Optimal. Leaf size=105 \[ \frac {4 b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {\frac {c (d+e x)}{c d+e}} \Pi \left (2;\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{e \sqrt {d+e x}}-\frac {2 \left (a+b \text {sech}^{-1}(c x)\right )}{e \sqrt {d+e x}} \]
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Rubi [A] time = 0.18, antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {6288, 932, 168, 538, 537} \[ \frac {4 b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {\frac {c (d+e x)}{c d+e}} \Pi \left (2;\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{e \sqrt {d+e x}}-\frac {2 \left (a+b \text {sech}^{-1}(c x)\right )}{e \sqrt {d+e x}} \]
Antiderivative was successfully verified.
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Rule 168
Rule 537
Rule 538
Rule 932
Rule 6288
Rubi steps
\begin {align*} \int \frac {a+b \text {sech}^{-1}(c x)}{(d+e x)^{3/2}} \, dx &=-\frac {2 \left (a+b \text {sech}^{-1}(c x)\right )}{e \sqrt {d+e x}}-\frac {\left (2 b \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}{e}\\ &=-\frac {2 \left (a+b \text {sech}^{-1}(c x)\right )}{e \sqrt {d+e x}}-\frac {\left (2 b \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}{e}\\ &=-\frac {2 \left (a+b \text {sech}^{-1}(c x)\right )}{e \sqrt {d+e x}}+\frac {\left (4 b \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 )}{e}\\ &=-\frac {2 \left (a+b \text {sech}^{-1}(c x)\right )}{e \sqrt {d+e x}}+\frac {\left (4 b \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 )}{e \sqrt {d+e x}}\\ &=-\frac {2 \left (a+b \text {sech}^{-1}(c x)\right )}{e \sqrt {d+e x}}+\frac {4 b \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 )}{e \sqrt {d+e x}}\\ \end {align*}
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Mathematica [C] time = 11.34, size = 1675, normalized size = 15.95 \[ -\frac {2 a}{e \sqrt {d+e x}}-\frac {2 b \text {sech}^{-1}(c x)}{e \sqrt {d+e x}}+\frac {4 i b \left (2 \sqrt {-\frac {i \left (c \sqrt {\frac {1-c x}{c x+1}} d+\sqrt {-c d-e} \sqrt {c d-e}-e \sqrt {\frac {1-c x}{c x+1}}\right )}{\left (-i c d+i e+\sqrt {-c d-e} \sqrt {c d-e}\right ) \left (\sqrt {\frac {1-c x}{c x+1}}-i\right )}} \sqrt {-\frac {i \left (-c \sqrt {\frac {1-c x}{c x+1}} d+\sqrt {-c d-e} \sqrt {c d-e}+e \sqrt {\frac {1-c x}{c x+1}}\right )}{\left (i c d-i e+\sqrt {-c d-e} \sqrt {c d-e}\right ) \left (\sqrt {\frac {1-c x}{c x+1}}-i\right )}} \left (\frac {1-c x}{c x+1}+1\right ) F\left (\sin ^{-1}\left (\sqrt {\frac {\left (\sqrt {-c d-e}-i \sqrt {c d-e}\right ) \left (\sqrt {\frac {1-c x}{c x+1}}+i\right )}{\left (\sqrt {-c d-e}+i \sqrt {c d-e}\right ) \left (\sqrt {\frac {1-c x}{c x+1}}-i\right )}}\right )|\frac {\left (\sqrt {-c d-e}+i \sqrt {c d-e}\right )^2}{\left (\sqrt {-c d-e}-i \sqrt {c d-e}\right )^2}\right )+\sqrt {\frac {\left (\sqrt {-c d-e}-i \sqrt {c d-e}\right ) \left (\sqrt {\frac {1-c x}{c x+1}}+i\right )}{\left (\sqrt {-c d-e}+i \sqrt {c d-e}\right ) \left (\sqrt {\frac {1-c x}{c x+1}}-i\right )}} \sqrt {\frac {1-c x}{c x+1}+1} \sqrt {\frac {-\frac {(1-c x) e}{c x+1}+e+c d \left (\frac {1-c x}{c x+1}+1\right )}{c d+e}} F\left (i \sinh ^{-1}\left (\sqrt {\frac {1-c x}{c x+1}}\right )|\frac {c d-e}{c d+e}\right )+2 i \sqrt {-\frac {i \left (c \sqrt {\frac {1-c x}{c x+1}} d+\sqrt {-c d-e} \sqrt {c d-e}-e \sqrt {\frac {1-c x}{c x+1}}\right )}{\left (-i c d+i e+\sqrt {-c d-e} \sqrt {c d-e}\right ) \left (\sqrt {\frac {1-c x}{c x+1}}-i\right )}} \sqrt {-\frac {i \left (-c \sqrt {\frac {1-c x}{c x+1}} d+\sqrt {-c d-e} \sqrt {c d-e}+e \sqrt {\frac {1-c x}{c x+1}}\right )}{\left (i c d-i e+\sqrt {-c d-e} \sqrt {c d-e}\right ) \left (\sqrt {\frac {1-c x}{c x+1}}-i\right )}} \left (\frac {1-c x}{c x+1}+1\right ) \left (\Pi \left (\frac {i \sqrt {-c d-e}-\sqrt {c d-e}}{\sqrt {-c d-e}-i \sqrt {c d-e}};\sin ^{-1}\left (\sqrt {\frac {\left (\sqrt {-c d-e}-i \sqrt {c d-e}\right ) \left (\sqrt {\frac {1-c x}{c x+1}}+i\right )}{\left (\sqrt {-c d-e}+i \sqrt {c d-e}\right ) \left (\sqrt {\frac {1-c x}{c x+1}}-i\right )}}\right )|\frac {\left (\sqrt {-c d-e}+i \sqrt {c d-e}\right )^2}{\left (\sqrt {-c d-e}-i \sqrt {c d-e}\right )^2}\right )-\Pi \left (\frac {\sqrt {c d-e}-i \sqrt {-c d-e}}{\sqrt {-c d-e}-i \sqrt {c d-e}};\sin ^{-1}\left (\sqrt {\frac {\left (\sqrt {-c d-e}-i \sqrt {c d-e}\right ) \left (\sqrt {\frac {1-c x}{c x+1}}+i\right )}{\left (\sqrt {-c d-e}+i \sqrt {c d-e}\right ) \left (\sqrt {\frac {1-c x}{c x+1}}-i\right )}}\right )|\frac {\left (\sqrt {-c d-e}+i \sqrt {c d-e}\right )^2}{\left (\sqrt {-c d-e}-i \sqrt {c d-e}\right )^2}\right )\right )\right )}{e \sqrt {\frac {\left (\sqrt {-c d-e}-i \sqrt {c d-e}\right ) \left (\sqrt {\frac {1-c x}{c x+1}}+i\right )}{\left (\sqrt {-c d-e}+i \sqrt {c d-e}\right ) \left (\sqrt {\frac {1-c x}{c x+1}}-i\right )}} \left (\frac {1-c x}{c x+1}+1\right ) \sqrt {\frac {c d+\frac {c (1-c x) d}{c x+1}+e-\frac {e (1-c x)}{c x+1}}{\frac {(1-c x) c}{c x+1}+c}}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 7.57, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {e x + d} {\left (b \operatorname {arsech}\left (c x\right ) + a\right )}}{e^{2} x^{2} + 2 \, d e x + d^{2}}, x\right ) \]
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 \[ \int \frac {b \operatorname {arsech}\left (c x\right ) + a}{{\left (e x + d\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.09, size = 253, normalized size = 2.41 \[ \frac {-\frac {2 a}{\sqrt {e x +d}}+2 b \left (-\frac {\mathrm {arcsech}\left (c x \right )}{\sqrt {e x +d}}-\frac {2 c \,e^{2} \sqrt {-\frac {\left (e x +d \right ) c -c d -e}{c x e}}\, x \sqrt {\frac {\left (e x +d \right ) c -c d +e}{c x e}}\, \EllipticPi \left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d +e}}, \frac {c d +e}{c d}, \frac {\sqrt {\frac {c}{c d -e}}}{\sqrt {\frac {c}{c d +e}}}\right ) \sqrt {-\frac {\left (e x +d \right ) c -c d +e}{c d -e}}\, \sqrt {-\frac {\left (e x +d \right ) c -c d -e}{c d +e}}}{d \sqrt {\frac {c}{c d +e}}\, \left (\left (e x +d \right )^{2} c^{2}-2 \left (e x +d \right ) c^{2} d +c^{2} d^{2}-e^{2}\right )}\right )}{e} \]
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 = -0.01 \[ \int \frac {a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )}{{\left (d+e\,x\right )}^{3/2}} \,d x \]
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 \[ \int \frac {a + b \operatorname {asech}{\left (c x \right )}}{\left (d + e x\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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