Optimal. Leaf size=262 \[ \frac{\left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{5 e}-\frac{b x \left (15 c^4 d^2+10 c^2 d e+3 e^2\right ) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{c^2 x^2-1}}{c \sqrt{d+e x^2}}\right )}{40 c^4 \sqrt{e} \sqrt{c^2 x^2}}+\frac{b c d^{5/2} x \tan ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{c^2 x^2-1}}\right )}{5 e \sqrt{c^2 x^2}}-\frac{b x \sqrt{c^2 x^2-1} \left (d+e x^2\right )^{3/2}}{20 c \sqrt{c^2 x^2}}-\frac{b x \sqrt{c^2 x^2-1} \left (7 c^2 d+3 e\right ) \sqrt{d+e x^2}}{40 c^3 \sqrt{c^2 x^2}} \]
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Rubi [A] time = 0.270425, antiderivative size = 262, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 10, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.476, Rules used = {5236, 446, 102, 154, 157, 63, 217, 206, 93, 204} \[ \frac{\left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{5 e}-\frac{b x \left (15 c^4 d^2+10 c^2 d e+3 e^2\right ) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{c^2 x^2-1}}{c \sqrt{d+e x^2}}\right )}{40 c^4 \sqrt{e} \sqrt{c^2 x^2}}+\frac{b c d^{5/2} x \tan ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{c^2 x^2-1}}\right )}{5 e \sqrt{c^2 x^2}}-\frac{b x \sqrt{c^2 x^2-1} \left (d+e x^2\right )^{3/2}}{20 c \sqrt{c^2 x^2}}-\frac{b x \sqrt{c^2 x^2-1} \left (7 c^2 d+3 e\right ) \sqrt{d+e x^2}}{40 c^3 \sqrt{c^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 5236
Rule 446
Rule 102
Rule 154
Rule 157
Rule 63
Rule 217
Rule 206
Rule 93
Rule 204
Rubi steps
\begin{align*} \int x \left (d+e x^2\right )^{3/2} \left (a+b \sec ^{-1}(c x)\right ) \, dx &=\frac{\left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{5 e}-\frac{(b c x) \int \frac{\left (d+e x^2\right )^{5/2}}{x \sqrt{-1+c^2 x^2}} \, dx}{5 e \sqrt{c^2 x^2}}\\ &=\frac{\left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{5 e}-\frac{(b c x) \operatorname{Subst}\left (\int \frac{(d+e x)^{5/2}}{x \sqrt{-1+c^2 x}} \, dx,x,x^2\right )}{10 e \sqrt{c^2 x^2}}\\ &=-\frac{b x \sqrt{-1+c^2 x^2} \left (d+e x^2\right )^{3/2}}{20 c \sqrt{c^2 x^2}}+\frac{\left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{5 e}-\frac{(b x) \operatorname{Subst}\left (\int \frac{\sqrt{d+e x} \left (2 c^2 d^2+\frac{1}{2} e \left (7 c^2 d+3 e\right ) x\right )}{x \sqrt{-1+c^2 x}} \, dx,x,x^2\right )}{20 c e \sqrt{c^2 x^2}}\\ &=-\frac{b \left (7 c^2 d+3 e\right ) x \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}}{40 c^3 \sqrt{c^2 x^2}}-\frac{b x \sqrt{-1+c^2 x^2} \left (d+e x^2\right )^{3/2}}{20 c \sqrt{c^2 x^2}}+\frac{\left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{5 e}-\frac{(b x) \operatorname{Subst}\left (\int \frac{2 c^4 d^3+\frac{1}{4} e \left (15 c^4 d^2+10 c^2 d e+3 e^2\right ) x}{x \sqrt{-1+c^2 x} \sqrt{d+e x}} \, dx,x,x^2\right )}{20 c^3 e \sqrt{c^2 x^2}}\\ &=-\frac{b \left (7 c^2 d+3 e\right ) x \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}}{40 c^3 \sqrt{c^2 x^2}}-\frac{b x \sqrt{-1+c^2 x^2} \left (d+e x^2\right )^{3/2}}{20 c \sqrt{c^2 x^2}}+\frac{\left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{5 e}-\frac{\left (b c d^3 x\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{-1+c^2 x} \sqrt{d+e x}} \, dx,x,x^2\right )}{10 e \sqrt{c^2 x^2}}-\frac{\left (b \left (15 c^4 d^2+10 c^2 d e+3 e^2\right ) x\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{-1+c^2 x} \sqrt{d+e x}} \, dx,x,x^2\right )}{80 c^3 \sqrt{c^2 x^2}}\\ &=-\frac{b \left (7 c^2 d+3 e\right ) x \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}}{40 c^3 \sqrt{c^2 x^2}}-\frac{b x \sqrt{-1+c^2 x^2} \left (d+e x^2\right )^{3/2}}{20 c \sqrt{c^2 x^2}}+\frac{\left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{5 e}-\frac{\left (b c d^3 x\right ) \operatorname{Subst}\left (\int \frac{1}{-d-x^2} \, dx,x,\frac{\sqrt{d+e x^2}}{\sqrt{-1+c^2 x^2}}\right )}{5 e \sqrt{c^2 x^2}}-\frac{\left (b \left (15 c^4 d^2+10 c^2 d e+3 e^2\right ) x\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{d+\frac{e}{c^2}+\frac{e x^2}{c^2}}} \, dx,x,\sqrt{-1+c^2 x^2}\right )}{40 c^5 \sqrt{c^2 x^2}}\\ &=-\frac{b \left (7 c^2 d+3 e\right ) x \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}}{40 c^3 \sqrt{c^2 x^2}}-\frac{b x \sqrt{-1+c^2 x^2} \left (d+e x^2\right )^{3/2}}{20 c \sqrt{c^2 x^2}}+\frac{\left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{5 e}+\frac{b c d^{5/2} x \tan ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{-1+c^2 x^2}}\right )}{5 e \sqrt{c^2 x^2}}-\frac{\left (b \left (15 c^4 d^2+10 c^2 d e+3 e^2\right ) x\right ) \operatorname{Subst}\left (\int \frac{1}{1-\frac{e x^2}{c^2}} \, dx,x,\frac{\sqrt{-1+c^2 x^2}}{\sqrt{d+e x^2}}\right )}{40 c^5 \sqrt{c^2 x^2}}\\ &=-\frac{b \left (7 c^2 d+3 e\right ) x \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}}{40 c^3 \sqrt{c^2 x^2}}-\frac{b x \sqrt{-1+c^2 x^2} \left (d+e x^2\right )^{3/2}}{20 c \sqrt{c^2 x^2}}+\frac{\left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{5 e}+\frac{b c d^{5/2} x \tan ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{-1+c^2 x^2}}\right )}{5 e \sqrt{c^2 x^2}}-\frac{b \left (15 c^4 d^2+10 c^2 d e+3 e^2\right ) x \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{-1+c^2 x^2}}{c \sqrt{d+e x^2}}\right )}{40 c^4 \sqrt{e} \sqrt{c^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.714011, size = 305, normalized size = 1.16 \[ \frac{\sqrt{d+e x^2} \left (8 a c^3 \left (d+e x^2\right )^2-b e x \sqrt{1-\frac{1}{c^2 x^2}} \left (c^2 \left (9 d+2 e x^2\right )+3 e\right )+8 b c^3 \sec ^{-1}(c x) \left (d+e x^2\right )^2\right )}{40 c^3 e}-\frac{b x \sqrt{1-\frac{1}{c^2 x^2}} \left (\sqrt{c^2} \sqrt{e} \sqrt{c^2 d+e} \left (15 c^4 d^2+10 c^2 d e+3 e^2\right ) \sqrt{\frac{c^2 \left (d+e x^2\right )}{c^2 d+e}} \sinh ^{-1}\left (\frac{c \sqrt{e} \sqrt{c^2 x^2-1}}{\sqrt{c^2} \sqrt{c^2 d+e}}\right )+8 c^7 d^{5/2} \sqrt{d+e x^2} \tan ^{-1}\left (\frac{\sqrt{d} \sqrt{c^2 x^2-1}}{\sqrt{d+e x^2}}\right )\right )}{40 c^6 e \sqrt{c^2 x^2-1} \sqrt{d+e x^2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.191, size = 0, normalized size = 0. \begin{align*} \int x \left ( e{x}^{2}+d \right ) ^{{\frac{3}{2}}} \left ( a+b{\rm arcsec} \left (cx\right ) \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 10.902, size = 3004, normalized size = 11.47 \begin{align*} \text{result too large to display} \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{\left (e x^{2} + d\right )}^{\frac{3}{2}}{\left (b \operatorname{arcsec}\left (c x\right ) + a\right )} x\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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