3.908 \(\int \frac{x^{-1+3 n}}{\left (a+b x^n\right )^{5/2} \sqrt{c+d x^n}} \, dx\)

Optimal. Leaf size=147 \[ -\frac{2 a^2 \sqrt{c+d x^n}}{3 b^2 n (b c-a d) \left (a+b x^n\right )^{3/2}}+\frac{2 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x^n}}{\sqrt{b} \sqrt{c+d x^n}}\right )}{b^{5/2} \sqrt{d} n}+\frac{4 a (3 b c-2 a d) \sqrt{c+d x^n}}{3 b^2 n (b c-a d)^2 \sqrt{a+b x^n}} \]

[Out]

(-2*a^2*Sqrt[c + d*x^n])/(3*b^2*(b*c - a*d)*n*(a + b*x^n)^(3/2)) + (4*a*(3*b*c -
2*a*d)*Sqrt[c + d*x^n])/(3*b^2*(b*c - a*d)^2*n*Sqrt[a + b*x^n]) + (2*ArcTanh[(S
qrt[d]*Sqrt[a + b*x^n])/(Sqrt[b]*Sqrt[c + d*x^n])])/(b^(5/2)*Sqrt[d]*n)

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Rubi [A]  time = 0.407181, antiderivative size = 147, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{2 a^2 \sqrt{c+d x^n}}{3 b^2 n (b c-a d) \left (a+b x^n\right )^{3/2}}+\frac{2 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x^n}}{\sqrt{b} \sqrt{c+d x^n}}\right )}{b^{5/2} \sqrt{d} n}+\frac{4 a (3 b c-2 a d) \sqrt{c+d x^n}}{3 b^2 n (b c-a d)^2 \sqrt{a+b x^n}} \]

Antiderivative was successfully verified.

[In]  Int[x^(-1 + 3*n)/((a + b*x^n)^(5/2)*Sqrt[c + d*x^n]),x]

[Out]

(-2*a^2*Sqrt[c + d*x^n])/(3*b^2*(b*c - a*d)*n*(a + b*x^n)^(3/2)) + (4*a*(3*b*c -
2*a*d)*Sqrt[c + d*x^n])/(3*b^2*(b*c - a*d)^2*n*Sqrt[a + b*x^n]) + (2*ArcTanh[(S
qrt[d]*Sqrt[a + b*x^n])/(Sqrt[b]*Sqrt[c + d*x^n])])/(b^(5/2)*Sqrt[d]*n)

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Rubi in Sympy [A]  time = 36.4581, size = 133, normalized size = 0.9 \[ \frac{2 a^{2} \sqrt{c + d x^{n}}}{3 b^{2} n \left (a + b x^{n}\right )^{\frac{3}{2}} \left (a d - b c\right )} - \frac{4 a \sqrt{c + d x^{n}} \left (2 a d - 3 b c\right )}{3 b^{2} n \sqrt{a + b x^{n}} \left (a d - b c\right )^{2}} + \frac{2 \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x^{n}}}{\sqrt{b} \sqrt{c + d x^{n}}} \right )}}{b^{\frac{5}{2}} \sqrt{d} n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*a**2*sqrt(c + d*x**n)/(3*b**2*n*(a + b*x**n)**(3/2)*(a*d - b*c)) - 4*a*sqrt(c
+ d*x**n)*(2*a*d - 3*b*c)/(3*b**2*n*sqrt(a + b*x**n)*(a*d - b*c)**2) + 2*atanh(s
qrt(d)*sqrt(a + b*x**n)/(sqrt(b)*sqrt(c + d*x**n)))/(b**(5/2)*sqrt(d)*n)

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Mathematica [A]  time = 0.39106, size = 136, normalized size = 0.93 \[ \frac{2 a \sqrt{c+d x^n} \left (-3 a^2 d+a b \left (5 c-4 d x^n\right )+6 b^2 c x^n\right )}{3 b^2 n (b c-a d)^2 \left (a+b x^n\right )^{3/2}}+\frac{\log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x^n} \sqrt{c+d x^n}+a d+b c+2 b d x^n\right )}{b^{5/2} \sqrt{d} n} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(-1 + 3*n)/((a + b*x^n)^(5/2)*Sqrt[c + d*x^n]),x]

[Out]

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

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Maple [F]  time = 0.096, size = 0, normalized size = 0. \[ \int{{x}^{-1+3\,n} \left ( a+b{x}^{n} \right ) ^{-{\frac{5}{2}}}{\frac{1}{\sqrt{c+d{x}^{n}}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

int(x^(-1+3*n)/(a+b*x^n)^(5/2)/(c+d*x^n)^(1/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{3 \, n - 1}}{{\left (b x^{n} + a\right )}^{\frac{5}{2}} \sqrt{d x^{n} + c}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(3*n - 1)/((b*x^n + a)^(5/2)*sqrt(d*x^n + c)),x, algorithm="maxima")

[Out]

integrate(x^(3*n - 1)/((b*x^n + a)^(5/2)*sqrt(d*x^n + c)), x)

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Fricas [A]  time = 0.47104, size = 1, normalized size = 0.01 \[ \left [\frac{4 \,{\left (2 \,{\left (3 \, a b^{2} c - 2 \, a^{2} b d\right )} \sqrt{b d} x^{n} +{\left (5 \, a^{2} b c - 3 \, a^{3} d\right )} \sqrt{b d}\right )} \sqrt{b x^{n} + a} \sqrt{d x^{n} + c} + 3 \,{\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2} +{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} x^{2 \, n} + 2 \,{\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} x^{n}\right )} \log \left (8 \, \sqrt{b d} b^{2} d^{2} x^{2 \, n} + 8 \,{\left (b^{2} c d + a b d^{2}\right )} \sqrt{b d} x^{n} + 4 \,{\left (2 \, b^{2} d^{2} x^{n} + b^{2} c d + a b d^{2}\right )} \sqrt{b x^{n} + a} \sqrt{d x^{n} + c} +{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} \sqrt{b d}\right )}{6 \,{\left ({\left (b^{6} c^{2} - 2 \, a b^{5} c d + a^{2} b^{4} d^{2}\right )} \sqrt{b d} n x^{2 \, n} + 2 \,{\left (a b^{5} c^{2} - 2 \, a^{2} b^{4} c d + a^{3} b^{3} d^{2}\right )} \sqrt{b d} n x^{n} +{\left (a^{2} b^{4} c^{2} - 2 \, a^{3} b^{3} c d + a^{4} b^{2} d^{2}\right )} \sqrt{b d} n\right )}}, \frac{2 \,{\left (2 \,{\left (3 \, a b^{2} c - 2 \, a^{2} b d\right )} \sqrt{-b d} x^{n} +{\left (5 \, a^{2} b c - 3 \, a^{3} d\right )} \sqrt{-b d}\right )} \sqrt{b x^{n} + a} \sqrt{d x^{n} + c} + 3 \,{\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2} +{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} x^{2 \, n} + 2 \,{\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} x^{n}\right )} \arctan \left (\frac{2 \, \sqrt{-b d} b d x^{n} +{\left (b c + a d\right )} \sqrt{-b d}}{2 \, \sqrt{b x^{n} + a} \sqrt{d x^{n} + c} b d}\right )}{3 \,{\left ({\left (b^{6} c^{2} - 2 \, a b^{5} c d + a^{2} b^{4} d^{2}\right )} \sqrt{-b d} n x^{2 \, n} + 2 \,{\left (a b^{5} c^{2} - 2 \, a^{2} b^{4} c d + a^{3} b^{3} d^{2}\right )} \sqrt{-b d} n x^{n} +{\left (a^{2} b^{4} c^{2} - 2 \, a^{3} b^{3} c d + a^{4} b^{2} d^{2}\right )} \sqrt{-b d} n\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(3*n - 1)/((b*x^n + a)^(5/2)*sqrt(d*x^n + c)),x, algorithm="fricas")

[Out]

[1/6*(4*(2*(3*a*b^2*c - 2*a^2*b*d)*sqrt(b*d)*x^n + (5*a^2*b*c - 3*a^3*d)*sqrt(b*
d))*sqrt(b*x^n + a)*sqrt(d*x^n + c) + 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^(2*n) + 2*(a*b^3*c^2 - 2*a^2*b^2*c*d + a^
3*b*d^2)*x^n)*log(8*sqrt(b*d)*b^2*d^2*x^(2*n) + 8*(b^2*c*d + a*b*d^2)*sqrt(b*d)*
x^n + 4*(2*b^2*d^2*x^n + b^2*c*d + a*b*d^2)*sqrt(b*x^n + a)*sqrt(d*x^n + c) + (b
^2*c^2 + 6*a*b*c*d + a^2*d^2)*sqrt(b*d)))/((b^6*c^2 - 2*a*b^5*c*d + a^2*b^4*d^2)
*sqrt(b*d)*n*x^(2*n) + 2*(a*b^5*c^2 - 2*a^2*b^4*c*d + a^3*b^3*d^2)*sqrt(b*d)*n*x
^n + (a^2*b^4*c^2 - 2*a^3*b^3*c*d + a^4*b^2*d^2)*sqrt(b*d)*n), 1/3*(2*(2*(3*a*b^
2*c - 2*a^2*b*d)*sqrt(-b*d)*x^n + (5*a^2*b*c - 3*a^3*d)*sqrt(-b*d))*sqrt(b*x^n +
a)*sqrt(d*x^n + c) + 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^(2*n) + 2*(a*b^3*c^2 - 2*a^2*b^2*c*d + a^3*b*d^2)*x^n)*ar
ctan(1/2*(2*sqrt(-b*d)*b*d*x^n + (b*c + a*d)*sqrt(-b*d))/(sqrt(b*x^n + a)*sqrt(d
*x^n + c)*b*d)))/((b^6*c^2 - 2*a*b^5*c*d + a^2*b^4*d^2)*sqrt(-b*d)*n*x^(2*n) + 2
*(a*b^5*c^2 - 2*a^2*b^4*c*d + a^3*b^3*d^2)*sqrt(-b*d)*n*x^n + (a^2*b^4*c^2 - 2*a
^3*b^3*c*d + a^4*b^2*d^2)*sqrt(-b*d)*n)]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{3 \, n - 1}}{{\left (b x^{n} + a\right )}^{\frac{5}{2}} \sqrt{d x^{n} + c}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(3*n - 1)/((b*x^n + a)^(5/2)*sqrt(d*x^n + c)),x, algorithm="giac")

[Out]

integrate(x^(3*n - 1)/((b*x^n + a)^(5/2)*sqrt(d*x^n + c)), x)