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

Optimal. Leaf size=152 \[ -\frac{\sqrt{d x-c} \left (2 a d^2+3 b c^2\right )}{2 d^5 \sqrt{c+d x}}-\frac{c \left (2 a d^2+3 b c^2\right )}{2 d^5 \sqrt{d x-c} \sqrt{c+d x}}+\frac{\left (2 a d^2+3 b c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d x-c}}{\sqrt{c+d x}}\right )}{d^5}+\frac{b x^3}{2 d^2 \sqrt{d x-c} \sqrt{c+d x}} \]

[Out]

-(c*(3*b*c^2 + 2*a*d^2))/(2*d^5*Sqrt[-c + d*x]*Sqrt[c + d*x]) + (b*x^3)/(2*d^2*S
qrt[-c + d*x]*Sqrt[c + d*x]) - ((3*b*c^2 + 2*a*d^2)*Sqrt[-c + d*x])/(2*d^5*Sqrt[
c + d*x]) + ((3*b*c^2 + 2*a*d^2)*ArcTanh[Sqrt[-c + d*x]/Sqrt[c + d*x]])/d^5

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Rubi [A]  time = 0.380678, antiderivative size = 152, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.194 \[ -\frac{\sqrt{d x-c} \left (2 a d^2+3 b c^2\right )}{2 d^5 \sqrt{c+d x}}-\frac{c \left (2 a d^2+3 b c^2\right )}{2 d^5 \sqrt{d x-c} \sqrt{c+d x}}+\frac{\left (2 a d^2+3 b c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d x-c}}{\sqrt{c+d x}}\right )}{d^5}+\frac{b x^3}{2 d^2 \sqrt{d x-c} \sqrt{c+d x}} \]

Antiderivative was successfully verified.

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

[Out]

-(c*(3*b*c^2 + 2*a*d^2))/(2*d^5*Sqrt[-c + d*x]*Sqrt[c + d*x]) + (b*x^3)/(2*d^2*S
qrt[-c + d*x]*Sqrt[c + d*x]) - ((3*b*c^2 + 2*a*d^2)*Sqrt[-c + d*x])/(2*d^5*Sqrt[
c + d*x]) + ((3*b*c^2 + 2*a*d^2)*ArcTanh[Sqrt[-c + d*x]/Sqrt[c + d*x]])/d^5

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Rubi in Sympy [A]  time = 24.6745, size = 134, normalized size = 0.88 \[ \frac{b x^{3}}{2 d^{2} \sqrt{- c + d x} \sqrt{c + d x}} - \frac{c \left (2 a d^{2} + 3 b c^{2}\right )}{2 d^{5} \sqrt{- c + d x} \sqrt{c + d x}} - \frac{\sqrt{- c + d x} \left (2 a d^{2} + 3 b c^{2}\right )}{2 d^{5} \sqrt{c + d x}} + \frac{\left (2 a d^{2} + 3 b c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{- c + d x}}{\sqrt{c + d x}} \right )}}{d^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b*x**3/(2*d**2*sqrt(-c + d*x)*sqrt(c + d*x)) - c*(2*a*d**2 + 3*b*c**2)/(2*d**5*s
qrt(-c + d*x)*sqrt(c + d*x)) - sqrt(-c + d*x)*(2*a*d**2 + 3*b*c**2)/(2*d**5*sqrt
(c + d*x)) + (2*a*d**2 + 3*b*c**2)*atanh(sqrt(-c + d*x)/sqrt(c + d*x))/d**5

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Mathematica [A]  time = 0.16081, size = 121, normalized size = 0.8 \[ \frac{d x \sqrt{d x-c} \sqrt{c+d x} \left (-2 a d^2-3 b c^2+b d^2 x^2\right )-\left (c^2-d^2 x^2\right ) \left (2 a d^2+3 b c^2\right ) \log \left (\sqrt{d x-c} \sqrt{c+d x}+d x\right )}{2 d^5 (d x-c) (c+d x)} \]

Antiderivative was successfully verified.

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

[Out]

(d*x*Sqrt[-c + d*x]*Sqrt[c + d*x]*(-3*b*c^2 - 2*a*d^2 + b*d^2*x^2) - (3*b*c^2 +
2*a*d^2)*(c^2 - d^2*x^2)*Log[d*x + Sqrt[-c + d*x]*Sqrt[c + d*x]])/(2*d^5*(-c + d
*x)*(c + d*x))

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Maple [C]  time = 0.033, size = 254, normalized size = 1.7 \[{\frac{{\it csgn} \left ( d \right ) }{2\,{d}^{5}} \left ({\it csgn} \left ( d \right ){x}^{3}b{d}^{3}\sqrt{{d}^{2}{x}^{2}-{c}^{2}}+2\,\ln \left ( \left ({\it csgn} \left ( d \right ) \sqrt{{d}^{2}{x}^{2}-{c}^{2}}+dx \right ){\it csgn} \left ( d \right ) \right ){x}^{2}a{d}^{4}+3\,\ln \left ( \left ({\it csgn} \left ( d \right ) \sqrt{{d}^{2}{x}^{2}-{c}^{2}}+dx \right ){\it csgn} \left ( d \right ) \right ){x}^{2}b{c}^{2}{d}^{2}-2\,ax\sqrt{{d}^{2}{x}^{2}-{c}^{2}}{d}^{3}{\it csgn} \left ( d \right ) -3\,b{c}^{2}x\sqrt{{d}^{2}{x}^{2}-{c}^{2}}{\it csgn} \left ( d \right ) d-2\,a{c}^{2}\ln \left ( \left ({\it csgn} \left ( d \right ) \sqrt{{d}^{2}{x}^{2}-{c}^{2}}+dx \right ){\it csgn} \left ( d \right ) \right ){d}^{2}-3\,b{c}^{4}\ln \left ( \left ({\it csgn} \left ( d \right ) \sqrt{{d}^{2}{x}^{2}-{c}^{2}}+dx \right ){\it csgn} \left ( d \right ) \right ) \right ){\frac{1}{\sqrt{{d}^{2}{x}^{2}-{c}^{2}}}}{\frac{1}{\sqrt{dx+c}}}{\frac{1}{\sqrt{dx-c}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.41408, size = 211, normalized size = 1.39 \[ \frac{b x^{3}}{2 \, \sqrt{d^{2} x^{2} - c^{2}} d^{2}} - \frac{3 \, b c^{2} x}{2 \, \sqrt{d^{2} x^{2} - c^{2}} d^{4}} - \frac{a x}{\sqrt{d^{2} x^{2} - c^{2}} d^{2}} + \frac{3 \, b c^{2} \log \left (2 \, d^{2} x + 2 \, \sqrt{d^{2} x^{2} - c^{2}} \sqrt{d^{2}}\right )}{2 \, \sqrt{d^{2}} d^{4}} + \frac{a \log \left (2 \, d^{2} x + 2 \, \sqrt{d^{2} x^{2} - c^{2}} \sqrt{d^{2}}\right )}{\sqrt{d^{2}} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*b*x^3/(sqrt(d^2*x^2 - c^2)*d^2) - 3/2*b*c^2*x/(sqrt(d^2*x^2 - c^2)*d^4) - a*
x/(sqrt(d^2*x^2 - c^2)*d^2) + 3/2*b*c^2*log(2*d^2*x + 2*sqrt(d^2*x^2 - c^2)*sqrt
(d^2))/(sqrt(d^2)*d^4) + a*log(2*d^2*x + 2*sqrt(d^2*x^2 - c^2)*sqrt(d^2))/(sqrt(
d^2)*d^2)

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Fricas [A]  time = 0.246244, size = 448, normalized size = 2.95 \[ -\frac{4 \, b d^{6} x^{6} - 7 \, b c^{2} d^{4} x^{4} + 2 \, b c^{6} + 2 \, a c^{4} d^{2} -{\left (b c^{4} d^{2} + 4 \, a c^{2} d^{4}\right )} x^{2} -{\left (4 \, b d^{5} x^{5} - 5 \, b c^{2} d^{3} x^{3} -{\left (3 \, b c^{4} d + 4 \, a c^{2} d^{3}\right )} x\right )} \sqrt{d x + c} \sqrt{d x - c} +{\left (3 \, b c^{6} + 2 \, a c^{4} d^{2} + 4 \,{\left (3 \, b c^{2} d^{4} + 2 \, a d^{6}\right )} x^{4} - 5 \,{\left (3 \, b c^{4} d^{2} + 2 \, a c^{2} d^{4}\right )} x^{2} -{\left (4 \,{\left (3 \, b c^{2} d^{3} + 2 \, a d^{5}\right )} x^{3} - 3 \,{\left (3 \, b c^{4} d + 2 \, a c^{2} d^{3}\right )} x\right )} \sqrt{d x + c} \sqrt{d x - c}\right )} \log \left (-d x + \sqrt{d x + c} \sqrt{d x - c}\right )}{2 \,{\left (4 \, d^{9} x^{4} - 5 \, c^{2} d^{7} x^{2} + c^{4} d^{5} -{\left (4 \, d^{8} x^{3} - 3 \, c^{2} d^{6} x\right )} \sqrt{d x + c} \sqrt{d x - c}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(4*b*d^6*x^6 - 7*b*c^2*d^4*x^4 + 2*b*c^6 + 2*a*c^4*d^2 - (b*c^4*d^2 + 4*a*c
^2*d^4)*x^2 - (4*b*d^5*x^5 - 5*b*c^2*d^3*x^3 - (3*b*c^4*d + 4*a*c^2*d^3)*x)*sqrt
(d*x + c)*sqrt(d*x - c) + (3*b*c^6 + 2*a*c^4*d^2 + 4*(3*b*c^2*d^4 + 2*a*d^6)*x^4
- 5*(3*b*c^4*d^2 + 2*a*c^2*d^4)*x^2 - (4*(3*b*c^2*d^3 + 2*a*d^5)*x^3 - 3*(3*b*c
^4*d + 2*a*c^2*d^3)*x)*sqrt(d*x + c)*sqrt(d*x - c))*log(-d*x + sqrt(d*x + c)*sqr
t(d*x - c)))/(4*d^9*x^4 - 5*c^2*d^7*x^2 + c^4*d^5 - (4*d^8*x^3 - 3*c^2*d^6*x)*sq
rt(d*x + c)*sqrt(d*x - c))

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Sympy [A]  time = 80.9724, size = 212, normalized size = 1.39 \[ a \left (\frac{{G_{6, 6}^{6, 2}\left (\begin{matrix} - \frac{1}{4}, \frac{1}{4} & - \frac{1}{2}, \frac{1}{2}, 1, 1 \\- \frac{1}{4}, 0, \frac{1}{4}, \frac{1}{2}, 1, 0 & \end{matrix} \middle |{\frac{c^{2}}{d^{2} x^{2}}} \right )}}{2 \pi ^{\frac{3}{2}} d^{3}} + \frac{i{G_{6, 6}^{2, 6}\left (\begin{matrix} - \frac{3}{2}, -1, - \frac{3}{4}, - \frac{1}{2}, - \frac{1}{4}, 1 & \\- \frac{3}{4}, - \frac{1}{4} & - \frac{3}{2}, -1, 0, 0 \end{matrix} \middle |{\frac{c^{2} e^{2 i \pi }}{d^{2} x^{2}}} \right )}}{2 \pi ^{\frac{3}{2}} d^{3}}\right ) + b \left (\frac{c^{2}{G_{6, 6}^{6, 2}\left (\begin{matrix} - \frac{5}{4}, - \frac{3}{4} & - \frac{3}{2}, - \frac{1}{2}, 0, 1 \\- \frac{5}{4}, -1, - \frac{3}{4}, - \frac{1}{2}, 0, 0 & \end{matrix} \middle |{\frac{c^{2}}{d^{2} x^{2}}} \right )}}{2 \pi ^{\frac{3}{2}} d^{5}} + \frac{i c^{2}{G_{6, 6}^{2, 6}\left (\begin{matrix} - \frac{5}{2}, -2, - \frac{7}{4}, - \frac{3}{2}, - \frac{5}{4}, 1 & \\- \frac{7}{4}, - \frac{5}{4} & - \frac{5}{2}, -2, -1, 0 \end{matrix} \middle |{\frac{c^{2} e^{2 i \pi }}{d^{2} x^{2}}} \right )}}{2 \pi ^{\frac{3}{2}} d^{5}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*(meijerg(((-1/4, 1/4), (-1/2, 1/2, 1, 1)), ((-1/4, 0, 1/4, 1/2, 1, 0), ()), c*
*2/(d**2*x**2))/(2*pi**(3/2)*d**3) + I*meijerg(((-3/2, -1, -3/4, -1/2, -1/4, 1),
()), ((-3/4, -1/4), (-3/2, -1, 0, 0)), c**2*exp_polar(2*I*pi)/(d**2*x**2))/(2*p
i**(3/2)*d**3)) + b*(c**2*meijerg(((-5/4, -3/4), (-3/2, -1/2, 0, 1)), ((-5/4, -1
, -3/4, -1/2, 0, 0), ()), c**2/(d**2*x**2))/(2*pi**(3/2)*d**5) + I*c**2*meijerg(
((-5/2, -2, -7/4, -3/2, -5/4, 1), ()), ((-7/4, -5/4), (-5/2, -2, -1, 0)), c**2*e
xp_polar(2*I*pi)/(d**2*x**2))/(2*pi**(3/2)*d**5))

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GIAC/XCAS [A]  time = 0.241076, size = 208, normalized size = 1.37 \[ -\frac{{\left ({\left (3 \, b d^{15} - \frac{{\left (d x + c\right )} b d^{15}}{c}\right )}{\left (d x + c\right )} - \frac{b c^{2} d^{15} - a d^{17}}{c}\right )} \sqrt{d x + c}}{768 \, \sqrt{d x - c}} - \frac{{\left (3 \, b c^{2} d^{15} + 2 \, a d^{17}\right )}{\rm ln}\left ({\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{2}\right )}{768 \, c} - \frac{2 \,{\left (b c^{3} + a c d^{2}\right )}}{{\left ({\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{2} + 2 \, c\right )} d^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/768*((3*b*d^15 - (d*x + c)*b*d^15/c)*(d*x + c) - (b*c^2*d^15 - a*d^17)/c)*sqr
t(d*x + c)/sqrt(d*x - c) - 1/768*(3*b*c^2*d^15 + 2*a*d^17)*ln((sqrt(d*x + c) - s
qrt(d*x - c))^2)/c - 2*(b*c^3 + a*c*d^2)/(((sqrt(d*x + c) - sqrt(d*x - c))^2 + 2
*c)*d^5)