Optimal. Leaf size=221 \[ \frac{\left (a^2 d^2+2 a b c d+5 b^2 c^2\right ) \sqrt{a+b x^n} \sqrt{c+d x^n}}{8 b^2 d^3 n}-\frac{(b c-a d) \left (a^2 d^2+2 a b c d+5 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x^n}}{\sqrt{b} \sqrt{c+d x^n}}\right )}{8 b^{5/2} d^{7/2} n}-\frac{(3 a d+5 b c) \left (a+b x^n\right )^{3/2} \sqrt{c+d x^n}}{12 b^2 d^2 n}+\frac{x^n \left (a+b x^n\right )^{3/2} \sqrt{c+d x^n}}{3 b d n} \]
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Rubi [A] time = 0.245118, antiderivative size = 221, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.233, Rules used = {446, 90, 80, 50, 63, 217, 206} \[ \frac{\left (a^2 d^2+2 a b c d+5 b^2 c^2\right ) \sqrt{a+b x^n} \sqrt{c+d x^n}}{8 b^2 d^3 n}-\frac{(b c-a d) \left (a^2 d^2+2 a b c d+5 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x^n}}{\sqrt{b} \sqrt{c+d x^n}}\right )}{8 b^{5/2} d^{7/2} n}-\frac{(3 a d+5 b c) \left (a+b x^n\right )^{3/2} \sqrt{c+d x^n}}{12 b^2 d^2 n}+\frac{x^n \left (a+b x^n\right )^{3/2} \sqrt{c+d x^n}}{3 b d n} \]
Antiderivative was successfully verified.
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Rule 446
Rule 90
Rule 80
Rule 50
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 \left (a+b x^n\right )^{3/2} \sqrt{c+d x^n}}{3 b d n}+\frac{\operatorname{Subst}\left (\int \frac{\sqrt{a+b x} \left (-a c-\frac{1}{2} (5 b c+3 a d) x\right )}{\sqrt{c+d x}} \, dx,x,x^n\right )}{3 b d n}\\ &=-\frac{(5 b c+3 a d) \left (a+b x^n\right )^{3/2} \sqrt{c+d x^n}}{12 b^2 d^2 n}+\frac{x^n \left (a+b x^n\right )^{3/2} \sqrt{c+d x^n}}{3 b d n}+\frac{\left (5 b^2 c^2+2 a b c d+a^2 d^2\right ) \operatorname{Subst}\left (\int \frac{\sqrt{a+b x}}{\sqrt{c+d x}} \, dx,x,x^n\right )}{8 b^2 d^2 n}\\ &=\frac{\left (5 b^2 c^2+2 a b c d+a^2 d^2\right ) \sqrt{a+b x^n} \sqrt{c+d x^n}}{8 b^2 d^3 n}-\frac{(5 b c+3 a d) \left (a+b x^n\right )^{3/2} \sqrt{c+d x^n}}{12 b^2 d^2 n}+\frac{x^n \left (a+b x^n\right )^{3/2} \sqrt{c+d x^n}}{3 b d n}-\frac{\left ((b c-a d) \left (5 b^2 c^2+2 a b c d+a^2 d^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x} \sqrt{c+d x}} \, dx,x,x^n\right )}{16 b^2 d^3 n}\\ &=\frac{\left (5 b^2 c^2+2 a b c d+a^2 d^2\right ) \sqrt{a+b x^n} \sqrt{c+d x^n}}{8 b^2 d^3 n}-\frac{(5 b c+3 a d) \left (a+b x^n\right )^{3/2} \sqrt{c+d x^n}}{12 b^2 d^2 n}+\frac{x^n \left (a+b x^n\right )^{3/2} \sqrt{c+d x^n}}{3 b d n}-\frac{\left ((b c-a d) \left (5 b^2 c^2+2 a b c d+a^2 d^2\right )\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 )}{8 b^3 d^3 n}\\ &=\frac{\left (5 b^2 c^2+2 a b c d+a^2 d^2\right ) \sqrt{a+b x^n} \sqrt{c+d x^n}}{8 b^2 d^3 n}-\frac{(5 b c+3 a d) \left (a+b x^n\right )^{3/2} \sqrt{c+d x^n}}{12 b^2 d^2 n}+\frac{x^n \left (a+b x^n\right )^{3/2} \sqrt{c+d x^n}}{3 b d n}-\frac{\left ((b c-a d) \left (5 b^2 c^2+2 a b c d+a^2 d^2\right )\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 )}{8 b^3 d^3 n}\\ &=\frac{\left (5 b^2 c^2+2 a b c d+a^2 d^2\right ) \sqrt{a+b x^n} \sqrt{c+d x^n}}{8 b^2 d^3 n}-\frac{(5 b c+3 a d) \left (a+b x^n\right )^{3/2} \sqrt{c+d x^n}}{12 b^2 d^2 n}+\frac{x^n \left (a+b x^n\right )^{3/2} \sqrt{c+d x^n}}{3 b d n}-\frac{(b c-a d) \left (5 b^2 c^2+2 a b c d+a^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x^n}}{\sqrt{b} \sqrt{c+d x^n}}\right )}{8 b^{5/2} d^{7/2} n}\\ \end{align*}
Mathematica [A] time = 0.496274, size = 191, normalized size = 0.86 \[ \frac{b \sqrt{d} \sqrt{a+b x^n} \left (c+d x^n\right ) \left (-3 a^2 d^2+2 a b d \left (d x^n-2 c\right )+b^2 \left (15 c^2-10 c d x^n+8 d^2 x^{2 n}\right )\right )-3 (b c-a d)^{3/2} \left (a^2 d^2+2 a b c d+5 b^2 c^2\right ) \sqrt{\frac{b \left (c+d x^n\right )}{b c-a d}} \sinh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x^n}}{\sqrt{b c-a d}}\right )}{24 b^3 d^{7/2} n \sqrt{c+d x^n}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.065, size = 0, normalized size = 0. \begin{align*} \int{{x}^{-1+3\,n}\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{\sqrt{b x^{n} + a} x^{3 \, n - 1}}{\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 = 1.20916, size = 1019, normalized size = 4.61 \begin{align*} \left [-\frac{3 \,{\left (5 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - a^{2} b c d^{2} - a^{3} d^{3}\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 (8 \, b^{3} d^{3} x^{2 \, n} + 15 \, b^{3} c^{2} d - 4 \, a b^{2} c d^{2} - 3 \, a^{2} b d^{3} - 2 \,{\left (5 \, b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{n}\right )} \sqrt{b x^{n} + a} \sqrt{d x^{n} + c}}{96 \, b^{3} d^{4} n}, \frac{3 \,{\left (5 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - a^{2} b c d^{2} - a^{3} d^{3}\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 (8 \, b^{3} d^{3} x^{2 \, n} + 15 \, b^{3} c^{2} d - 4 \, a b^{2} c d^{2} - 3 \, a^{2} b d^{3} - 2 \,{\left (5 \, b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{n}\right )} \sqrt{b x^{n} + a} \sqrt{d x^{n} + c}}{48 \, b^{3} d^{4} 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{\sqrt{b x^{n} + a} x^{3 \, n - 1}}{\sqrt{d x^{n} + c}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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