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

Optimal. Leaf size=199 \[ -\frac{(b c-a d)^2 (a d+5 b c) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x^n}}{\sqrt{b} \sqrt{c+d x^n}}\right )}{8 b^{3/2} d^{7/2} n}+\frac{(b c-a d) (a d+5 b c) \sqrt{a+b x^n} \sqrt{c+d x^n}}{8 b d^3 n}-\frac{(a d+5 b c) \left (a+b x^n\right )^{3/2} \sqrt{c+d x^n}}{12 b d^2 n}+\frac{\left (a+b x^n\right )^{5/2} \sqrt{c+d x^n}}{3 b d n} \]

[Out]

((b*c - a*d)*(5*b*c + a*d)*Sqrt[a + b*x^n]*Sqrt[c + d*x^n])/(8*b*d^3*n) - ((5*b*
c + a*d)*(a + b*x^n)^(3/2)*Sqrt[c + d*x^n])/(12*b*d^2*n) + ((a + b*x^n)^(5/2)*Sq
rt[c + d*x^n])/(3*b*d*n) - ((b*c - a*d)^2*(5*b*c + a*d)*ArcTanh[(Sqrt[d]*Sqrt[a
+ b*x^n])/(Sqrt[b]*Sqrt[c + d*x^n])])/(8*b^(3/2)*d^(7/2)*n)

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Rubi [A]  time = 0.445682, antiderivative size = 199, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{(b c-a d)^2 (a d+5 b c) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x^n}}{\sqrt{b} \sqrt{c+d x^n}}\right )}{8 b^{3/2} d^{7/2} n}+\frac{(b c-a d) (a d+5 b c) \sqrt{a+b x^n} \sqrt{c+d x^n}}{8 b d^3 n}-\frac{(a d+5 b c) \left (a+b x^n\right )^{3/2} \sqrt{c+d x^n}}{12 b d^2 n}+\frac{\left (a+b x^n\right )^{5/2} \sqrt{c+d x^n}}{3 b d n} \]

Antiderivative was successfully verified.

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

[Out]

((b*c - a*d)*(5*b*c + a*d)*Sqrt[a + b*x^n]*Sqrt[c + d*x^n])/(8*b*d^3*n) - ((5*b*
c + a*d)*(a + b*x^n)^(3/2)*Sqrt[c + d*x^n])/(12*b*d^2*n) + ((a + b*x^n)^(5/2)*Sq
rt[c + d*x^n])/(3*b*d*n) - ((b*c - a*d)^2*(5*b*c + a*d)*ArcTanh[(Sqrt[d]*Sqrt[a
+ b*x^n])/(Sqrt[b]*Sqrt[c + d*x^n])])/(8*b^(3/2)*d^(7/2)*n)

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Rubi in Sympy [A]  time = 38.3812, size = 170, normalized size = 0.85 \[ \frac{\left (a + b x^{n}\right )^{\frac{5}{2}} \sqrt{c + d x^{n}}}{3 b d n} - \frac{\left (a + b x^{n}\right )^{\frac{3}{2}} \sqrt{c + d x^{n}} \left (a d + 5 b c\right )}{12 b d^{2} n} - \frac{\sqrt{a + b x^{n}} \sqrt{c + d x^{n}} \left (a d - b c\right ) \left (a d + 5 b c\right )}{8 b d^{3} n} - \frac{\left (a d - b c\right )^{2} \left (a d + 5 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x^{n}}}{\sqrt{b} \sqrt{c + d x^{n}}} \right )}}{8 b^{\frac{3}{2}} d^{\frac{7}{2}} n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(a + b*x**n)**(5/2)*sqrt(c + d*x**n)/(3*b*d*n) - (a + b*x**n)**(3/2)*sqrt(c + d*
x**n)*(a*d + 5*b*c)/(12*b*d**2*n) - sqrt(a + b*x**n)*sqrt(c + d*x**n)*(a*d - b*c
)*(a*d + 5*b*c)/(8*b*d**3*n) - (a*d - b*c)**2*(a*d + 5*b*c)*atanh(sqrt(d)*sqrt(a
+ b*x**n)/(sqrt(b)*sqrt(c + d*x**n)))/(8*b**(3/2)*d**(7/2)*n)

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Mathematica [A]  time = 0.341184, size = 172, normalized size = 0.86 \[ \frac{2 \sqrt{b} \sqrt{d} \sqrt{a+b x^n} \sqrt{c+d x^n} \left (3 a^2 d^2+2 a b d \left (7 d x^n-11 c\right )+b^2 \left (15 c^2-10 c d x^n+8 d^2 x^{2 n}\right )\right )-3 (b c-a d)^2 (a d+5 b c) \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x^n} \sqrt{c+d x^n}+a d+b c+2 b d x^n\right )}{48 b^{3/2} d^{7/2} n} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[b]*Sqrt[d]*Sqrt[a + b*x^n]*Sqrt[c + d*x^n]*(3*a^2*d^2 + 2*a*b*d*(-11*c +
7*d*x^n) + b^2*(15*c^2 - 10*c*d*x^n + 8*d^2*x^(2*n))) - 3*(b*c - a*d)^2*(5*b*c
+ 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]])/(48*b^(3/2)*d^(7/2)*n)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

int(x^(-1+2*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. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.374091, size = 1, normalized size = 0.01 \[ \left [\frac{4 \,{\left (8 \, \sqrt{b d} b^{2} d^{2} x^{2 \, n} - 2 \,{\left (5 \, b^{2} c d - 7 \, a b d^{2}\right )} \sqrt{b d} x^{n} +{\left (15 \, b^{2} c^{2} - 22 \, a b c d + 3 \, a^{2} d^{2}\right )} \sqrt{b d}\right )} \sqrt{b x^{n} + a} \sqrt{d x^{n} + c} + 3 \,{\left (5 \, b^{3} c^{3} - 9 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} \log \left (8 \, \sqrt{b d} b^{2} d^{2} x^{2 \, n} + 8 \,{\left (b^{2} c d + a b d^{2}\right )} \sqrt{b d} x^{n} - 4 \,{\left (2 \, b^{2} d^{2} x^{n} + b^{2} c d + a b d^{2}\right )} \sqrt{b x^{n} + a} \sqrt{d x^{n} + c} +{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} \sqrt{b d}\right )}{96 \, \sqrt{b d} b d^{3} n}, \frac{2 \,{\left (8 \, \sqrt{-b d} b^{2} d^{2} x^{2 \, n} - 2 \,{\left (5 \, b^{2} c d - 7 \, a b d^{2}\right )} \sqrt{-b d} x^{n} +{\left (15 \, b^{2} c^{2} - 22 \, a b c d + 3 \, a^{2} d^{2}\right )} \sqrt{-b d}\right )} \sqrt{b x^{n} + a} \sqrt{d x^{n} + c} - 3 \,{\left (5 \, b^{3} c^{3} - 9 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} \arctan \left (\frac{2 \, \sqrt{-b d} b d x^{n} +{\left (b c + a d\right )} \sqrt{-b d}}{2 \, \sqrt{b x^{n} + a} \sqrt{d x^{n} + c} b d}\right )}{48 \, \sqrt{-b d} b d^{3} n}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/96*(4*(8*sqrt(b*d)*b^2*d^2*x^(2*n) - 2*(5*b^2*c*d - 7*a*b*d^2)*sqrt(b*d)*x^n
+ (15*b^2*c^2 - 22*a*b*c*d + 3*a^2*d^2)*sqrt(b*d))*sqrt(b*x^n + a)*sqrt(d*x^n +
c) + 3*(5*b^3*c^3 - 9*a*b^2*c^2*d + 3*a^2*b*c*d^2 + a^3*d^3)*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)*sqr
t(b*d)))/(sqrt(b*d)*b*d^3*n), 1/48*(2*(8*sqrt(-b*d)*b^2*d^2*x^(2*n) - 2*(5*b^2*c
*d - 7*a*b*d^2)*sqrt(-b*d)*x^n + (15*b^2*c^2 - 22*a*b*c*d + 3*a^2*d^2)*sqrt(-b*d
))*sqrt(b*x^n + a)*sqrt(d*x^n + c) - 3*(5*b^3*c^3 - 9*a*b^2*c^2*d + 3*a^2*b*c*d^
2 + a^3*d^3)*arctan(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)))/(sqrt(-b*d)*b*d^3*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+2*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. \[ \int \frac{{\left (b x^{n} + a\right )}^{\frac{3}{2}} x^{2 \, n - 1}}{\sqrt{d x^{n} + c}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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