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

Optimal. Leaf size=137 \[ -\frac{a^2 \sqrt{c+d x^2}}{3 b^2 \left (a+b x^2\right )^{3/2} (b c-a d)}+\frac{\tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x^2}}{\sqrt{b} \sqrt{c+d x^2}}\right )}{b^{5/2} \sqrt{d}}+\frac{2 a \sqrt{c+d x^2} (3 b c-2 a d)}{3 b^2 \sqrt{a+b x^2} (b c-a d)^2} \]

[Out]

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

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Rubi [A]  time = 0.37428, antiderivative size = 137, 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 \[ -\frac{a^2 \sqrt{c+d x^2}}{3 b^2 \left (a+b x^2\right )^{3/2} (b c-a d)}+\frac{\tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x^2}}{\sqrt{b} \sqrt{c+d x^2}}\right )}{b^{5/2} \sqrt{d}}+\frac{2 a \sqrt{c+d x^2} (3 b c-2 a d)}{3 b^2 \sqrt{a+b x^2} (b c-a d)^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 38.348, size = 124, normalized size = 0.91 \[ \frac{a^{2} \sqrt{c + d x^{2}}}{3 b^{2} \left (a + b x^{2}\right )^{\frac{3}{2}} \left (a d - b c\right )} - \frac{2 a \sqrt{c + d x^{2}} \left (2 a d - 3 b c\right )}{3 b^{2} \sqrt{a + b x^{2}} \left (a d - b c\right )^{2}} + \frac{\operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x^{2}}}{\sqrt{b} \sqrt{c + d x^{2}}} \right )}}{b^{\frac{5}{2}} \sqrt{d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.328669, size = 133, normalized size = 0.97 \[ \frac{a \sqrt{c+d x^2} \left (-3 a^2 d+a b \left (5 c-4 d x^2\right )+6 b^2 c x^2\right )}{3 b^2 \left (a+b x^2\right )^{3/2} (b c-a d)^2}+\frac{\log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x^2} \sqrt{c+d x^2}+a d+b c+2 b d x^2\right )}{2 b^{5/2} \sqrt{d}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.054, size = 609, normalized size = 4.5 \[{\frac{1}{6\,{b}^{2} \left ( ad-bc \right ) ^{2} \left ( bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac \right ) } \left ( -8\,\sqrt{bd}{x}^{4}{a}^{2}b{d}^{2}+12\,{x}^{4}a{b}^{2}cd\sqrt{bd}+3\,\ln \left ( 1/2\,{\frac{2\,bd{x}^{2}+2\,\sqrt{ \left ( b{x}^{2}+a \right ) \left ( d{x}^{2}+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) \sqrt{ \left ( b{x}^{2}+a \right ) \left ( d{x}^{2}+c \right ) }{x}^{2}{a}^{2}b{d}^{2}-6\,\ln \left ( 1/2\,{\frac{2\,bd{x}^{2}+2\,\sqrt{ \left ( b{x}^{2}+a \right ) \left ( d{x}^{2}+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) \sqrt{ \left ( b{x}^{2}+a \right ) \left ( d{x}^{2}+c \right ) }{x}^{2}a{b}^{2}cd+3\,\ln \left ( 1/2\,{\frac{2\,bd{x}^{2}+2\,\sqrt{ \left ( b{x}^{2}+a \right ) \left ( d{x}^{2}+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) \sqrt{ \left ( b{x}^{2}+a \right ) \left ( d{x}^{2}+c \right ) }{x}^{2}{b}^{3}{c}^{2}-6\,{x}^{2}{a}^{3}{d}^{2}\sqrt{bd}+2\,\sqrt{bd}{x}^{2}{a}^{2}bcd+12\,{x}^{2}a{b}^{2}{c}^{2}\sqrt{bd}+3\,\ln \left ( 1/2\,{\frac{2\,bd{x}^{2}+2\,\sqrt{ \left ( b{x}^{2}+a \right ) \left ( d{x}^{2}+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) \sqrt{ \left ( b{x}^{2}+a \right ) \left ( d{x}^{2}+c \right ) }{a}^{3}{d}^{2}-6\,\ln \left ( 1/2\,{\frac{2\,bd{x}^{2}+2\,\sqrt{ \left ( b{x}^{2}+a \right ) \left ( d{x}^{2}+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) \sqrt{ \left ( b{x}^{2}+a \right ) \left ( d{x}^{2}+c \right ) }{a}^{2}bcd+3\,\ln \left ( 1/2\,{\frac{2\,bd{x}^{2}+2\,\sqrt{ \left ( b{x}^{2}+a \right ) \left ( d{x}^{2}+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) \sqrt{ \left ( b{x}^{2}+a \right ) \left ( d{x}^{2}+c \right ) }a{b}^{2}{c}^{2}-6\,{a}^{3}cd\sqrt{bd}+10\,\sqrt{bd}{a}^{2}b{c}^{2} \right ) \sqrt{d{x}^{2}+c}{\frac{1}{\sqrt{b{x}^{2}+a}}}{\frac{1}{\sqrt{bd}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*(-8*(b*d)^(1/2)*x^4*a^2*b*d^2+12*x^4*a*b^2*c*d*(b*d)^(1/2)+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))*((b*x^2+a)*(d
*x^2+c))^(1/2)*x^2*a^2*b*d^2-6*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))*((b*x^2+a)*(d*x^2+c))^(1/2)*x^2*a*b^2*c*d+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))*(
(b*x^2+a)*(d*x^2+c))^(1/2)*x^2*b^3*c^2-6*x^2*a^3*d^2*(b*d)^(1/2)+2*(b*d)^(1/2)*x
^2*a^2*b*c*d+12*x^2*a*b^2*c^2*(b*d)^(1/2)+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))*((b*x^2+a)*(d*x^2+c))^(1/2)*a^3*d^
2-6*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))*((b*x^2+a)*(d*x^2+c))^(1/2)*a^2*b*c*d+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))*((b*x^2+a)*(d*x^2+c))^(1/2)*a*b^
2*c^2-6*a^3*c*d*(b*d)^(1/2)+10*(b*d)^(1/2)*a^2*b*c^2)*(d*x^2+c)^(1/2)/b^2/(b*x^2
+a)^(1/2)/(b*d)^(1/2)/(a*d-b*c)^2/(b*d*x^4+a*d*x^2+b*c*x^2+a*c)

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.401892, size = 1, normalized size = 0.01 \[ \left [\frac{4 \,{\left (5 \, a^{2} b c - 3 \, a^{3} d + 2 \,{\left (3 \, a b^{2} c - 2 \, a^{2} b d\right )} x^{2}\right )} \sqrt{b x^{2} + a} \sqrt{d x^{2} + c} \sqrt{b d} + 3 \,{\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2} +{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} x^{4} + 2 \,{\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} x^{2}\right )} \log \left (4 \,{\left (2 \, b^{2} d^{2} x^{2} + b^{2} c d + a b d^{2}\right )} \sqrt{b x^{2} + a} \sqrt{d x^{2} + c} +{\left (8 \, b^{2} d^{2} x^{4} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 8 \,{\left (b^{2} c d + a b d^{2}\right )} x^{2}\right )} \sqrt{b d}\right )}{12 \,{\left (a^{2} b^{4} c^{2} - 2 \, a^{3} b^{3} c d + a^{4} b^{2} d^{2} +{\left (b^{6} c^{2} - 2 \, a b^{5} c d + a^{2} b^{4} d^{2}\right )} x^{4} + 2 \,{\left (a b^{5} c^{2} - 2 \, a^{2} b^{4} c d + a^{3} b^{3} d^{2}\right )} x^{2}\right )} \sqrt{b d}}, \frac{2 \,{\left (5 \, a^{2} b c - 3 \, a^{3} d + 2 \,{\left (3 \, a b^{2} c - 2 \, a^{2} b d\right )} x^{2}\right )} \sqrt{b x^{2} + a} \sqrt{d x^{2} + c} \sqrt{-b d} + 3 \,{\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2} +{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} x^{4} + 2 \,{\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} x^{2}\right )} \arctan \left (\frac{{\left (2 \, b d x^{2} + b c + a d\right )} \sqrt{-b d}}{2 \, \sqrt{b x^{2} + a} \sqrt{d x^{2} + c} b d}\right )}{6 \,{\left (a^{2} b^{4} c^{2} - 2 \, a^{3} b^{3} c d + a^{4} b^{2} d^{2} +{\left (b^{6} c^{2} - 2 \, a b^{5} c d + a^{2} b^{4} d^{2}\right )} x^{4} + 2 \,{\left (a b^{5} c^{2} - 2 \, a^{2} b^{4} c d + a^{3} b^{3} d^{2}\right )} x^{2}\right )} \sqrt{-b d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/12*(4*(5*a^2*b*c - 3*a^3*d + 2*(3*a*b^2*c - 2*a^2*b*d)*x^2)*sqrt(b*x^2 + a)*s
qrt(d*x^2 + c)*sqrt(b*d) + 3*(a^2*b^2*c^2 - 2*a^3*b*c*d + a^4*d^2 + (b^4*c^2 - 2
*a*b^3*c*d + a^2*b^2*d^2)*x^4 + 2*(a*b^3*c^2 - 2*a^2*b^2*c*d + a^3*b*d^2)*x^2)*l
og(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^2*b^4*c^2 - 2*a^3*b^3*c*d + a^4*b^2*d^2 + (b^6*c^2 - 2*a*b^5*c*d + a^2*b^
4*d^2)*x^4 + 2*(a*b^5*c^2 - 2*a^2*b^4*c*d + a^3*b^3*d^2)*x^2)*sqrt(b*d)), 1/6*(2
*(5*a^2*b*c - 3*a^3*d + 2*(3*a*b^2*c - 2*a^2*b*d)*x^2)*sqrt(b*x^2 + a)*sqrt(d*x^
2 + c)*sqrt(-b*d) + 3*(a^2*b^2*c^2 - 2*a^3*b*c*d + a^4*d^2 + (b^4*c^2 - 2*a*b^3*
c*d + a^2*b^2*d^2)*x^4 + 2*(a*b^3*c^2 - 2*a^2*b^2*c*d + a^3*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^2*b^4*c^2 - 2*a^3*b^3*c*d + a^4*b^2*d^2 + (b^6*c^2 - 2*a*b^5*c*d + a^2*b^4*d^2
)*x^4 + 2*(a*b^5*c^2 - 2*a^2*b^4*c*d + a^3*b^3*d^2)*x^2)*sqrt(-b*d))]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.599341, 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)^(5/2)*sqrt(d*x^2 + c)),x, algorithm="giac")

[Out]

sage0*x