Optimal. Leaf size=418 \[ -\frac{d \left (d+e x^2\right )^{5/2} \left (a+b \text{sech}^{-1}(c x)\right )}{5 e^2}+\frac{\left (d+e x^2\right )^{7/2} \left (a+b \text{sech}^{-1}(c x)\right )}{7 e^2}+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \left (3 c^4 d^2-38 c^2 d e-25 e^2\right ) \sqrt{d+e x^2}}{560 c^6 e}+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \left (-35 c^4 d^2 e+35 c^6 d^3-63 c^2 d e^2-25 e^3\right ) \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{1-c^2 x^2}}{c \sqrt{d+e x^2}}\right )}{560 c^7 e^{3/2}}+\frac{2 b d^{7/2} \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{1-c^2 x^2}}\right )}{35 e^2}-\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \left (d+e x^2\right )^{5/2}}{42 c^2 e}-\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \left (13 c^2 d+25 e\right ) \left (d+e x^2\right )^{3/2}}{840 c^4 e} \]
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Rubi [A] time = 0.529505, antiderivative size = 418, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 12, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.522, Rules used = {266, 43, 6301, 12, 573, 154, 157, 63, 217, 203, 93, 207} \[ -\frac{d \left (d+e x^2\right )^{5/2} \left (a+b \text{sech}^{-1}(c x)\right )}{5 e^2}+\frac{\left (d+e x^2\right )^{7/2} \left (a+b \text{sech}^{-1}(c x)\right )}{7 e^2}+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \left (3 c^4 d^2-38 c^2 d e-25 e^2\right ) \sqrt{d+e x^2}}{560 c^6 e}+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \left (-35 c^4 d^2 e+35 c^6 d^3-63 c^2 d e^2-25 e^3\right ) \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{1-c^2 x^2}}{c \sqrt{d+e x^2}}\right )}{560 c^7 e^{3/2}}+\frac{2 b d^{7/2} \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{1-c^2 x^2}}\right )}{35 e^2}-\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \left (d+e x^2\right )^{5/2}}{42 c^2 e}-\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \left (13 c^2 d+25 e\right ) \left (d+e x^2\right )^{3/2}}{840 c^4 e} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rule 6301
Rule 12
Rule 573
Rule 154
Rule 157
Rule 63
Rule 217
Rule 203
Rule 93
Rule 207
Rubi steps
\begin{align*} \int x^3 \left (d+e x^2\right )^{3/2} \left (a+b \text{sech}^{-1}(c x)\right ) \, dx &=-\frac{d \left (d+e x^2\right )^{5/2} \left (a+b \text{sech}^{-1}(c x)\right )}{5 e^2}+\frac{\left (d+e x^2\right )^{7/2} \left (a+b \text{sech}^{-1}(c x)\right )}{7 e^2}+\left (b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{\left (d+e x^2\right )^{5/2} \left (-2 d+5 e x^2\right )}{35 e^2 x \sqrt{1-c^2 x^2}} \, dx\\ &=-\frac{d \left (d+e x^2\right )^{5/2} \left (a+b \text{sech}^{-1}(c x)\right )}{5 e^2}+\frac{\left (d+e x^2\right )^{7/2} \left (a+b \text{sech}^{-1}(c x)\right )}{7 e^2}+\frac{\left (b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{\left (d+e x^2\right )^{5/2} \left (-2 d+5 e x^2\right )}{x \sqrt{1-c^2 x^2}} \, dx}{35 e^2}\\ &=-\frac{d \left (d+e x^2\right )^{5/2} \left (a+b \text{sech}^{-1}(c x)\right )}{5 e^2}+\frac{\left (d+e x^2\right )^{7/2} \left (a+b \text{sech}^{-1}(c x)\right )}{7 e^2}+\frac{\left (b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \frac{(d+e x)^{5/2} (-2 d+5 e x)}{x \sqrt{1-c^2 x}} \, dx,x,x^2\right )}{70 e^2}\\ &=-\frac{b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2} \left (d+e x^2\right )^{5/2}}{42 c^2 e}-\frac{d \left (d+e x^2\right )^{5/2} \left (a+b \text{sech}^{-1}(c x)\right )}{5 e^2}+\frac{\left (d+e x^2\right )^{7/2} \left (a+b \text{sech}^{-1}(c x)\right )}{7 e^2}-\frac{\left (b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \frac{(d+e x)^{3/2} \left (6 c^2 d^2-\frac{1}{2} e \left (13 c^2 d+25 e\right ) x\right )}{x \sqrt{1-c^2 x}} \, dx,x,x^2\right )}{210 c^2 e^2}\\ &=-\frac{b \left (13 c^2 d+25 e\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{840 c^4 e}-\frac{b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2} \left (d+e x^2\right )^{5/2}}{42 c^2 e}-\frac{d \left (d+e x^2\right )^{5/2} \left (a+b \text{sech}^{-1}(c x)\right )}{5 e^2}+\frac{\left (d+e x^2\right )^{7/2} \left (a+b \text{sech}^{-1}(c x)\right )}{7 e^2}+\frac{\left (b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{d+e x} \left (-12 c^4 d^3-\frac{3}{4} e \left (3 c^4 d^2-38 c^2 d e-25 e^2\right ) x\right )}{x \sqrt{1-c^2 x}} \, dx,x,x^2\right )}{420 c^4 e^2}\\ &=\frac{b \left (3 c^4 d^2-38 c^2 d e-25 e^2\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2} \sqrt{d+e x^2}}{560 c^6 e}-\frac{b \left (13 c^2 d+25 e\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{840 c^4 e}-\frac{b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2} \left (d+e x^2\right )^{5/2}}{42 c^2 e}-\frac{d \left (d+e x^2\right )^{5/2} \left (a+b \text{sech}^{-1}(c x)\right )}{5 e^2}+\frac{\left (d+e x^2\right )^{7/2} \left (a+b \text{sech}^{-1}(c x)\right )}{7 e^2}-\frac{\left (b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \frac{12 c^6 d^4+\frac{3}{8} e \left (35 c^6 d^3-35 c^4 d^2 e-63 c^2 d e^2-25 e^3\right ) x}{x \sqrt{1-c^2 x} \sqrt{d+e x}} \, dx,x,x^2\right )}{420 c^6 e^2}\\ &=\frac{b \left (3 c^4 d^2-38 c^2 d e-25 e^2\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2} \sqrt{d+e x^2}}{560 c^6 e}-\frac{b \left (13 c^2 d+25 e\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{840 c^4 e}-\frac{b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2} \left (d+e x^2\right )^{5/2}}{42 c^2 e}-\frac{d \left (d+e x^2\right )^{5/2} \left (a+b \text{sech}^{-1}(c x)\right )}{5 e^2}+\frac{\left (d+e x^2\right )^{7/2} \left (a+b \text{sech}^{-1}(c x)\right )}{7 e^2}-\frac{\left (b d^4 \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-c^2 x} \sqrt{d+e x}} \, dx,x,x^2\right )}{35 e^2}-\frac{\left (b \left (35 c^6 d^3-35 c^4 d^2 e-63 c^2 d e^2-25 e^3\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-c^2 x} \sqrt{d+e x}} \, dx,x,x^2\right )}{1120 c^6 e}\\ &=\frac{b \left (3 c^4 d^2-38 c^2 d e-25 e^2\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2} \sqrt{d+e x^2}}{560 c^6 e}-\frac{b \left (13 c^2 d+25 e\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{840 c^4 e}-\frac{b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2} \left (d+e x^2\right )^{5/2}}{42 c^2 e}-\frac{d \left (d+e x^2\right )^{5/2} \left (a+b \text{sech}^{-1}(c x)\right )}{5 e^2}+\frac{\left (d+e x^2\right )^{7/2} \left (a+b \text{sech}^{-1}(c x)\right )}{7 e^2}-\frac{\left (2 b d^4 \sqrt{\frac{1}{1+c x}} \sqrt{1+c 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 )}{35 e^2}+\frac{\left (b \left (35 c^6 d^3-35 c^4 d^2 e-63 c^2 d e^2-25 e^3\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c 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 )}{560 c^8 e}\\ &=\frac{b \left (3 c^4 d^2-38 c^2 d e-25 e^2\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2} \sqrt{d+e x^2}}{560 c^6 e}-\frac{b \left (13 c^2 d+25 e\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{840 c^4 e}-\frac{b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2} \left (d+e x^2\right )^{5/2}}{42 c^2 e}-\frac{d \left (d+e x^2\right )^{5/2} \left (a+b \text{sech}^{-1}(c x)\right )}{5 e^2}+\frac{\left (d+e x^2\right )^{7/2} \left (a+b \text{sech}^{-1}(c x)\right )}{7 e^2}+\frac{2 b d^{7/2} \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{1-c^2 x^2}}\right )}{35 e^2}+\frac{\left (b \left (35 c^6 d^3-35 c^4 d^2 e-63 c^2 d e^2-25 e^3\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c 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 )}{560 c^8 e}\\ &=\frac{b \left (3 c^4 d^2-38 c^2 d e-25 e^2\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2} \sqrt{d+e x^2}}{560 c^6 e}-\frac{b \left (13 c^2 d+25 e\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{840 c^4 e}-\frac{b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2} \left (d+e x^2\right )^{5/2}}{42 c^2 e}-\frac{d \left (d+e x^2\right )^{5/2} \left (a+b \text{sech}^{-1}(c x)\right )}{5 e^2}+\frac{\left (d+e x^2\right )^{7/2} \left (a+b \text{sech}^{-1}(c x)\right )}{7 e^2}+\frac{b \left (35 c^6 d^3-35 c^4 d^2 e-63 c^2 d e^2-25 e^3\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{1-c^2 x^2}}{c \sqrt{d+e x^2}}\right )}{560 c^7 e^{3/2}}+\frac{2 b d^{7/2} \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{1-c^2 x^2}}\right )}{35 e^2}\\ \end{align*}
Mathematica [A] time = 2.8559, size = 382, normalized size = 0.91 \[ \frac{b \sqrt{\frac{1-c x}{c x+1}} \sqrt{c^2 x^2-1} \left (\sqrt{c^2} \sqrt{e} \sqrt{c^2 d+e} \left (-35 c^4 d^2 e+35 c^6 d^3-63 c^2 d e^2-25 e^3\right ) \sqrt{\frac{c^2 \left (d+e x^2\right )}{c^2 d+e}} \sinh ^{-1}\left (\frac{c \sqrt{e} \sqrt{c^2 x^2-1}}{\sqrt{c^2} \sqrt{c^2 d+e}}\right )+32 c^9 d^{7/2} \sqrt{d+e x^2} \tan ^{-1}\left (\frac{\sqrt{d} \sqrt{c^2 x^2-1}}{\sqrt{d+e x^2}}\right )\right )}{560 c^9 e^2 (c x-1) \sqrt{d+e x^2}}-\frac{\sqrt{d+e x^2} \left (48 a c^6 \left (2 d-5 e x^2\right ) \left (d+e x^2\right )^2+b e \sqrt{\frac{1-c x}{c x+1}} (c x+1) \left (c^4 \left (57 d^2+106 d e x^2+40 e^2 x^4\right )+2 c^2 e \left (82 d+25 e x^2\right )+75 e^2\right )+48 b c^6 \text{sech}^{-1}(c x) \left (2 d-5 e x^2\right ) \left (d+e x^2\right )^2\right )}{1680 c^6 e^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.135, size = 0, normalized size = 0. \begin{align*} \int{x}^{3} \left ( e{x}^{2}+d \right ) ^{{\frac{3}{2}}} \left ( a+b{\rm arcsech} \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 = 17.7434, size = 4365, normalized size = 10.44 \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{arsech}\left (c x\right ) + a\right )} x^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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