Optimal. Leaf size=374 \[ -\frac{d \left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{5 e^2}+\frac{\left (d+e x^2\right )^{7/2} \left (a+b \sec ^{-1}(c x)\right )}{7 e^2}+\frac{b x \sqrt{c^2 x^2-1} \left (3 c^4 d^2-38 c^2 d e-25 e^2\right ) \sqrt{d+e x^2}}{560 c^5 e \sqrt{c^2 x^2}}-\frac{2 b c d^{7/2} x \tan ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{c^2 x^2-1}}\right )}{35 e^2 \sqrt{c^2 x^2}}+\frac{b x \left (-35 c^4 d^2 e+35 c^6 d^3-63 c^2 d e^2-25 e^3\right ) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{c^2 x^2-1}}{c \sqrt{d+e x^2}}\right )}{560 c^6 e^{3/2} \sqrt{c^2 x^2}}-\frac{b x \sqrt{c^2 x^2-1} \left (d+e x^2\right )^{5/2}}{42 c e \sqrt{c^2 x^2}}-\frac{b x \sqrt{c^2 x^2-1} \left (13 c^2 d+25 e\right ) \left (d+e x^2\right )^{3/2}}{840 c^3 e \sqrt{c^2 x^2}} \]
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Rubi [A] time = 0.507682, antiderivative size = 374, 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, 5238, 12, 573, 154, 157, 63, 217, 206, 93, 204} \[ -\frac{d \left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{5 e^2}+\frac{\left (d+e x^2\right )^{7/2} \left (a+b \sec ^{-1}(c x)\right )}{7 e^2}+\frac{b x \sqrt{c^2 x^2-1} \left (3 c^4 d^2-38 c^2 d e-25 e^2\right ) \sqrt{d+e x^2}}{560 c^5 e \sqrt{c^2 x^2}}-\frac{2 b c d^{7/2} x \tan ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{c^2 x^2-1}}\right )}{35 e^2 \sqrt{c^2 x^2}}+\frac{b x \left (-35 c^4 d^2 e+35 c^6 d^3-63 c^2 d e^2-25 e^3\right ) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{c^2 x^2-1}}{c \sqrt{d+e x^2}}\right )}{560 c^6 e^{3/2} \sqrt{c^2 x^2}}-\frac{b x \sqrt{c^2 x^2-1} \left (d+e x^2\right )^{5/2}}{42 c e \sqrt{c^2 x^2}}-\frac{b x \sqrt{c^2 x^2-1} \left (13 c^2 d+25 e\right ) \left (d+e x^2\right )^{3/2}}{840 c^3 e \sqrt{c^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rule 5238
Rule 12
Rule 573
Rule 154
Rule 157
Rule 63
Rule 217
Rule 206
Rule 93
Rule 204
Rubi steps
\begin{align*} \int x^3 \left (d+e x^2\right )^{3/2} \left (a+b \sec ^{-1}(c x)\right ) \, dx &=-\frac{d \left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{5 e^2}+\frac{\left (d+e x^2\right )^{7/2} \left (a+b \sec ^{-1}(c x)\right )}{7 e^2}-\frac{(b c x) \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}{\sqrt{c^2 x^2}}\\ &=-\frac{d \left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{5 e^2}+\frac{\left (d+e x^2\right )^{7/2} \left (a+b \sec ^{-1}(c x)\right )}{7 e^2}-\frac{(b c x) \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 \sqrt{c^2 x^2}}\\ &=-\frac{d \left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{5 e^2}+\frac{\left (d+e x^2\right )^{7/2} \left (a+b \sec ^{-1}(c x)\right )}{7 e^2}-\frac{(b c x) \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 \sqrt{c^2 x^2}}\\ &=-\frac{b x \sqrt{-1+c^2 x^2} \left (d+e x^2\right )^{5/2}}{42 c e \sqrt{c^2 x^2}}-\frac{d \left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{5 e^2}+\frac{\left (d+e x^2\right )^{7/2} \left (a+b \sec ^{-1}(c x)\right )}{7 e^2}-\frac{(b x) \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 e^2 \sqrt{c^2 x^2}}\\ &=-\frac{b \left (13 c^2 d+25 e\right ) x \sqrt{-1+c^2 x^2} \left (d+e x^2\right )^{3/2}}{840 c^3 e \sqrt{c^2 x^2}}-\frac{b x \sqrt{-1+c^2 x^2} \left (d+e x^2\right )^{5/2}}{42 c e \sqrt{c^2 x^2}}-\frac{d \left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{5 e^2}+\frac{\left (d+e x^2\right )^{7/2} \left (a+b \sec ^{-1}(c x)\right )}{7 e^2}-\frac{(b x) \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^3 e^2 \sqrt{c^2 x^2}}\\ &=\frac{b \left (3 c^4 d^2-38 c^2 d e-25 e^2\right ) x \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}}{560 c^5 e \sqrt{c^2 x^2}}-\frac{b \left (13 c^2 d+25 e\right ) x \sqrt{-1+c^2 x^2} \left (d+e x^2\right )^{3/2}}{840 c^3 e \sqrt{c^2 x^2}}-\frac{b x \sqrt{-1+c^2 x^2} \left (d+e x^2\right )^{5/2}}{42 c e \sqrt{c^2 x^2}}-\frac{d \left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{5 e^2}+\frac{\left (d+e x^2\right )^{7/2} \left (a+b \sec ^{-1}(c x)\right )}{7 e^2}-\frac{(b x) \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^5 e^2 \sqrt{c^2 x^2}}\\ &=\frac{b \left (3 c^4 d^2-38 c^2 d e-25 e^2\right ) x \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}}{560 c^5 e \sqrt{c^2 x^2}}-\frac{b \left (13 c^2 d+25 e\right ) x \sqrt{-1+c^2 x^2} \left (d+e x^2\right )^{3/2}}{840 c^3 e \sqrt{c^2 x^2}}-\frac{b x \sqrt{-1+c^2 x^2} \left (d+e x^2\right )^{5/2}}{42 c e \sqrt{c^2 x^2}}-\frac{d \left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{5 e^2}+\frac{\left (d+e x^2\right )^{7/2} \left (a+b \sec ^{-1}(c x)\right )}{7 e^2}+\frac{\left (b c d^4 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 \sqrt{c^2 x^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 ) x\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{-1+c^2 x} \sqrt{d+e x}} \, dx,x,x^2\right )}{1120 c^5 e \sqrt{c^2 x^2}}\\ &=\frac{b \left (3 c^4 d^2-38 c^2 d e-25 e^2\right ) x \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}}{560 c^5 e \sqrt{c^2 x^2}}-\frac{b \left (13 c^2 d+25 e\right ) x \sqrt{-1+c^2 x^2} \left (d+e x^2\right )^{3/2}}{840 c^3 e \sqrt{c^2 x^2}}-\frac{b x \sqrt{-1+c^2 x^2} \left (d+e x^2\right )^{5/2}}{42 c e \sqrt{c^2 x^2}}-\frac{d \left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{5 e^2}+\frac{\left (d+e x^2\right )^{7/2} \left (a+b \sec ^{-1}(c x)\right )}{7 e^2}+\frac{\left (2 b c d^4 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 \sqrt{c^2 x^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 ) 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^7 e \sqrt{c^2 x^2}}\\ &=\frac{b \left (3 c^4 d^2-38 c^2 d e-25 e^2\right ) x \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}}{560 c^5 e \sqrt{c^2 x^2}}-\frac{b \left (13 c^2 d+25 e\right ) x \sqrt{-1+c^2 x^2} \left (d+e x^2\right )^{3/2}}{840 c^3 e \sqrt{c^2 x^2}}-\frac{b x \sqrt{-1+c^2 x^2} \left (d+e x^2\right )^{5/2}}{42 c e \sqrt{c^2 x^2}}-\frac{d \left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{5 e^2}+\frac{\left (d+e x^2\right )^{7/2} \left (a+b \sec ^{-1}(c x)\right )}{7 e^2}-\frac{2 b c d^{7/2} x \tan ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{-1+c^2 x^2}}\right )}{35 e^2 \sqrt{c^2 x^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 ) 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^7 e \sqrt{c^2 x^2}}\\ &=\frac{b \left (3 c^4 d^2-38 c^2 d e-25 e^2\right ) x \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}}{560 c^5 e \sqrt{c^2 x^2}}-\frac{b \left (13 c^2 d+25 e\right ) x \sqrt{-1+c^2 x^2} \left (d+e x^2\right )^{3/2}}{840 c^3 e \sqrt{c^2 x^2}}-\frac{b x \sqrt{-1+c^2 x^2} \left (d+e x^2\right )^{5/2}}{42 c e \sqrt{c^2 x^2}}-\frac{d \left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{5 e^2}+\frac{\left (d+e x^2\right )^{7/2} \left (a+b \sec ^{-1}(c x)\right )}{7 e^2}-\frac{2 b c d^{7/2} x \tan ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{-1+c^2 x^2}}\right )}{35 e^2 \sqrt{c^2 x^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 ) x \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{-1+c^2 x^2}}{c \sqrt{d+e x^2}}\right )}{560 c^6 e^{3/2} \sqrt{c^2 x^2}}\\ \end{align*}
Mathematica [C] time = 0.595271, size = 339, normalized size = 0.91 \[ -\frac{\sqrt{d+e x^2} \left (48 a c^5 \left (2 d-5 e x^2\right ) \left (d+e x^2\right )^2+b e x \sqrt{1-\frac{1}{c^2 x^2}} \left (c^4 \left (57 d^2+106 d e x^2+40 e^2 x^4\right )+2 c^2 e \left (82 d+25 e x^2\right )+75 e^2\right )+48 b c^5 \sec ^{-1}(c x) \left (2 d-5 e x^2\right ) \left (d+e x^2\right )^2\right )}{1680 c^5 e^2}-\frac{b \left (e x^4 \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{1-c^2 x^2} \left (35 c^4 d^2 e-35 c^6 d^3+63 c^2 d e^2+25 e^3\right ) \sqrt{\frac{e x^2}{d}+1} F_1\left (1;\frac{1}{2},\frac{1}{2};2;c^2 x^2,-\frac{e x^2}{d}\right )+32 c^4 d^4 \left (c^2 x^2-1\right ) \sqrt{\frac{d}{e x^2}+1} F_1\left (1;\frac{1}{2},\frac{1}{2};2;\frac{1}{c^2 x^2},-\frac{d}{e x^2}\right )\right )}{1120 c^5 e^2 x \left (c^2 x^2-1\right ) \sqrt{d+e x^2}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 1.519, size = 0, normalized size = 0. \begin{align*} \int{x}^{3} \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 = 31.2762, size = 3771, normalized size = 10.08 \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^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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