3.269 \(\int \frac{x^4 \left (a+b x^2\right )}{(-c+d x)^{3/2} (c+d x)^{3/2}} \, dx\)

Optimal. Leaf size=161 \[ \frac{3 c^2 \left (4 a d^2+5 b c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d x-c}}{\sqrt{c+d x}}\right )}{4 d^7}+\frac{3 x \sqrt{d x-c} \sqrt{c+d x} \left (4 a d^2+5 b c^2\right )}{8 d^6}-\frac{x^3 \left (4 a d^2+5 b c^2\right )}{4 d^4 \sqrt{d x-c} \sqrt{c+d x}}+\frac{b x^5}{4 d^2 \sqrt{d x-c} \sqrt{c+d x}} \]

[Out]

-((5*b*c^2 + 4*a*d^2)*x^3)/(4*d^4*Sqrt[-c + d*x]*Sqrt[c + d*x]) + (b*x^5)/(4*d^2
*Sqrt[-c + d*x]*Sqrt[c + d*x]) + (3*(5*b*c^2 + 4*a*d^2)*x*Sqrt[-c + d*x]*Sqrt[c
+ d*x])/(8*d^6) + (3*c^2*(5*b*c^2 + 4*a*d^2)*ArcTanh[Sqrt[-c + d*x]/Sqrt[c + d*x
]])/(4*d^7)

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Rubi [A]  time = 0.417564, antiderivative size = 161, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.226 \[ \frac{3 c^2 \left (4 a d^2+5 b c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d x-c}}{\sqrt{c+d x}}\right )}{4 d^7}+\frac{3 x \sqrt{d x-c} \sqrt{c+d x} \left (4 a d^2+5 b c^2\right )}{8 d^6}-\frac{x^3 \left (4 a d^2+5 b c^2\right )}{4 d^4 \sqrt{d x-c} \sqrt{c+d x}}+\frac{b x^5}{4 d^2 \sqrt{d x-c} \sqrt{c+d x}} \]

Antiderivative was successfully verified.

[In]  Int[(x^4*(a + b*x^2))/((-c + d*x)^(3/2)*(c + d*x)^(3/2)),x]

[Out]

-((5*b*c^2 + 4*a*d^2)*x^3)/(4*d^4*Sqrt[-c + d*x]*Sqrt[c + d*x]) + (b*x^5)/(4*d^2
*Sqrt[-c + d*x]*Sqrt[c + d*x]) + (3*(5*b*c^2 + 4*a*d^2)*x*Sqrt[-c + d*x]*Sqrt[c
+ d*x])/(8*d^6) + (3*c^2*(5*b*c^2 + 4*a*d^2)*ArcTanh[Sqrt[-c + d*x]/Sqrt[c + d*x
]])/(4*d^7)

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Rubi in Sympy [A]  time = 26.8087, size = 146, normalized size = 0.91 \[ \frac{b x^{5}}{4 d^{2} \sqrt{- c + d x} \sqrt{c + d x}} + \frac{3 c^{2} \left (4 a d^{2} + 5 b c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{- c + d x}}{\sqrt{c + d x}} \right )}}{4 d^{7}} - \frac{x^{3} \left (4 a d^{2} + 5 b c^{2}\right )}{4 d^{4} \sqrt{- c + d x} \sqrt{c + d x}} + \frac{3 x \sqrt{- c + d x} \sqrt{c + d x} \left (4 a d^{2} + 5 b c^{2}\right )}{8 d^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4*(b*x**2+a)/(d*x-c)**(3/2)/(d*x+c)**(3/2),x)

[Out]

b*x**5/(4*d**2*sqrt(-c + d*x)*sqrt(c + d*x)) + 3*c**2*(4*a*d**2 + 5*b*c**2)*atan
h(sqrt(-c + d*x)/sqrt(c + d*x))/(4*d**7) - x**3*(4*a*d**2 + 5*b*c**2)/(4*d**4*sq
rt(-c + d*x)*sqrt(c + d*x)) + 3*x*sqrt(-c + d*x)*sqrt(c + d*x)*(4*a*d**2 + 5*b*c
**2)/(8*d**6)

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Mathematica [A]  time = 0.196303, size = 150, normalized size = 0.93 \[ \frac{d x \sqrt{d x-c} \sqrt{c+d x} \left (4 a d^2 \left (d^2 x^2-3 c^2\right )+b \left (-15 c^4+5 c^2 d^2 x^2+2 d^4 x^4\right )\right )-3 c^2 \left (c^2-d^2 x^2\right ) \left (4 a d^2+5 b c^2\right ) \log \left (\sqrt{d x-c} \sqrt{c+d x}+d x\right )}{8 d^7 (d x-c) (c+d x)} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^4*(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]*(4*a*d^2*(-3*c^2 + d^2*x^2) + b*(-15*c^4 + 5*c
^2*d^2*x^2 + 2*d^4*x^4)) - 3*c^2*(5*b*c^2 + 4*a*d^2)*(c^2 - d^2*x^2)*Log[d*x + S
qrt[-c + d*x]*Sqrt[c + d*x]])/(8*d^7*(-c + d*x)*(c + d*x))

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Maple [C]  time = 0.047, size = 316, normalized size = 2. \[{\frac{{\it csgn} \left ( d \right ) }{8\,{d}^{7}} \left ( 2\,{\it csgn} \left ( d \right ){x}^{5}b{d}^{5}\sqrt{{d}^{2}{x}^{2}-{c}^{2}}+4\,{\it csgn} \left ( d \right ){x}^{3}a{d}^{5}\sqrt{{d}^{2}{x}^{2}-{c}^{2}}+5\,{\it csgn} \left ( d \right ){x}^{3}b{c}^{2}{d}^{3}\sqrt{{d}^{2}{x}^{2}-{c}^{2}}+12\,\ln \left ( \left ({\it csgn} \left ( d \right ) \sqrt{{d}^{2}{x}^{2}-{c}^{2}}+dx \right ){\it csgn} \left ( d \right ) \right ){x}^{2}a{c}^{2}{d}^{4}+15\,\ln \left ( \left ({\it csgn} \left ( d \right ) \sqrt{{d}^{2}{x}^{2}-{c}^{2}}+dx \right ){\it csgn} \left ( d \right ) \right ){x}^{2}b{c}^{4}{d}^{2}-12\,a{c}^{2}x\sqrt{{d}^{2}{x}^{2}-{c}^{2}}{d}^{3}{\it csgn} \left ( d \right ) -15\,b{c}^{4}x\sqrt{{d}^{2}{x}^{2}-{c}^{2}}{\it csgn} \left ( d \right ) d-12\,a{c}^{4}\ln \left ( \left ({\it csgn} \left ( d \right ) \sqrt{{d}^{2}{x}^{2}-{c}^{2}}+dx \right ){\it csgn} \left ( d \right ) \right ){d}^{2}-15\,b{c}^{6}\ln \left ( \left ({\it csgn} \left ( d \right ) \sqrt{{d}^{2}{x}^{2}-{c}^{2}}+dx \right ){\it csgn} \left ( d \right ) \right ) \right ){\frac{1}{\sqrt{{d}^{2}{x}^{2}-{c}^{2}}}}{\frac{1}{\sqrt{dx+c}}}{\frac{1}{\sqrt{dx-c}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4*(b*x^2+a)/(d*x-c)^(3/2)/(d*x+c)^(3/2),x)

[Out]

1/8*(2*csgn(d)*x^5*b*d^5*(d^2*x^2-c^2)^(1/2)+4*csgn(d)*x^3*a*d^5*(d^2*x^2-c^2)^(
1/2)+5*csgn(d)*x^3*b*c^2*d^3*(d^2*x^2-c^2)^(1/2)+12*ln((csgn(d)*(d^2*x^2-c^2)^(1
/2)+d*x)*csgn(d))*x^2*a*c^2*d^4+15*ln((csgn(d)*(d^2*x^2-c^2)^(1/2)+d*x)*csgn(d))
*x^2*b*c^4*d^2-12*a*c^2*x*(d^2*x^2-c^2)^(1/2)*d^3*csgn(d)-15*b*c^4*x*(d^2*x^2-c^
2)^(1/2)*csgn(d)*d-12*a*c^4*ln((csgn(d)*(d^2*x^2-c^2)^(1/2)+d*x)*csgn(d))*d^2-15
*b*c^6*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^7/(d*x+c)^(1/2)/(d*x-c)^(1/2)

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Maxima [A]  time = 1.43494, size = 289, normalized size = 1.8 \[ \frac{b x^{5}}{4 \, \sqrt{d^{2} x^{2} - c^{2}} d^{2}} + \frac{5 \, b c^{2} x^{3}}{8 \, \sqrt{d^{2} x^{2} - c^{2}} d^{4}} + \frac{a x^{3}}{2 \, \sqrt{d^{2} x^{2} - c^{2}} d^{2}} - \frac{15 \, b c^{4} x}{8 \, \sqrt{d^{2} x^{2} - c^{2}} d^{6}} - \frac{3 \, a c^{2} x}{2 \, \sqrt{d^{2} x^{2} - c^{2}} d^{4}} + \frac{15 \, b c^{4} \log \left (2 \, d^{2} x + 2 \, \sqrt{d^{2} x^{2} - c^{2}} \sqrt{d^{2}}\right )}{8 \, \sqrt{d^{2}} d^{6}} + \frac{3 \, a c^{2} \log \left (2 \, d^{2} x + 2 \, \sqrt{d^{2} x^{2} - c^{2}} \sqrt{d^{2}}\right )}{2 \, \sqrt{d^{2}} d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)*x^4/((d*x + c)^(3/2)*(d*x - c)^(3/2)),x, algorithm="maxima")

[Out]

1/4*b*x^5/(sqrt(d^2*x^2 - c^2)*d^2) + 5/8*b*c^2*x^3/(sqrt(d^2*x^2 - c^2)*d^4) +
1/2*a*x^3/(sqrt(d^2*x^2 - c^2)*d^2) - 15/8*b*c^4*x/(sqrt(d^2*x^2 - c^2)*d^6) - 3
/2*a*c^2*x/(sqrt(d^2*x^2 - c^2)*d^4) + 15/8*b*c^4*log(2*d^2*x + 2*sqrt(d^2*x^2 -
c^2)*sqrt(d^2))/(sqrt(d^2)*d^6) + 3/2*a*c^2*log(2*d^2*x + 2*sqrt(d^2*x^2 - c^2)
*sqrt(d^2))/(sqrt(d^2)*d^4)

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Fricas [A]  time = 0.244879, size = 702, normalized size = 4.36 \[ -\frac{32 \, b d^{10} x^{10} - 8 \, b c^{10} - 8 \, a c^{8} d^{2} + 8 \,{\left (5 \, b c^{2} d^{8} + 8 \, a d^{10}\right )} x^{8} - 2 \,{\left (101 \, b c^{4} d^{6} + 72 \, a c^{2} d^{8}\right )} x^{6} +{\left (101 \, b c^{6} d^{4} + 36 \, a c^{4} d^{6}\right )} x^{4} +{\left (29 \, b c^{8} d^{2} + 44 \, a c^{6} d^{4}\right )} x^{2} -{\left (32 \, b d^{9} x^{9} + 8 \,{\left (7 \, b c^{2} d^{7} + 8 \, a d^{9}\right )} x^{7} - 2 \,{\left (85 \, b c^{4} d^{5} + 56 \, a c^{2} d^{7}\right )} x^{5} +{\left (25 \, b c^{6} d^{3} - 12 \, a c^{4} d^{5}\right )} x^{3} +{\left (25 \, b c^{8} d + 28 \, a c^{6} d^{3}\right )} x\right )} \sqrt{d x + c} \sqrt{d x - c} - 3 \,{\left (5 \, b c^{10} + 4 \, a c^{8} d^{2} - 16 \,{\left (5 \, b c^{4} d^{6} + 4 \, a c^{2} d^{8}\right )} x^{6} + 28 \,{\left (5 \, b c^{6} d^{4} + 4 \, a c^{4} d^{6}\right )} x^{4} - 13 \,{\left (5 \, b c^{8} d^{2} + 4 \, a c^{6} d^{4}\right )} x^{2} +{\left (16 \,{\left (5 \, b c^{4} d^{5} + 4 \, a c^{2} d^{7}\right )} x^{5} - 20 \,{\left (5 \, b c^{6} d^{3} + 4 \, a c^{4} d^{5}\right )} x^{3} + 5 \,{\left (5 \, b c^{8} d + 4 \, a c^{6} d^{3}\right )} x\right )} \sqrt{d x + c} \sqrt{d x - c}\right )} \log \left (-d x + \sqrt{d x + c} \sqrt{d x - c}\right )}{8 \,{\left (16 \, d^{13} x^{6} - 28 \, c^{2} d^{11} x^{4} + 13 \, c^{4} d^{9} x^{2} - c^{6} d^{7} -{\left (16 \, d^{12} x^{5} - 20 \, c^{2} d^{10} x^{3} + 5 \, c^{4} d^{8} x\right )} \sqrt{d x + c} \sqrt{d x - c}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)*x^4/((d*x + c)^(3/2)*(d*x - c)^(3/2)),x, algorithm="fricas")

[Out]

-1/8*(32*b*d^10*x^10 - 8*b*c^10 - 8*a*c^8*d^2 + 8*(5*b*c^2*d^8 + 8*a*d^10)*x^8 -
2*(101*b*c^4*d^6 + 72*a*c^2*d^8)*x^6 + (101*b*c^6*d^4 + 36*a*c^4*d^6)*x^4 + (29
*b*c^8*d^2 + 44*a*c^6*d^4)*x^2 - (32*b*d^9*x^9 + 8*(7*b*c^2*d^7 + 8*a*d^9)*x^7 -
2*(85*b*c^4*d^5 + 56*a*c^2*d^7)*x^5 + (25*b*c^6*d^3 - 12*a*c^4*d^5)*x^3 + (25*b
*c^8*d + 28*a*c^6*d^3)*x)*sqrt(d*x + c)*sqrt(d*x - c) - 3*(5*b*c^10 + 4*a*c^8*d^
2 - 16*(5*b*c^4*d^6 + 4*a*c^2*d^8)*x^6 + 28*(5*b*c^6*d^4 + 4*a*c^4*d^6)*x^4 - 13
*(5*b*c^8*d^2 + 4*a*c^6*d^4)*x^2 + (16*(5*b*c^4*d^5 + 4*a*c^2*d^7)*x^5 - 20*(5*b
*c^6*d^3 + 4*a*c^4*d^5)*x^3 + 5*(5*b*c^8*d + 4*a*c^6*d^3)*x)*sqrt(d*x + c)*sqrt(
d*x - c))*log(-d*x + sqrt(d*x + c)*sqrt(d*x - c)))/(16*d^13*x^6 - 28*c^2*d^11*x^
4 + 13*c^4*d^9*x^2 - c^6*d^7 - (16*d^12*x^5 - 20*c^2*d^10*x^3 + 5*c^4*d^8*x)*sqr
t(d*x + c)*sqrt(d*x - c))

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Sympy [A]  time = 102.515, size = 233, normalized size = 1.45 \[ a \left (\frac{c^{2}{G_{6, 6}^{6, 2}\left (\begin{matrix} - \frac{5}{4}, - \frac{3}{4} & - \frac{3}{2}, - \frac{1}{2}, 0, 1 \\- \frac{5}{4}, -1, - \frac{3}{4}, - \frac{1}{2}, 0, 0 & \end{matrix} \middle |{\frac{c^{2}}{d^{2} x^{2}}} \right )}}{2 \pi ^{\frac{3}{2}} d^{5}} + \frac{i c^{2}{G_{6, 6}^{2, 6}\left (\begin{matrix} - \frac{5}{2}, -2, - \frac{7}{4}, - \frac{3}{2}, - \frac{5}{4}, 1 & \\- \frac{7}{4}, - \frac{5}{4} & - \frac{5}{2}, -2, -1, 0 \end{matrix} \middle |{\frac{c^{2} e^{2 i \pi }}{d^{2} x^{2}}} \right )}}{2 \pi ^{\frac{3}{2}} d^{5}}\right ) + b \left (\frac{c^{4}{G_{6, 6}^{6, 2}\left (\begin{matrix} - \frac{9}{4}, - \frac{7}{4} & - \frac{5}{2}, - \frac{3}{2}, -1, 1 \\- \frac{9}{4}, -2, - \frac{7}{4}, - \frac{3}{2}, -1, 0 & \end{matrix} \middle |{\frac{c^{2}}{d^{2} x^{2}}} \right )}}{2 \pi ^{\frac{3}{2}} d^{7}} + \frac{i c^{4}{G_{6, 6}^{2, 6}\left (\begin{matrix} - \frac{7}{2}, -3, - \frac{11}{4}, - \frac{5}{2}, - \frac{9}{4}, 1 & \\- \frac{11}{4}, - \frac{9}{4} & - \frac{7}{2}, -3, -2, 0 \end{matrix} \middle |{\frac{c^{2} e^{2 i \pi }}{d^{2} x^{2}}} \right )}}{2 \pi ^{\frac{3}{2}} d^{7}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4*(b*x**2+a)/(d*x-c)**(3/2)/(d*x+c)**(3/2),x)

[Out]

a*(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*exp_polar(2*I*pi)/(
d**2*x**2))/(2*pi**(3/2)*d**5)) + b*(c**4*meijerg(((-9/4, -7/4), (-5/2, -3/2, -1
, 1)), ((-9/4, -2, -7/4, -3/2, -1, 0), ()), c**2/(d**2*x**2))/(2*pi**(3/2)*d**7)
+ I*c**4*meijerg(((-7/2, -3, -11/4, -5/2, -9/4, 1), ()), ((-11/4, -9/4), (-7/2,
-3, -2, 0)), c**2*exp_polar(2*I*pi)/(d**2*x**2))/(2*pi**(3/2)*d**7))

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GIAC/XCAS [A]  time = 0.243284, size = 290, normalized size = 1.8 \[ -\frac{1}{688128} \,{\left (5 \, b c^{3} d^{35} + 4 \, a c d^{37}\right )}{\rm ln}\left ({\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{2}\right ) - \frac{{\left ({\left ({\left (2 \,{\left (5 \, b d^{35} - \frac{{\left (d x + c\right )} b d^{35}}{c}\right )}{\left (d x + c\right )} - \frac{25 \, b c^{2} d^{35} + 4 \, a d^{37}}{c}\right )}{\left (d x + c\right )} + \frac{35 \, b c^{3} d^{35} + 12 \, a c d^{37}}{c}\right )}{\left (d x + c\right )} - \frac{2 \,{\left (7 \, b c^{4} d^{35} + 2 \, a c^{2} d^{37}\right )}}{c}\right )} \sqrt{d x + c}}{2064384 \, \sqrt{d x - c}} - \frac{2 \,{\left (b c^{5} + a c^{3} d^{2}\right )}}{{\left ({\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{2} + 2 \, c\right )} d^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)*x^4/((d*x + c)^(3/2)*(d*x - c)^(3/2)),x, algorithm="giac")

[Out]

-1/688128*(5*b*c^3*d^35 + 4*a*c*d^37)*ln((sqrt(d*x + c) - sqrt(d*x - c))^2) - 1/
2064384*(((2*(5*b*d^35 - (d*x + c)*b*d^35/c)*(d*x + c) - (25*b*c^2*d^35 + 4*a*d^
37)/c)*(d*x + c) + (35*b*c^3*d^35 + 12*a*c*d^37)/c)*(d*x + c) - 2*(7*b*c^4*d^35
+ 2*a*c^2*d^37)/c)*sqrt(d*x + c)/sqrt(d*x - c) - 2*(b*c^5 + a*c^3*d^2)/(((sqrt(d
*x + c) - sqrt(d*x - c))^2 + 2*c)*d^7)