3.660 \(\int \frac{x^4 \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=202 \[ \frac{\left (8 a^2 d^2-40 a b c d+35 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{8 d^{9/2}}-\frac{x \sqrt{c+d x^2} \left (8 a^2 d^2-40 a b c d+35 b^2 c^2\right )}{8 c d^4}+\frac{x^3 \left (8 a^2 d^2-40 a b c d+35 b^2 c^2\right )}{12 c d^3 \sqrt{c+d x^2}}+\frac{x^5 (b c-a d)^2}{3 c d^2 \left (c+d x^2\right )^{3/2}}+\frac{b^2 x^5}{4 d^2 \sqrt{c+d x^2}} \]

[Out]

((b*c - a*d)^2*x^5)/(3*c*d^2*(c + d*x^2)^(3/2)) + ((35*b^2*c^2 - 40*a*b*c*d + 8*
a^2*d^2)*x^3)/(12*c*d^3*Sqrt[c + d*x^2]) + (b^2*x^5)/(4*d^2*Sqrt[c + d*x^2]) - (
(35*b^2*c^2 - 40*a*b*c*d + 8*a^2*d^2)*x*Sqrt[c + d*x^2])/(8*c*d^4) + ((35*b^2*c^
2 - 40*a*b*c*d + 8*a^2*d^2)*ArcTanh[(Sqrt[d]*x)/Sqrt[c + d*x^2]])/(8*d^(9/2))

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Rubi [A]  time = 0.415568, antiderivative size = 202, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{\left (8 a^2 d^2-40 a b c d+35 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{8 d^{9/2}}-\frac{x \sqrt{c+d x^2} \left (8 a^2 d^2-40 a b c d+35 b^2 c^2\right )}{8 c d^4}+\frac{x^3 \left (8 a^2 d^2-40 a b c d+35 b^2 c^2\right )}{12 c d^3 \sqrt{c+d x^2}}+\frac{x^5 (b c-a d)^2}{3 c d^2 \left (c+d x^2\right )^{3/2}}+\frac{b^2 x^5}{4 d^2 \sqrt{c+d x^2}} \]

Antiderivative was successfully verified.

[In]  Int[(x^4*(a + b*x^2)^2)/(c + d*x^2)^(5/2),x]

[Out]

((b*c - a*d)^2*x^5)/(3*c*d^2*(c + d*x^2)^(3/2)) + ((35*b^2*c^2 - 40*a*b*c*d + 8*
a^2*d^2)*x^3)/(12*c*d^3*Sqrt[c + d*x^2]) + (b^2*x^5)/(4*d^2*Sqrt[c + d*x^2]) - (
(35*b^2*c^2 - 40*a*b*c*d + 8*a^2*d^2)*x*Sqrt[c + d*x^2])/(8*c*d^4) + ((35*b^2*c^
2 - 40*a*b*c*d + 8*a^2*d^2)*ArcTanh[(Sqrt[d]*x)/Sqrt[c + d*x^2]])/(8*d^(9/2))

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Rubi in Sympy [A]  time = 46.5344, size = 190, normalized size = 0.94 \[ \frac{b^{2} x^{5}}{4 d^{2} \sqrt{c + d x^{2}}} + \frac{\left (8 a^{2} d^{2} - 40 a b c d + 35 b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} x}{\sqrt{c + d x^{2}}} \right )}}{8 d^{\frac{9}{2}}} + \frac{x^{5} \left (a d - b c\right )^{2}}{3 c d^{2} \left (c + d x^{2}\right )^{\frac{3}{2}}} + \frac{x^{3} \left (8 a^{2} d^{2} - 40 a b c d + 35 b^{2} c^{2}\right )}{12 c d^{3} \sqrt{c + d x^{2}}} - \frac{x \sqrt{c + d x^{2}} \left (8 a^{2} d^{2} - 40 a b c d + 35 b^{2} c^{2}\right )}{8 c d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4*(b*x**2+a)**2/(d*x**2+c)**(5/2),x)

[Out]

b**2*x**5/(4*d**2*sqrt(c + d*x**2)) + (8*a**2*d**2 - 40*a*b*c*d + 35*b**2*c**2)*
atanh(sqrt(d)*x/sqrt(c + d*x**2))/(8*d**(9/2)) + x**5*(a*d - b*c)**2/(3*c*d**2*(
c + d*x**2)**(3/2)) + x**3*(8*a**2*d**2 - 40*a*b*c*d + 35*b**2*c**2)/(12*c*d**3*
sqrt(c + d*x**2)) - x*sqrt(c + d*x**2)*(8*a**2*d**2 - 40*a*b*c*d + 35*b**2*c**2)
/(8*c*d**4)

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Mathematica [A]  time = 0.227294, size = 156, normalized size = 0.77 \[ \frac{\left (8 a^2 d^2-40 a b c d+35 b^2 c^2\right ) \log \left (\sqrt{d} \sqrt{c+d x^2}+d x\right )}{8 d^{9/2}}+\frac{x \left (-8 a^2 d^2 \left (3 c+4 d x^2\right )+8 a b d \left (15 c^2+20 c d x^2+3 d^2 x^4\right )+b^2 \left (-\left (105 c^3+140 c^2 d x^2+21 c d^2 x^4-6 d^3 x^6\right )\right )\right )}{24 d^4 \left (c+d x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^4*(a + b*x^2)^2)/(c + d*x^2)^(5/2),x]

[Out]

(x*(-8*a^2*d^2*(3*c + 4*d*x^2) + 8*a*b*d*(15*c^2 + 20*c*d*x^2 + 3*d^2*x^4) - b^2
*(105*c^3 + 140*c^2*d*x^2 + 21*c*d^2*x^4 - 6*d^3*x^6)))/(24*d^4*(c + d*x^2)^(3/2
)) + ((35*b^2*c^2 - 40*a*b*c*d + 8*a^2*d^2)*Log[d*x + Sqrt[d]*Sqrt[c + d*x^2]])/
(8*d^(9/2))

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Maple [A]  time = 0.029, size = 255, normalized size = 1.3 \[ -{\frac{{a}^{2}{x}^{3}}{3\,d} \left ( d{x}^{2}+c \right ) ^{-{\frac{3}{2}}}}-{\frac{{a}^{2}x}{{d}^{2}}{\frac{1}{\sqrt{d{x}^{2}+c}}}}+{{a}^{2}\ln \left ( x\sqrt{d}+\sqrt{d{x}^{2}+c} \right ){d}^{-{\frac{5}{2}}}}+{\frac{{b}^{2}{x}^{7}}{4\,d} \left ( d{x}^{2}+c \right ) ^{-{\frac{3}{2}}}}-{\frac{7\,{b}^{2}c{x}^{5}}{8\,{d}^{2}} \left ( d{x}^{2}+c \right ) ^{-{\frac{3}{2}}}}-{\frac{35\,{b}^{2}{c}^{2}{x}^{3}}{24\,{d}^{3}} \left ( d{x}^{2}+c \right ) ^{-{\frac{3}{2}}}}-{\frac{35\,{b}^{2}{c}^{2}x}{8\,{d}^{4}}{\frac{1}{\sqrt{d{x}^{2}+c}}}}+{\frac{35\,{b}^{2}{c}^{2}}{8}\ln \left ( x\sqrt{d}+\sqrt{d{x}^{2}+c} \right ){d}^{-{\frac{9}{2}}}}+{\frac{ab{x}^{5}}{d} \left ( d{x}^{2}+c \right ) ^{-{\frac{3}{2}}}}+{\frac{5\,abc{x}^{3}}{3\,{d}^{2}} \left ( d{x}^{2}+c \right ) ^{-{\frac{3}{2}}}}+5\,{\frac{abcx}{{d}^{3}\sqrt{d{x}^{2}+c}}}-5\,{\frac{abc\ln \left ( x\sqrt{d}+\sqrt{d{x}^{2}+c} \right ) }{{d}^{7/2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4*(b*x^2+a)^2/(d*x^2+c)^(5/2),x)

[Out]

-1/3*a^2*x^3/d/(d*x^2+c)^(3/2)-a^2/d^2*x/(d*x^2+c)^(1/2)+a^2/d^(5/2)*ln(x*d^(1/2
)+(d*x^2+c)^(1/2))+1/4*b^2*x^7/d/(d*x^2+c)^(3/2)-7/8*b^2*c/d^2*x^5/(d*x^2+c)^(3/
2)-35/24*b^2*c^2/d^3*x^3/(d*x^2+c)^(3/2)-35/8*b^2*c^2/d^4*x/(d*x^2+c)^(1/2)+35/8
*b^2*c^2/d^(9/2)*ln(x*d^(1/2)+(d*x^2+c)^(1/2))+a*b*x^5/d/(d*x^2+c)^(3/2)+5/3*a*b
*c/d^2*x^3/(d*x^2+c)^(3/2)+5*a*b*c/d^3*x/(d*x^2+c)^(1/2)-5*a*b*c/d^(7/2)*ln(x*d^
(1/2)+(d*x^2+c)^(1/2))

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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*x^4/(d*x^2 + c)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.308755, size = 1, normalized size = 0. \[ \left [\frac{2 \,{\left (6 \, b^{2} d^{3} x^{7} - 3 \,{\left (7 \, b^{2} c d^{2} - 8 \, a b d^{3}\right )} x^{5} - 4 \,{\left (35 \, b^{2} c^{2} d - 40 \, a b c d^{2} + 8 \, a^{2} d^{3}\right )} x^{3} - 3 \,{\left (35 \, b^{2} c^{3} - 40 \, a b c^{2} d + 8 \, a^{2} c d^{2}\right )} x\right )} \sqrt{d x^{2} + c} \sqrt{d} + 3 \,{\left (35 \, b^{2} c^{4} - 40 \, a b c^{3} d + 8 \, a^{2} c^{2} d^{2} +{\left (35 \, b^{2} c^{2} d^{2} - 40 \, a b c d^{3} + 8 \, a^{2} d^{4}\right )} x^{4} + 2 \,{\left (35 \, b^{2} c^{3} d - 40 \, a b c^{2} d^{2} + 8 \, a^{2} c d^{3}\right )} x^{2}\right )} \log \left (-2 \, \sqrt{d x^{2} + c} d x -{\left (2 \, d x^{2} + c\right )} \sqrt{d}\right )}{48 \,{\left (d^{6} x^{4} + 2 \, c d^{5} x^{2} + c^{2} d^{4}\right )} \sqrt{d}}, \frac{{\left (6 \, b^{2} d^{3} x^{7} - 3 \,{\left (7 \, b^{2} c d^{2} - 8 \, a b d^{3}\right )} x^{5} - 4 \,{\left (35 \, b^{2} c^{2} d - 40 \, a b c d^{2} + 8 \, a^{2} d^{3}\right )} x^{3} - 3 \,{\left (35 \, b^{2} c^{3} - 40 \, a b c^{2} d + 8 \, a^{2} c d^{2}\right )} x\right )} \sqrt{d x^{2} + c} \sqrt{-d} + 3 \,{\left (35 \, b^{2} c^{4} - 40 \, a b c^{3} d + 8 \, a^{2} c^{2} d^{2} +{\left (35 \, b^{2} c^{2} d^{2} - 40 \, a b c d^{3} + 8 \, a^{2} d^{4}\right )} x^{4} + 2 \,{\left (35 \, b^{2} c^{3} d - 40 \, a b c^{2} d^{2} + 8 \, a^{2} c d^{3}\right )} x^{2}\right )} \arctan \left (\frac{\sqrt{-d} x}{\sqrt{d x^{2} + c}}\right )}{24 \,{\left (d^{6} x^{4} + 2 \, c d^{5} x^{2} + c^{2} d^{4}\right )} \sqrt{-d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2*x^4/(d*x^2 + c)^(5/2),x, algorithm="fricas")

[Out]

[1/48*(2*(6*b^2*d^3*x^7 - 3*(7*b^2*c*d^2 - 8*a*b*d^3)*x^5 - 4*(35*b^2*c^2*d - 40
*a*b*c*d^2 + 8*a^2*d^3)*x^3 - 3*(35*b^2*c^3 - 40*a*b*c^2*d + 8*a^2*c*d^2)*x)*sqr
t(d*x^2 + c)*sqrt(d) + 3*(35*b^2*c^4 - 40*a*b*c^3*d + 8*a^2*c^2*d^2 + (35*b^2*c^
2*d^2 - 40*a*b*c*d^3 + 8*a^2*d^4)*x^4 + 2*(35*b^2*c^3*d - 40*a*b*c^2*d^2 + 8*a^2
*c*d^3)*x^2)*log(-2*sqrt(d*x^2 + c)*d*x - (2*d*x^2 + c)*sqrt(d)))/((d^6*x^4 + 2*
c*d^5*x^2 + c^2*d^4)*sqrt(d)), 1/24*((6*b^2*d^3*x^7 - 3*(7*b^2*c*d^2 - 8*a*b*d^3
)*x^5 - 4*(35*b^2*c^2*d - 40*a*b*c*d^2 + 8*a^2*d^3)*x^3 - 3*(35*b^2*c^3 - 40*a*b
*c^2*d + 8*a^2*c*d^2)*x)*sqrt(d*x^2 + c)*sqrt(-d) + 3*(35*b^2*c^4 - 40*a*b*c^3*d
+ 8*a^2*c^2*d^2 + (35*b^2*c^2*d^2 - 40*a*b*c*d^3 + 8*a^2*d^4)*x^4 + 2*(35*b^2*c
^3*d - 40*a*b*c^2*d^2 + 8*a^2*c*d^3)*x^2)*arctan(sqrt(-d)*x/sqrt(d*x^2 + c)))/((
d^6*x^4 + 2*c*d^5*x^2 + c^2*d^4)*sqrt(-d))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{4} \left (a + b x^{2}\right )^{2}}{\left (c + d x^{2}\right )^{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4*(b*x**2+a)**2/(d*x**2+c)**(5/2),x)

[Out]

Integral(x**4*(a + b*x**2)**2/(c + d*x**2)**(5/2), x)

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GIAC/XCAS [A]  time = 0.247294, size = 257, normalized size = 1.27 \[ \frac{{\left ({\left (3 \,{\left (\frac{2 \, b^{2} x^{2}}{d} - \frac{7 \, b^{2} c^{2} d^{5} - 8 \, a b c d^{6}}{c d^{7}}\right )} x^{2} - \frac{4 \,{\left (35 \, b^{2} c^{3} d^{4} - 40 \, a b c^{2} d^{5} + 8 \, a^{2} c d^{6}\right )}}{c d^{7}}\right )} x^{2} - \frac{3 \,{\left (35 \, b^{2} c^{4} d^{3} - 40 \, a b c^{3} d^{4} + 8 \, a^{2} c^{2} d^{5}\right )}}{c d^{7}}\right )} x}{24 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}}} - \frac{{\left (35 \, b^{2} c^{2} - 40 \, a b c d + 8 \, a^{2} d^{2}\right )}{\rm ln}\left ({\left | -\sqrt{d} x + \sqrt{d x^{2} + c} \right |}\right )}{8 \, d^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2*x^4/(d*x^2 + c)^(5/2),x, algorithm="giac")

[Out]

1/24*((3*(2*b^2*x^2/d - (7*b^2*c^2*d^5 - 8*a*b*c*d^6)/(c*d^7))*x^2 - 4*(35*b^2*c
^3*d^4 - 40*a*b*c^2*d^5 + 8*a^2*c*d^6)/(c*d^7))*x^2 - 3*(35*b^2*c^4*d^3 - 40*a*b
*c^3*d^4 + 8*a^2*c^2*d^5)/(c*d^7))*x/(d*x^2 + c)^(3/2) - 1/8*(35*b^2*c^2 - 40*a*
b*c*d + 8*a^2*d^2)*ln(abs(-sqrt(d)*x + sqrt(d*x^2 + c)))/d^(9/2)