Optimal. Leaf size=199 \[ -\frac{(b c-a d)^2 (a d+5 b c) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x^n}}{\sqrt{b} \sqrt{c+d x^n}}\right )}{8 b^{3/2} d^{7/2} n}-\frac{(a d+5 b c) \left (a+b x^n\right )^{3/2} \sqrt{c+d x^n}}{12 b d^2 n}+\frac{(b c-a d) (a d+5 b c) \sqrt{a+b x^n} \sqrt{c+d x^n}}{8 b d^3 n}+\frac{\left (a+b x^n\right )^{5/2} \sqrt{c+d x^n}}{3 b d n} \]
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Rubi [A] time = 0.164262, antiderivative size = 199, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {446, 80, 50, 63, 217, 206} \[ -\frac{(b c-a d)^2 (a d+5 b c) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x^n}}{\sqrt{b} \sqrt{c+d x^n}}\right )}{8 b^{3/2} d^{7/2} n}-\frac{(a d+5 b c) \left (a+b x^n\right )^{3/2} \sqrt{c+d x^n}}{12 b d^2 n}+\frac{(b c-a d) (a d+5 b c) \sqrt{a+b x^n} \sqrt{c+d x^n}}{8 b d^3 n}+\frac{\left (a+b x^n\right )^{5/2} \sqrt{c+d x^n}}{3 b d n} \]
Antiderivative was successfully verified.
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Rule 446
Rule 80
Rule 50
Rule 63
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \frac{x^{-1+2 n} \left (a+b x^n\right )^{3/2}}{\sqrt{c+d x^n}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x (a+b x)^{3/2}}{\sqrt{c+d x}} \, dx,x,x^n\right )}{n}\\ &=\frac{\left (a+b x^n\right )^{5/2} \sqrt{c+d x^n}}{3 b d n}-\frac{(5 b c+a d) \operatorname{Subst}\left (\int \frac{(a+b x)^{3/2}}{\sqrt{c+d x}} \, dx,x,x^n\right )}{6 b d n}\\ &=-\frac{(5 b c+a d) \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}+\frac{((b c-a d) (5 b c+a d)) \operatorname{Subst}\left (\int \frac{\sqrt{a+b x}}{\sqrt{c+d x}} \, dx,x,x^n\right )}{8 b d^2 n}\\ &=\frac{(b c-a d) (5 b c+a d) \sqrt{a+b x^n} \sqrt{c+d x^n}}{8 b d^3 n}-\frac{(5 b c+a d) \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}-\frac{\left ((b c-a d)^2 (5 b c+a d)\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x} \sqrt{c+d x}} \, dx,x,x^n\right )}{16 b d^3 n}\\ &=\frac{(b c-a d) (5 b c+a d) \sqrt{a+b x^n} \sqrt{c+d x^n}}{8 b d^3 n}-\frac{(5 b c+a d) \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}-\frac{\left ((b c-a d)^2 (5 b c+a d)\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^2 d^3 n}\\ &=\frac{(b c-a d) (5 b c+a d) \sqrt{a+b x^n} \sqrt{c+d x^n}}{8 b d^3 n}-\frac{(5 b c+a d) \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}-\frac{\left ((b c-a d)^2 (5 b c+a d)\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^2 d^3 n}\\ &=\frac{(b c-a d) (5 b c+a d) \sqrt{a+b x^n} \sqrt{c+d x^n}}{8 b d^3 n}-\frac{(5 b c+a d) \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}-\frac{(b c-a d)^2 (5 b c+a d) \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}\\ \end{align*}
Mathematica [A] time = 0.557486, size = 178, normalized size = 0.89 \[ \frac{b \sqrt{d} \sqrt{a+b x^n} \left (c+d x^n\right ) \left (3 a^2 d^2+2 a b d \left (7 d x^n-11 c\right )+b^2 \left (15 c^2-10 c d x^n+8 d^2 x^{2 n}\right )\right )-3 (b c-a d)^{5/2} (a d+5 b c) \sqrt{\frac{b \left (c+d x^n\right )}{b c-a d}} \sinh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x^n}}{\sqrt{b c-a d}}\right )}{24 b^2 d^{7/2} n \sqrt{c+d x^n}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.066, size = 0, normalized size = 0. \begin{align*} \int{{x}^{-1+2\,n} \left ( a+b{x}^{n} \right ) ^{{\frac{3}{2}}}{\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{{\left (b x^{n} + a\right )}^{\frac{3}{2}} x^{2 \, 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 = 1.22346, size = 1031, normalized size = 5.18 \begin{align*} \left [\frac{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 )} \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 - 22 \, a b^{2} c d^{2} + 3 \, a^{2} b d^{3} - 2 \,{\left (5 \, b^{3} c d^{2} - 7 \, a b^{2} d^{3}\right )} x^{n}\right )} \sqrt{b x^{n} + a} \sqrt{d x^{n} + c}}{96 \, b^{2} d^{4} n}, \frac{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 )} \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 - 22 \, a b^{2} c d^{2} + 3 \, a^{2} b d^{3} - 2 \,{\left (5 \, b^{3} c d^{2} - 7 \, a b^{2} d^{3}\right )} x^{n}\right )} \sqrt{b x^{n} + a} \sqrt{d x^{n} + c}}{48 \, b^{2} 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{{\left (b x^{n} + a\right )}^{\frac{3}{2}} x^{2 \, 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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