Optimal. Leaf size=343 \[ -\frac {4 b e \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {d+e x} \sqrt {1-c^2 x^2}}{15 c^2}+\frac {2 (d+e x)^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e}-\frac {28 b d \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {d+e x} E\left (\text {ArcSin}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{15 c \sqrt {\frac {c (d+e x)}{c d+e}}}-\frac {4 b \left (2 c^2 d^2+e^2\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {\frac {c (d+e x)}{c d+e}} F\left (\text {ArcSin}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{15 c^3 \sqrt {d+e x}}-\frac {4 b d^3 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {\frac {c (d+e x)}{c d+e}} \Pi \left (2;\text {ArcSin}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{5 e \sqrt {d+e x}} \]
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Rubi [A]
time = 0.45, 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 = {6423, 972,
733, 430, 946, 174, 552, 551, 858, 435, 945, 1598} \begin {gather*} \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 (\text {ArcSin}\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;\text {ArcSin}\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 (\text {ArcSin}\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 e \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2} \sqrt {d+e x}}{15 c^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 174
Rule 430
Rule 435
Rule 551
Rule 552
Rule 733
Rule 858
Rule 945
Rule 946
Rule 972
Rule 1598
Rule 6423
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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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] Result contains complex when optimal does not.
time = 18.89, size = 2653, normalized size = 7.73 \begin {gather*} \text {Result too large to show} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(817\) vs.
\(2(302)=604\).
time = 0.49, size = 818, normalized size = 2.38
method | result | size |
derivativedivides | \(\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 {-c \left (e x +d \right )+c d +e}{c e x}}\, x \sqrt {-\frac {-c \left (e x +d \right )+c d -e}{c e x}}\, \left (\sqrt {\frac {c}{c d +e}}\, c^{2} \left (e x +d \right )^{\frac {5}{2}}-2 \sqrt {\frac {c}{c d +e}}\, c^{2} d \left (e x +d \right )^{\frac {3}{2}}+9 \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c d +e}}\, \sqrt {\frac {-c \left (e x +d \right )+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 {-c \left (e x +d \right )+c d +e}{c d +e}}\, \sqrt {\frac {-c \left (e x +d \right )+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 {-c \left (e x +d \right )+c d +e}{c d +e}}\, \sqrt {\frac {-c \left (e x +d \right )+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}+\sqrt {\frac {c}{c d +e}}\, c^{2} d^{2} \sqrt {e x +d}-7 \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c d +e}}\, \sqrt {\frac {-c \left (e x +d \right )+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 {-c \left (e x +d \right )+c d +e}{c d +e}}\, \sqrt {\frac {-c \left (e x +d \right )+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 \left (e x +d \right )+c d +e}{c d +e}}\, \sqrt {\frac {-c \left (e x +d \right )+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}}\, e^{2} \sqrt {e x +d}\right )}{15 c \sqrt {\frac {c}{c d +e}}\, \left (c^{2} \left (e x +d \right )^{2}-2 c^{2} d \left (e x +d \right )+c^{2} d^{2}-e^{2}\right )}\right )}{e}\) | \(818\) |
default | \(\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 {-c \left (e x +d \right )+c d +e}{c e x}}\, x \sqrt {-\frac {-c \left (e x +d \right )+c d -e}{c e x}}\, \left (\sqrt {\frac {c}{c d +e}}\, c^{2} \left (e x +d \right )^{\frac {5}{2}}-2 \sqrt {\frac {c}{c d +e}}\, c^{2} d \left (e x +d \right )^{\frac {3}{2}}+9 \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c d +e}}\, \sqrt {\frac {-c \left (e x +d \right )+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 {-c \left (e x +d \right )+c d +e}{c d +e}}\, \sqrt {\frac {-c \left (e x +d \right )+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 {-c \left (e x +d \right )+c d +e}{c d +e}}\, \sqrt {\frac {-c \left (e x +d \right )+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}+\sqrt {\frac {c}{c d +e}}\, c^{2} d^{2} \sqrt {e x +d}-7 \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c d +e}}\, \sqrt {\frac {-c \left (e x +d \right )+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 {-c \left (e x +d \right )+c d +e}{c d +e}}\, \sqrt {\frac {-c \left (e x +d \right )+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 \left (e x +d \right )+c d +e}{c d +e}}\, \sqrt {\frac {-c \left (e x +d \right )+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}}\, e^{2} \sqrt {e x +d}\right )}{15 c \sqrt {\frac {c}{c d +e}}\, \left (c^{2} \left (e x +d \right )^{2}-2 c^{2} d \left (e x +d \right )+c^{2} d^{2}-e^{2}\right )}\right )}{e}\) | \(818\) |
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 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \left (a + b \operatorname {asech}{\left (c x \right )}\right ) \left (d + e x\right )^{\frac {3}{2}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right )\,{\left (d+e\,x\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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