^8)/(c^9*d^11))^(3/4) + (9261*b^6*c^6 + 7938*a*b^5*c^5*d + 8
883*a^2*b^4*c^4*d^2 + 3996*a^3*b^3*c^3*d^3 + 2115*a^4*b^2*c^2*d^4 + 450*a^5*b*c*
d^5 + 125*a^6*d^6)*sqrt(x)) - 4*((11*b^2*c^2*d - 6*a*b*c*d^2 - 5*a^2*d^3)*x^3 +
(7*b^2*c^3 + 2*a*b*c^2*d - 9*a^2*c*d^2)*x)*sqrt(x))/(c^2*d^4*x^4 + 2*c^3*d^3*x^2
+ c^4*d^2)

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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**(1/2)/(d*x**2+c)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.283365, size = 562, normalized size = 1.54 \[ -\frac{11 \, b^{2} c^{2} d x^{\frac{7}{2}} - 6 \, a b c d^{2} x^{\frac{7}{2}} - 5 \, a^{2} d^{3} x^{\frac{7}{2}} + 7 \, b^{2} c^{3} x^{\frac{3}{2}} + 2 \, a b c^{2} d x^{\frac{3}{2}} - 9 \, a^{2} c d^{2} x^{\frac{3}{2}}}{16 \,{\left (d x^{2} + c\right )}^{2} c^{2} d^{2}} + \frac{\sqrt{2}{\left (21 \, \left (c d^{3}\right )^{\frac{3}{4}} b^{2} c^{2} + 6 \, \left (c d^{3}\right )^{\frac{3}{4}} a b c d + 5 \, \left (c d^{3}\right )^{\frac{3}{4}} a^{2} d^{2}\right )} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{c}{d}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{c}{d}\right )^{\frac{1}{4}}}\right )}{64 \, c^{3} d^{5}} + \frac{\sqrt{2}{\left (21 \, \left (c d^{3}\right )^{\frac{3}{4}} b^{2} c^{2} + 6 \, \left (c d^{3}\right )^{\frac{3}{4}} a b c d + 5 \, \left (c d^{3}\right )^{\frac{3}{4}} a^{2} d^{2}\right )} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{c}{d}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{c}{d}\right )^{\frac{1}{4}}}\right )}{64 \, c^{3} d^{5}} - \frac{\sqrt{2}{\left (21 \, \left (c d^{3}\right )^{\frac{3}{4}} b^{2} c^{2} + 6 \, \left (c d^{3}\right )^{\frac{3}{4}} a b c d + 5 \, \left (c d^{3}\right )^{\frac{3}{4}} a^{2} d^{2}\right )}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{c}{d}\right )^{\frac{1}{4}} + x + \sqrt{\frac{c}{d}}\right )}{128 \, c^{3} d^{5}} + \frac{\sqrt{2}{\left (21 \, \left (c d^{3}\right )^{\frac{3}{4}} b^{2} c^{2} + 6 \, \left (c d^{3}\right )^{\frac{3}{4}} a b c d + 5 \, \left (c d^{3}\right )^{\frac{3}{4}} a^{2} d^{2}\right )}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{c}{d}\right )^{\frac{1}{4}} + x + \sqrt{\frac{c}{d}}\right )}{128 \, c^{3} d^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/16*(11*b^2*c^2*d*x^(7/2) - 6*a*b*c*d^2*x^(7/2) - 5*a^2*d^3*x^(7/2) + 7*b^2*c^
3*x^(3/2) + 2*a*b*c^2*d*x^(3/2) - 9*a^2*c*d^2*x^(3/2))/((d*x^2 + c)^2*c^2*d^2) +
1/64*sqrt(2)*(21*(c*d^3)^(3/4)*b^2*c^2 + 6*(c*d^3)^(3/4)*a*b*c*d + 5*(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^3*d^5) + 1/64*sqrt(2)*(21*(c*d^3)^(3/4)*b^2*c^2 + 6*(c*d^3)^(3/4)*a*b*c*d + 5*
(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^3*d^5) - 1/128*sqrt(2)*(21*(c*d^3)^(3/4)*b^2*c^2 + 6*(c*d^3)^(3/4)*
a*b*c*d + 5*(c*d^3)^(3/4)*a^2*d^2)*ln(sqrt(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt(c/d
))/(c^3*d^5) + 1/128*sqrt(2)*(21*(c*d^3)^(3/4)*b^2*c^2 + 6*(c*d^3)^(3/4)*a*b*c*d
+ 5*(c*d^3)^(3/4)*a^2*d^2)*ln(-sqrt(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt(c/d))/(c^
3*d^5)