Optimal. Leaf size=150 \[ -\frac{3 (a d+b c) \sqrt{a+b x^n} \sqrt{c+d x^n}}{4 b^2 d^2 n}-\frac{\left (4 a b c d-3 (a d+b c)^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x^n}}{\sqrt{b} \sqrt{c+d x^n}}\right )}{4 b^{5/2} d^{5/2} n}+\frac{x^n \sqrt{a+b x^n} \sqrt{c+d x^n}}{2 b d n} \]
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Rubi [A] time = 0.151673, antiderivative size = 150, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {446, 90, 80, 63, 217, 206} \[ -\frac{3 (a d+b c) \sqrt{a+b x^n} \sqrt{c+d x^n}}{4 b^2 d^2 n}-\frac{\left (4 a b c d-3 (a d+b c)^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x^n}}{\sqrt{b} \sqrt{c+d x^n}}\right )}{4 b^{5/2} d^{5/2} n}+\frac{x^n \sqrt{a+b x^n} \sqrt{c+d x^n}}{2 b d n} \]
Antiderivative was successfully verified.
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Rule 446
Rule 90
Rule 80
Rule 63
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \frac{x^{-1+3 n}}{\sqrt{a+b x^n} \sqrt{c+d x^n}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^2}{\sqrt{a+b x} \sqrt{c+d x}} \, dx,x,x^n\right )}{n}\\ &=\frac{x^n \sqrt{a+b x^n} \sqrt{c+d x^n}}{2 b d n}+\frac{\operatorname{Subst}\left (\int \frac{-a c-\frac{3}{2} (b c+a d) x}{\sqrt{a+b x} \sqrt{c+d x}} \, dx,x,x^n\right )}{2 b d n}\\ &=-\frac{3 (b c+a d) \sqrt{a+b x^n} \sqrt{c+d x^n}}{4 b^2 d^2 n}+\frac{x^n \sqrt{a+b x^n} \sqrt{c+d x^n}}{2 b d n}-\frac{\left (4 a b c d-3 (b c+a d)^2\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x} \sqrt{c+d x}} \, dx,x,x^n\right )}{8 b^2 d^2 n}\\ &=-\frac{3 (b c+a d) \sqrt{a+b x^n} \sqrt{c+d x^n}}{4 b^2 d^2 n}+\frac{x^n \sqrt{a+b x^n} \sqrt{c+d x^n}}{2 b d n}-\frac{\left (4 a b c d-3 (b c+a d)^2\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{c-\frac{a d}{b}+\frac{d x^2}{b}}} \, dx,x,\sqrt{a+b x^n}\right )}{4 b^3 d^2 n}\\ &=-\frac{3 (b c+a d) \sqrt{a+b x^n} \sqrt{c+d x^n}}{4 b^2 d^2 n}+\frac{x^n \sqrt{a+b x^n} \sqrt{c+d x^n}}{2 b d n}-\frac{\left (4 a b c d-3 (b c+a d)^2\right ) \operatorname{Subst}\left (\int \frac{1}{1-\frac{d x^2}{b}} \, dx,x,\frac{\sqrt{a+b x^n}}{\sqrt{c+d x^n}}\right )}{4 b^3 d^2 n}\\ &=-\frac{3 (b c+a d) \sqrt{a+b x^n} \sqrt{c+d x^n}}{4 b^2 d^2 n}+\frac{x^n \sqrt{a+b x^n} \sqrt{c+d x^n}}{2 b d n}-\frac{\left (4 a b c d-3 (b c+a d)^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x^n}}{\sqrt{b} \sqrt{c+d x^n}}\right )}{4 b^{5/2} d^{5/2} n}\\ \end{align*}
Mathematica [A] time = 0.304349, size = 157, normalized size = 1.05 \[ \frac{\sqrt{b c-a d} \left (3 a^2 d^2+2 a b c d+3 b^2 c^2\right ) \sqrt{\frac{b \left (c+d x^n\right )}{b c-a d}} \sinh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x^n}}{\sqrt{b c-a d}}\right )+b \sqrt{d} \sqrt{a+b x^n} \left (c+d x^n\right ) \left (-3 a d-3 b c+2 b d x^n\right )}{4 b^3 d^{5/2} n \sqrt{c+d x^n}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.062, size = 0, normalized size = 0. \begin{align*} \int{{x}^{-1+3\,n}{\frac{1}{\sqrt{a+b{x}^{n}}}}{\frac{1}{\sqrt{c+d{x}^{n}}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3 \, n - 1}}{\sqrt{b x^{n} + a} \sqrt{d x^{n} + c}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.16984, size = 810, normalized size = 5.4 \begin{align*} \left [\frac{{\left (3 \, b^{2} c^{2} + 2 \, a b c d + 3 \, a^{2} d^{2}\right )} \sqrt{b d} \log \left (8 \, b^{2} d^{2} x^{2 \, n} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \,{\left (2 \, \sqrt{b d} b d x^{n} +{\left (b c + a d\right )} \sqrt{b d}\right )} \sqrt{b x^{n} + a} \sqrt{d x^{n} + c} + 8 \,{\left (b^{2} c d + a b d^{2}\right )} x^{n}\right ) + 4 \,{\left (2 \, b^{2} d^{2} x^{n} - 3 \, b^{2} c d - 3 \, a b d^{2}\right )} \sqrt{b x^{n} + a} \sqrt{d x^{n} + c}}{16 \, b^{3} d^{3} n}, -\frac{{\left (3 \, b^{2} c^{2} + 2 \, a b c d + 3 \, a^{2} d^{2}\right )} \sqrt{-b d} \arctan \left (\frac{{\left (2 \, \sqrt{-b d} b d x^{n} +{\left (b c + a d\right )} \sqrt{-b d}\right )} \sqrt{b x^{n} + a} \sqrt{d x^{n} + c}}{2 \,{\left (b^{2} d^{2} x^{2 \, n} + a b c d +{\left (b^{2} c d + a b d^{2}\right )} x^{n}\right )}}\right ) - 2 \,{\left (2 \, b^{2} d^{2} x^{n} - 3 \, b^{2} c d - 3 \, a b d^{2}\right )} \sqrt{b x^{n} + a} \sqrt{d x^{n} + c}}{8 \, b^{3} d^{3} n}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3 \, n - 1}}{\sqrt{b x^{n} + a} \sqrt{d x^{n} + c}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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