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

Optimal. Leaf size=133 \[ -\frac{2 a^2 \sqrt{c+d x^n}}{b^2 n (b c-a d) \sqrt{a+b x^n}}-\frac{(3 a d+b c) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x^n}}{\sqrt{b} \sqrt{c+d x^n}}\right )}{b^{5/2} d^{3/2} n}+\frac{\sqrt{a+b x^n} \sqrt{c+d x^n}}{b^2 d n} \]

[Out]

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

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Rubi [A]  time = 0.452403, antiderivative size = 133, 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 \[ -\frac{2 a^2 \sqrt{c+d x^n}}{b^2 n (b c-a d) \sqrt{a+b x^n}}-\frac{(3 a d+b c) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x^n}}{\sqrt{b} \sqrt{c+d x^n}}\right )}{b^{5/2} d^{3/2} n}+\frac{\sqrt{a+b x^n} \sqrt{c+d x^n}}{b^2 d n} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 34.6073, size = 116, normalized size = 0.87 \[ \frac{2 a^{2} \sqrt{c + d x^{n}}}{b^{2} n \sqrt{a + b x^{n}} \left (a d - b c\right )} + \frac{\sqrt{a + b x^{n}} \sqrt{c + d x^{n}}}{b^{2} d n} - \frac{\left (3 a d + b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{c + d x^{n}}}{\sqrt{d} \sqrt{a + b x^{n}}} \right )}}{b^{\frac{5}{2}} d^{\frac{3}{2}} n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.414949, size = 132, normalized size = 0.99 \[ \frac{\sqrt{a+b x^n} \sqrt{c+d x^n} \left (\frac{2 a^2}{(a d-b c) \left (a+b x^n\right )}+\frac{1}{d}\right )}{b^2 n}-\frac{(3 a d+b c) \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x^n} \sqrt{c+d x^n}+a d+b c+2 b d x^n\right )}{2 b^{5/2} d^{3/2} n} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.085, size = 0, normalized size = 0. \[ \int{{x}^{-1+3\,n} \left ( a+b{x}^{n} \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{c+d{x}^{n}}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.436908, size = 1, normalized size = 0.01 \[ \left [\frac{4 \,{\left ({\left (b^{2} c - a b d\right )} \sqrt{b d} x^{n} +{\left (a b c - 3 \, a^{2} d\right )} \sqrt{b d}\right )} \sqrt{b x^{n} + a} \sqrt{d x^{n} + c} +{\left (a b^{2} c^{2} + 2 \, a^{2} b c d - 3 \, a^{3} d^{2} +{\left (b^{3} c^{2} + 2 \, a b^{2} c d - 3 \, a^{2} b d^{2}\right )} x^{n}\right )} \log \left (8 \, \sqrt{b d} b^{2} d^{2} x^{2 \, n} + 8 \,{\left (b^{2} c d + a b d^{2}\right )} \sqrt{b d} x^{n} - 4 \,{\left (2 \, b^{2} d^{2} x^{n} + b^{2} c d + a b d^{2}\right )} \sqrt{b x^{n} + a} \sqrt{d x^{n} + c} +{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} \sqrt{b d}\right )}{4 \,{\left ({\left (b^{4} c d - a b^{3} d^{2}\right )} \sqrt{b d} n x^{n} +{\left (a b^{3} c d - a^{2} b^{2} d^{2}\right )} \sqrt{b d} n\right )}}, \frac{2 \,{\left ({\left (b^{2} c - a b d\right )} \sqrt{-b d} x^{n} +{\left (a b c - 3 \, a^{2} d\right )} \sqrt{-b d}\right )} \sqrt{b x^{n} + a} \sqrt{d x^{n} + c} -{\left (a b^{2} c^{2} + 2 \, a^{2} b c d - 3 \, a^{3} d^{2} +{\left (b^{3} c^{2} + 2 \, a b^{2} c d - 3 \, a^{2} b d^{2}\right )} x^{n}\right )} \arctan \left (\frac{2 \, \sqrt{-b d} b d x^{n} +{\left (b c + a d\right )} \sqrt{-b d}}{2 \, \sqrt{b x^{n} + a} \sqrt{d x^{n} + c} b d}\right )}{2 \,{\left ({\left (b^{4} c d - a b^{3} d^{2}\right )} \sqrt{-b d} n x^{n} +{\left (a b^{3} c d - a^{2} b^{2} d^{2}\right )} \sqrt{-b d} n\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/4*(4*((b^2*c - a*b*d)*sqrt(b*d)*x^n + (a*b*c - 3*a^2*d)*sqrt(b*d))*sqrt(b*x^n
+ a)*sqrt(d*x^n + c) + (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^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^4*c*d - a*b^3*d^2)*sqr
t(b*d)*n*x^n + (a*b^3*c*d - a^2*b^2*d^2)*sqrt(b*d)*n), 1/2*(2*((b^2*c - a*b*d)*s
qrt(-b*d)*x^n + (a*b*c - 3*a^2*d)*sqrt(-b*d))*sqrt(b*x^n + a)*sqrt(d*x^n + c) -
(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^n
)*arctan(1/2*(2*sqrt(-b*d)*b*d*x^n + (b*c + a*d)*sqrt(-b*d))/(sqrt(b*x^n + a)*sq
rt(d*x^n + c)*b*d)))/((b^4*c*d - a*b^3*d^2)*sqrt(-b*d)*n*x^n + (a*b^3*c*d - a^2*
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)**(3/2)/(c+d*x**n)**(1/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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