Optimal. Leaf size=343 \[ \frac {2 (d+e x)^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e}-\frac {4 b e \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2} \sqrt {d+e x}}{15 c^2}-\frac {4 b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (2 c^2 d^2+e^2\right ) \sqrt {\frac {c (d+e x)}{c d+e}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{15 c^3 \sqrt {d+e x}}-\frac {4 b d^3 \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {\frac {c (d+e x)}{c d+e}} \Pi \left (2;\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{5 e \sqrt {d+e x}}-\frac {28 b d \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {d+e x} E\left (\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{15 c \sqrt {\frac {c (d+e x)}{c d+e}}} \]
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Rubi [A] time = 0.62, antiderivative size = 343, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 12, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {6288, 958, 719, 419, 932, 168, 538, 537, 844, 424, 931, 1584} \[ \frac {2 (d+e x)^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e}-\frac {4 b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (2 c^2 d^2+e^2\right ) \sqrt {\frac {c (d+e x)}{c d+e}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{15 c^3 \sqrt {d+e x}}-\frac {4 b e \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2} \sqrt {d+e x}}{15 c^2}-\frac {4 b d^3 \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {\frac {c (d+e x)}{c d+e}} \Pi \left (2;\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{5 e \sqrt {d+e x}}-\frac {28 b d \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {d+e x} E\left (\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{15 c \sqrt {\frac {c (d+e x)}{c d+e}}} \]
Antiderivative was successfully verified.
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Rule 168
Rule 419
Rule 424
Rule 537
Rule 538
Rule 719
Rule 844
Rule 931
Rule 932
Rule 958
Rule 1584
Rule 6288
Rubi steps
\begin {align*} \int (d+e x)^{3/2} \left (a+b \text {sech}^{-1}(c x)\right ) \, dx &=\frac {2 (d+e x)^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e}+\frac {\left (2 b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {(d+e x)^{5/2}}{x \sqrt {1-c^2 x^2}} \, dx}{5 e}\\ &=\frac {2 (d+e x)^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e}+\frac {\left (2 b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \left (\frac {3 d^2 e}{\sqrt {d+e x} \sqrt {1-c^2 x^2}}+\frac {d^3}{x \sqrt {d+e x} \sqrt {1-c^2 x^2}}+\frac {3 d e^2 x}{\sqrt {d+e x} \sqrt {1-c^2 x^2}}+\frac {e^3 x^2}{\sqrt {d+e x} \sqrt {1-c^2 x^2}}\right ) \, dx}{5 e}\\ &=\frac {2 (d+e x)^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e}+\frac {1}{5} \left (6 b d^2 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {1-c^2 x^2}} \, dx+\frac {\left (2 b d^3 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {1}{x \sqrt {d+e x} \sqrt {1-c^2 x^2}} \, dx}{5 e}+\frac {1}{5} \left (6 b d e \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {x}{\sqrt {d+e x} \sqrt {1-c^2 x^2}} \, dx+\frac {1}{5} \left (2 b e^2 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {x^2}{\sqrt {d+e x} \sqrt {1-c^2 x^2}} \, dx\\ &=-\frac {4 b e \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {d+e x} \sqrt {1-c^2 x^2}}{15 c^2}+\frac {2 (d+e x)^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e}+\frac {1}{5} \left (6 b d \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {\sqrt {d+e x}}{\sqrt {1-c^2 x^2}} \, dx-\frac {1}{5} \left (6 b d^2 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {1-c^2 x^2}} \, dx+\frac {\left (2 b d^3 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {1}{x \sqrt {1-c x} \sqrt {1+c x} \sqrt {d+e x}} \, dx}{5 e}+\frac {\left (2 b e \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {e x-2 c^2 d x^2}{x \sqrt {d+e x} \sqrt {1-c^2 x^2}} \, dx}{15 c^2}-\frac {\left (12 b d^2 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {-\frac {c^2 (d+e x)}{-c^2 d-c e}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+\frac {2 c e x^2}{-c^2 d-c e}}} \, dx,x,\frac {\sqrt {1-c x}}{\sqrt {2}}\right )}{5 c \sqrt {d+e x}}\\ &=-\frac {4 b e \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {d+e x} \sqrt {1-c^2 x^2}}{15 c^2}+\frac {2 (d+e x)^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e}-\frac {12 b d^2 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {\frac {c (d+e x)}{c d+e}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{5 c \sqrt {d+e x}}-\frac {\left (4 b d^3 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \sqrt {2-x^2} \sqrt {d+\frac {e}{c}-\frac {e x^2}{c}}} \, dx,x,\sqrt {1-c x}\right )}{5 e}+\frac {\left (2 b e \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {e-2 c^2 d x}{\sqrt {d+e x} \sqrt {1-c^2 x^2}} \, dx}{15 c^2}-\frac {\left (12 b d \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {d+e x}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+\frac {2 c e x^2}{-c^2 d-c e}}}{\sqrt {1-x^2}} \, dx,x,\frac {\sqrt {1-c x}}{\sqrt {2}}\right )}{5 c \sqrt {-\frac {c^2 (d+e x)}{-c^2 d-c e}}}+\frac {\left (12 b d^2 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {-\frac {c^2 (d+e x)}{-c^2 d-c e}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+\frac {2 c e x^2}{-c^2 d-c e}}} \, dx,x,\frac {\sqrt {1-c x}}{\sqrt {2}}\right )}{5 c \sqrt {d+e x}}\\ &=-\frac {4 b e \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {d+e x} \sqrt {1-c^2 x^2}}{15 c^2}+\frac {2 (d+e x)^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e}-\frac {12 b d \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {d+e x} E\left (\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{5 c \sqrt {\frac {c (d+e x)}{c d+e}}}-\frac {1}{15} \left (4 b d \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {\sqrt {d+e x}}{\sqrt {1-c^2 x^2}} \, dx+\frac {\left (2 b \left (2 c^2 d^2+e^2\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {1-c^2 x^2}} \, dx}{15 c^2}-\frac {\left (4 b d^3 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {\frac {c (d+e x)}{c d+e}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \sqrt {2-x^2} \sqrt {1-\frac {e x^2}{c \left (d+\frac {e}{c}\right )}}} \, dx,x,\sqrt {1-c x}\right )}{5 e \sqrt {d+e x}}\\ &=-\frac {4 b e \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {d+e x} \sqrt {1-c^2 x^2}}{15 c^2}+\frac {2 (d+e x)^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e}-\frac {12 b d \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {d+e x} E\left (\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{5 c \sqrt {\frac {c (d+e x)}{c d+e}}}-\frac {4 b d^3 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {\frac {c (d+e x)}{c d+e}} \Pi \left (2;\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{5 e \sqrt {d+e x}}+\frac {\left (8 b d \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {d+e x}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+\frac {2 c e x^2}{-c^2 d-c e}}}{\sqrt {1-x^2}} \, dx,x,\frac {\sqrt {1-c x}}{\sqrt {2}}\right )}{15 c \sqrt {-\frac {c^2 (d+e x)}{-c^2 d-c e}}}-\frac {\left (4 b \left (2 c^2 d^2+e^2\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {-\frac {c^2 (d+e x)}{-c^2 d-c e}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+\frac {2 c e x^2}{-c^2 d-c e}}} \, dx,x,\frac {\sqrt {1-c x}}{\sqrt {2}}\right )}{15 c^3 \sqrt {d+e x}}\\ &=-\frac {4 b e \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {d+e x} \sqrt {1-c^2 x^2}}{15 c^2}+\frac {2 (d+e x)^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e}-\frac {28 b d \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {d+e x} E\left (\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{15 c \sqrt {\frac {c (d+e x)}{c d+e}}}-\frac {4 b \left (2 c^2 d^2+e^2\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {\frac {c (d+e x)}{c d+e}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{15 c^3 \sqrt {d+e x}}-\frac {4 b d^3 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {\frac {c (d+e x)}{c d+e}} \Pi \left (2;\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{5 e \sqrt {d+e x}}\\ \end {align*}
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Mathematica [C] time = 10.78, size = 2653, normalized size = 7.73 \[ \text {Result too large to show} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 173.84, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a e x + a d + {\left (b e x + b d\right )} \operatorname {arsech}\left (c x\right )\right )} \sqrt {e x + d}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (e x + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arsech}\left (c x\right ) + a\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.23, size = 830, normalized size = 2.42 \[ \frac {\frac {2 \left (e x +d \right )^{\frac {5}{2}} a}{5}+2 b \left (\frac {\left (e x +d \right )^{\frac {5}{2}} \mathrm {arcsech}\left (c x \right )}{5}-\frac {2 e^{2} \sqrt {-\frac {\left (e x +d \right ) c -c d -e}{c x e}}\, x \sqrt {\frac {\left (e x +d \right ) c -c d +e}{c x e}}\, \left (\sqrt {\frac {c}{c d +e}}\, \left (e x +d \right )^{\frac {5}{2}} c^{2}+9 \sqrt {-\frac {\left (e x +d \right ) c -c d -e}{c d +e}}\, \sqrt {-\frac {\left (e x +d \right ) c -c d +e}{c d -e}}\, \EllipticF \left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d +e}}, \sqrt {\frac {c d +e}{c d -e}}\right ) c^{2} d^{2}-7 \sqrt {-\frac {\left (e x +d \right ) c -c d -e}{c d +e}}\, \sqrt {-\frac {\left (e x +d \right ) c -c d +e}{c d -e}}\, \EllipticE \left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d +e}}, \sqrt {\frac {c d +e}{c d -e}}\right ) c^{2} d^{2}-3 \sqrt {-\frac {\left (e x +d \right ) c -c d -e}{c d +e}}\, \sqrt {-\frac {\left (e x +d \right ) c -c d +e}{c d -e}}\, \EllipticPi \left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d +e}}, \frac {c d +e}{c d}, \frac {\sqrt {\frac {c}{c d -e}}}{\sqrt {\frac {c}{c d +e}}}\right ) c^{2} d^{2}-2 \sqrt {\frac {c}{c d +e}}\, \left (e x +d \right )^{\frac {3}{2}} c^{2} d -7 \sqrt {-\frac {\left (e x +d \right ) c -c d -e}{c d +e}}\, \sqrt {-\frac {\left (e x +d \right ) c -c d +e}{c d -e}}\, \EllipticF \left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d +e}}, \sqrt {\frac {c d +e}{c d -e}}\right ) c d e +7 \sqrt {-\frac {\left (e x +d \right ) c -c d -e}{c d +e}}\, \sqrt {-\frac {\left (e x +d \right ) c -c d +e}{c d -e}}\, \EllipticE \left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d +e}}, \sqrt {\frac {c d +e}{c d -e}}\right ) c d e +\sqrt {\frac {c}{c d +e}}\, \sqrt {e x +d}\, c^{2} d^{2}+\sqrt {-\frac {\left (e x +d \right ) c -c d -e}{c d +e}}\, \sqrt {-\frac {\left (e x +d \right ) c -c d +e}{c d -e}}\, \EllipticF \left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d +e}}, \sqrt {\frac {c d +e}{c d -e}}\right ) e^{2}-\sqrt {\frac {c}{c d +e}}\, \sqrt {e x +d}\, e^{2}\right )}{15 c \sqrt {\frac {c}{c d +e}}\, \left (\left (e x +d \right )^{2} c^{2}-2 \left (e x +d \right ) c^{2} d +c^{2} d^{2}-e^{2}\right )}\right )}{e} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right )\,{\left (d+e\,x\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b \operatorname {asech}{\left (c x \right )}\right ) \left (d + e x\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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