3.980 \(\int \frac{x^5}{\left (a+b x^2\right )^{3/2} \sqrt{c+d x^2}} \, dx\)

Optimal. Leaf size=129 \[ -\frac{a^2 \sqrt{c+d x^2}}{b^2 \sqrt{a+b x^2} (b c-a d)}-\frac{(3 a d+b c) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x^2}}{\sqrt{b} \sqrt{c+d x^2}}\right )}{2 b^{5/2} d^{3/2}}+\frac{\sqrt{a+b x^2} \sqrt{c+d x^2}}{2 b^2 d} \]

[Out]

-((a^2*Sqrt[c + d*x^2])/(b^2*(b*c - a*d)*Sqrt[a + b*x^2])) + (Sqrt[a + b*x^2]*Sq
rt[c + d*x^2])/(2*b^2*d) - ((b*c + 3*a*d)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x^2])/(Sqr
t[b]*Sqrt[c + d*x^2])])/(2*b^(5/2)*d^(3/2))

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Rubi [A]  time = 0.420238, antiderivative size = 129, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ -\frac{a^2 \sqrt{c+d x^2}}{b^2 \sqrt{a+b x^2} (b c-a d)}-\frac{(3 a d+b c) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x^2}}{\sqrt{b} \sqrt{c+d x^2}}\right )}{2 b^{5/2} d^{3/2}}+\frac{\sqrt{a+b x^2} \sqrt{c+d x^2}}{2 b^2 d} \]

Antiderivative was successfully verified.

[In]  Int[x^5/((a + b*x^2)^(3/2)*Sqrt[c + d*x^2]),x]

[Out]

-((a^2*Sqrt[c + d*x^2])/(b^2*(b*c - a*d)*Sqrt[a + b*x^2])) + (Sqrt[a + b*x^2]*Sq
rt[c + d*x^2])/(2*b^2*d) - ((b*c + 3*a*d)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x^2])/(Sqr
t[b]*Sqrt[c + d*x^2])])/(2*b^(5/2)*d^(3/2))

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Rubi in Sympy [A]  time = 35.8505, size = 112, normalized size = 0.87 \[ \frac{a^{2} \sqrt{c + d x^{2}}}{b^{2} \sqrt{a + b x^{2}} \left (a d - b c\right )} + \frac{\sqrt{a + b x^{2}} \sqrt{c + d x^{2}}}{2 b^{2} d} - \frac{\left (3 a d + b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{c + d x^{2}}}{\sqrt{d} \sqrt{a + b x^{2}}} \right )}}{2 b^{\frac{5}{2}} d^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5/(b*x**2+a)**(3/2)/(d*x**2+c)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.250161, size = 129, normalized size = 1. \[ \frac{\sqrt{a+b x^2} \sqrt{c+d x^2} \left (\frac{2 a^2}{\left (a+b x^2\right ) (a d-b c)}+\frac{1}{d}\right )}{2 b^2}-\frac{(3 a d+b c) \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x^2} \sqrt{c+d x^2}+a d+b c+2 b d x^2\right )}{4 b^{5/2} d^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^5/((a + b*x^2)^(3/2)*Sqrt[c + d*x^2]),x]

[Out]

(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]*(d^(-1) + (2*a^2)/((-(b*c) + a*d)*(a + b*x^2)))
)/(2*b^2) - ((b*c + 3*a*d)*Log[b*c + a*d + 2*b*d*x^2 + 2*Sqrt[b]*Sqrt[d]*Sqrt[a
+ b*x^2]*Sqrt[c + d*x^2]])/(4*b^(5/2)*d^(3/2))

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Maple [B]  time = 0.054, size = 511, normalized size = 4. \[ -{\frac{1}{4\,{b}^{2}d \left ( ad-bc \right ) } \left ( 3\,\ln \left ( 1/2\,{\frac{2\,bd{x}^{2}+2\,\sqrt{ \left ( b{x}^{2}+a \right ) \left ( d{x}^{2}+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){x}^{2}{a}^{2}b{d}^{2}-2\,\ln \left ( 1/2\,{\frac{2\,bd{x}^{2}+2\,\sqrt{ \left ( b{x}^{2}+a \right ) \left ( d{x}^{2}+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){x}^{2}a{b}^{2}cd-\ln \left ({\frac{1}{2} \left ( 2\,bd{x}^{2}+2\,\sqrt{ \left ( b{x}^{2}+a \right ) \left ( d{x}^{2}+c \right ) }\sqrt{bd}+ad+bc \right ){\frac{1}{\sqrt{bd}}}} \right ){x}^{2}{b}^{3}{c}^{2}-2\,\sqrt{bd}\sqrt{ \left ( b{x}^{2}+a \right ) \left ( d{x}^{2}+c \right ) }{x}^{2}abd+2\,\sqrt{bd}\sqrt{ \left ( b{x}^{2}+a \right ) \left ( d{x}^{2}+c \right ) }{x}^{2}{b}^{2}c+3\,{a}^{3}{d}^{2}\ln \left ( 1/2\,{\frac{2\,bd{x}^{2}+2\,\sqrt{ \left ( b{x}^{2}+a \right ) \left ( d{x}^{2}+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) -2\,\ln \left ( 1/2\,{\frac{2\,bd{x}^{2}+2\,\sqrt{ \left ( b{x}^{2}+a \right ) \left ( d{x}^{2}+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){a}^{2}bcd-\ln \left ({\frac{1}{2} \left ( 2\,bd{x}^{2}+2\,\sqrt{ \left ( b{x}^{2}+a \right ) \left ( d{x}^{2}+c \right ) }\sqrt{bd}+ad+bc \right ){\frac{1}{\sqrt{bd}}}} \right ) a{b}^{2}{c}^{2}-6\,{a}^{2}\sqrt{ \left ( b{x}^{2}+a \right ) \left ( d{x}^{2}+c \right ) }\sqrt{bd}d+2\,\sqrt{bd}\sqrt{ \left ( b{x}^{2}+a \right ) \left ( d{x}^{2}+c \right ) }abc \right ) \sqrt{d{x}^{2}+c}{\frac{1}{\sqrt{b{x}^{2}+a}}}{\frac{1}{\sqrt{bd}}}{\frac{1}{\sqrt{ \left ( b{x}^{2}+a \right ) \left ( d{x}^{2}+c \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5/(b*x^2+a)^(3/2)/(d*x^2+c)^(1/2),x)

[Out]

-1/4*(3*ln(1/2*(2*b*d*x^2+2*((b*x^2+a)*(d*x^2+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*
d)^(1/2))*x^2*a^2*b*d^2-2*ln(1/2*(2*b*d*x^2+2*((b*x^2+a)*(d*x^2+c))^(1/2)*(b*d)^
(1/2)+a*d+b*c)/(b*d)^(1/2))*x^2*a*b^2*c*d-ln(1/2*(2*b*d*x^2+2*((b*x^2+a)*(d*x^2+
c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*x^2*b^3*c^2-2*(b*d)^(1/2)*((b*x^2+a)
*(d*x^2+c))^(1/2)*x^2*a*b*d+2*(b*d)^(1/2)*((b*x^2+a)*(d*x^2+c))^(1/2)*x^2*b^2*c+
3*a^3*d^2*ln(1/2*(2*b*d*x^2+2*((b*x^2+a)*(d*x^2+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(
b*d)^(1/2))-2*ln(1/2*(2*b*d*x^2+2*((b*x^2+a)*(d*x^2+c))^(1/2)*(b*d)^(1/2)+a*d+b*
c)/(b*d)^(1/2))*a^2*b*c*d-ln(1/2*(2*b*d*x^2+2*((b*x^2+a)*(d*x^2+c))^(1/2)*(b*d)^
(1/2)+a*d+b*c)/(b*d)^(1/2))*a*b^2*c^2-6*a^2*((b*x^2+a)*(d*x^2+c))^(1/2)*(b*d)^(1
/2)*d+2*(b*d)^(1/2)*((b*x^2+a)*(d*x^2+c))^(1/2)*a*b*c)/b^2*(d*x^2+c)^(1/2)/(b*x^
2+a)^(1/2)/d/(b*d)^(1/2)/(a*d-b*c)/((b*x^2+a)*(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(x^5/((b*x^2 + a)^(3/2)*sqrt(d*x^2 + c)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.377386, size = 1, normalized size = 0.01 \[ \left [\frac{4 \,{\left (a b c - 3 \, a^{2} d +{\left (b^{2} c - a b d\right )} x^{2}\right )} \sqrt{b x^{2} + a} \sqrt{d x^{2} + c} \sqrt{b d} +{\left (a b^{2} c^{2} + 2 \, a^{2} b c d - 3 \, a^{3} d^{2} +{\left (b^{3} c^{2} + 2 \, a b^{2} c d - 3 \, a^{2} b d^{2}\right )} x^{2}\right )} \log \left (-4 \,{\left (2 \, b^{2} d^{2} x^{2} + b^{2} c d + a b d^{2}\right )} \sqrt{b x^{2} + a} \sqrt{d x^{2} + c} +{\left (8 \, b^{2} d^{2} x^{4} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 8 \,{\left (b^{2} c d + a b d^{2}\right )} x^{2}\right )} \sqrt{b d}\right )}{8 \,{\left (a b^{3} c d - a^{2} b^{2} d^{2} +{\left (b^{4} c d - a b^{3} d^{2}\right )} x^{2}\right )} \sqrt{b d}}, \frac{2 \,{\left (a b c - 3 \, a^{2} d +{\left (b^{2} c - a b d\right )} x^{2}\right )} \sqrt{b x^{2} + a} \sqrt{d x^{2} + c} \sqrt{-b d} -{\left (a b^{2} c^{2} + 2 \, a^{2} b c d - 3 \, a^{3} d^{2} +{\left (b^{3} c^{2} + 2 \, a b^{2} c d - 3 \, a^{2} b d^{2}\right )} x^{2}\right )} \arctan \left (\frac{{\left (2 \, b d x^{2} + b c + a d\right )} \sqrt{-b d}}{2 \, \sqrt{b x^{2} + a} \sqrt{d x^{2} + c} b d}\right )}{4 \,{\left (a b^{3} c d - a^{2} b^{2} d^{2} +{\left (b^{4} c d - a b^{3} d^{2}\right )} x^{2}\right )} \sqrt{-b d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/((b*x^2 + a)^(3/2)*sqrt(d*x^2 + c)),x, algorithm="fricas")

[Out]

[1/8*(4*(a*b*c - 3*a^2*d + (b^2*c - a*b*d)*x^2)*sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*
sqrt(b*d) + (a*b^2*c^2 + 2*a^2*b*c*d - 3*a^3*d^2 + (b^3*c^2 + 2*a*b^2*c*d - 3*a^
2*b*d^2)*x^2)*log(-4*(2*b^2*d^2*x^2 + b^2*c*d + a*b*d^2)*sqrt(b*x^2 + a)*sqrt(d*
x^2 + c) + (8*b^2*d^2*x^4 + b^2*c^2 + 6*a*b*c*d + a^2*d^2 + 8*(b^2*c*d + a*b*d^2
)*x^2)*sqrt(b*d)))/((a*b^3*c*d - a^2*b^2*d^2 + (b^4*c*d - a*b^3*d^2)*x^2)*sqrt(b
*d)), 1/4*(2*(a*b*c - 3*a^2*d + (b^2*c - a*b*d)*x^2)*sqrt(b*x^2 + a)*sqrt(d*x^2
+ c)*sqrt(-b*d) - (a*b^2*c^2 + 2*a^2*b*c*d - 3*a^3*d^2 + (b^3*c^2 + 2*a*b^2*c*d
- 3*a^2*b*d^2)*x^2)*arctan(1/2*(2*b*d*x^2 + b*c + a*d)*sqrt(-b*d)/(sqrt(b*x^2 +
a)*sqrt(d*x^2 + c)*b*d)))/((a*b^3*c*d - a^2*b^2*d^2 + (b^4*c*d - a*b^3*d^2)*x^2)
*sqrt(-b*d))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{5}}{\left (a + b x^{2}\right )^{\frac{3}{2}} \sqrt{c + d x^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5/(b*x**2+a)**(3/2)/(d*x**2+c)**(1/2),x)

[Out]

Integral(x**5/((a + b*x**2)**(3/2)*sqrt(c + d*x**2)), x)

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GIAC/XCAS [A]  time = 0.559617, size = 4, normalized size = 0.03 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/((b*x^2 + a)^(3/2)*sqrt(d*x^2 + c)),x, algorithm="giac")

[Out]

sage0*x