Optimal. Leaf size=609 \[ -\frac {2 \left (a+b \text {sech}^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}+\frac {16 b c^2 e \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2}}{15 \left (c^2 d^2-e^2\right )^2 \sqrt {d+e x}}+\frac {4 b e \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2}}{5 d^2 \left (c^2 d^2-e^2\right ) \sqrt {d+e x}}+\frac {4 b e \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2}}{15 d \left (c^2 d^2-e^2\right ) (d+e x)^{3/2}}+\frac {4 b c \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {\frac {c (d+e x)}{c d+e}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{15 d \left (c^2 d^2-e^2\right ) \sqrt {d+e x}}-\frac {4 b c \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {d+e x} E\left (\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{5 d^2 \left (c^2 d^2-e^2\right ) \sqrt {\frac {c (d+e x)}{c d+e}}}-\frac {16 b c^3 \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {d+e x} E\left (\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{15 \left (c^2 d^2-e^2\right )^2 \sqrt {\frac {c (d+e x)}{c d+e}}}+\frac {4 b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {\frac {c (d+e x)}{c d+e}} \Pi \left (2;\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{5 d^2 e \sqrt {d+e x}} \]
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Rubi [A] time = 0.59, antiderivative size = 609, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 13, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.722, Rules used = {6288, 958, 745, 835, 844, 719, 424, 419, 21, 932, 168, 538, 537} \[ -\frac {2 \left (a+b \text {sech}^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}+\frac {16 b c^2 e \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2}}{15 \left (c^2 d^2-e^2\right )^2 \sqrt {d+e x}}+\frac {4 b e \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2}}{5 d^2 \left (c^2 d^2-e^2\right ) \sqrt {d+e x}}+\frac {4 b e \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2}}{15 d \left (c^2 d^2-e^2\right ) (d+e x)^{3/2}}+\frac {4 b c \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {\frac {c (d+e x)}{c d+e}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{15 d \left (c^2 d^2-e^2\right ) \sqrt {d+e x}}-\frac {16 b c^3 \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {d+e x} E\left (\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{15 \left (c^2 d^2-e^2\right )^2 \sqrt {\frac {c (d+e x)}{c d+e}}}-\frac {4 b c \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {d+e x} E\left (\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{5 d^2 \left (c^2 d^2-e^2\right ) \sqrt {\frac {c (d+e x)}{c d+e}}}+\frac {4 b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {\frac {c (d+e x)}{c d+e}} \Pi \left (2;\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{5 d^2 e \sqrt {d+e x}} \]
Antiderivative was successfully verified.
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Rule 21
Rule 168
Rule 419
Rule 424
Rule 537
Rule 538
Rule 719
Rule 745
Rule 835
Rule 844
Rule 932
Rule 958
Rule 6288
Rubi steps
\begin {align*} \int \frac {a+b \text {sech}^{-1}(c x)}{(d+e x)^{7/2}} \, dx &=-\frac {2 \left (a+b \text {sech}^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}-\frac {\left (2 b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {1}{x (d+e x)^{5/2} \sqrt {1-c^2 x^2}} \, dx}{5 e}\\ &=-\frac {2 \left (a+b \text {sech}^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}-\frac {\left (2 b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \left (-\frac {e}{d (d+e x)^{5/2} \sqrt {1-c^2 x^2}}-\frac {e}{d^2 (d+e x)^{3/2} \sqrt {1-c^2 x^2}}+\frac {1}{d^2 x \sqrt {d+e x} \sqrt {1-c^2 x^2}}\right ) \, dx}{5 e}\\ &=-\frac {2 \left (a+b \text {sech}^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}+\frac {\left (2 b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {1}{(d+e x)^{3/2} \sqrt {1-c^2 x^2}} \, dx}{5 d^2}+\frac {\left (2 b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {1}{(d+e x)^{5/2} \sqrt {1-c^2 x^2}} \, dx}{5 d}-\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}{5 d^2 e}\\ &=\frac {4 b e \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{15 d \left (c^2 d^2-e^2\right ) (d+e x)^{3/2}}+\frac {4 b e \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{5 d^2 \left (c^2 d^2-e^2\right ) \sqrt {d+e x}}-\frac {2 \left (a+b \text {sech}^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}-\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}{5 d^2 e}-\frac {\left (4 b c^2 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {-\frac {d}{2}-\frac {e x}{2}}{\sqrt {d+e x} \sqrt {1-c^2 x^2}} \, dx}{5 d^2 \left (c^2 d^2-e^2\right )}-\frac {\left (4 b c^2 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {-\frac {3 d}{2}+\frac {e x}{2}}{(d+e x)^{3/2} \sqrt {1-c^2 x^2}} \, dx}{15 d \left (c^2 d^2-e^2\right )}\\ &=\frac {4 b e \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{15 d \left (c^2 d^2-e^2\right ) (d+e x)^{3/2}}+\frac {16 b c^2 e \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{15 \left (c^2 d^2-e^2\right )^2 \sqrt {d+e x}}+\frac {4 b e \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{5 d^2 \left (c^2 d^2-e^2\right ) \sqrt {d+e x}}-\frac {2 \left (a+b \text {sech}^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}+\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 )}{5 d^2 e}-\frac {\left (8 b c^2 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {\frac {1}{4} \left (-3 c^2 d^2-e^2\right )-c^2 d e x}{\sqrt {d+e x} \sqrt {1-c^2 x^2}} \, dx}{15 d \left (c^2 d^2-e^2\right )^2}+\frac {\left (2 b c^2 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {\sqrt {d+e x}}{\sqrt {1-c^2 x^2}} \, dx}{5 d^2 \left (c^2 d^2-e^2\right )}\\ &=\frac {4 b e \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{15 d \left (c^2 d^2-e^2\right ) (d+e x)^{3/2}}+\frac {16 b c^2 e \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{15 \left (c^2 d^2-e^2\right )^2 \sqrt {d+e x}}+\frac {4 b e \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{5 d^2 \left (c^2 d^2-e^2\right ) \sqrt {d+e x}}-\frac {2 \left (a+b \text {sech}^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}+\frac {\left (8 b c^4 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {\sqrt {d+e x}}{\sqrt {1-c^2 x^2}} \, dx}{15 \left (c^2 d^2-e^2\right )^2}-\frac {\left (2 b c^2 (c d-e) (c d+e) \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 d \left (c^2 d^2-e^2\right )^2}+\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 )}{5 d^2 e \sqrt {d+e x}}-\frac {\left (4 b c \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 d^2 \left (c^2 d^2-e^2\right ) \sqrt {-\frac {c^2 (d+e x)}{-c^2 d-c e}}}\\ &=\frac {4 b e \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{15 d \left (c^2 d^2-e^2\right ) (d+e x)^{3/2}}+\frac {16 b c^2 e \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{15 \left (c^2 d^2-e^2\right )^2 \sqrt {d+e x}}+\frac {4 b e \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{5 d^2 \left (c^2 d^2-e^2\right ) \sqrt {d+e x}}-\frac {2 \left (a+b \text {sech}^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}-\frac {4 b c \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 d^2 \left (c^2 d^2-e^2\right ) \sqrt {\frac {c (d+e x)}{c d+e}}}+\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 )}{5 d^2 e \sqrt {d+e x}}-\frac {\left (16 b c^3 \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 \left (c^2 d^2-e^2\right )^2 \sqrt {-\frac {c^2 (d+e x)}{-c^2 d-c e}}}+\frac {\left (4 b c (c d-e) (c d+e) \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 d \left (c^2 d^2-e^2\right )^2 \sqrt {d+e x}}\\ &=\frac {4 b e \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{15 d \left (c^2 d^2-e^2\right ) (d+e x)^{3/2}}+\frac {16 b c^2 e \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{15 \left (c^2 d^2-e^2\right )^2 \sqrt {d+e x}}+\frac {4 b e \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{5 d^2 \left (c^2 d^2-e^2\right ) \sqrt {d+e x}}-\frac {2 \left (a+b \text {sech}^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}-\frac {16 b c^3 \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 \left (c^2 d^2-e^2\right )^2 \sqrt {\frac {c (d+e x)}{c d+e}}}-\frac {4 b c \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 d^2 \left (c^2 d^2-e^2\right ) \sqrt {\frac {c (d+e x)}{c d+e}}}+\frac {4 b c \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 d \left (c^2 d^2-e^2\right ) \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 )}{5 d^2 e \sqrt {d+e x}}\\ \end {align*}
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Mathematica [C] time = 14.96, size = 8675, normalized size = 14.24 \[ \text {Result too large to show} \]
Warning: Unable to verify antiderivative.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
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 \[ \int \frac {b \operatorname {arsech}\left (c x\right ) + a}{{\left (e x + d\right )}^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.13, size = 1632, normalized size = 2.68 \[ \text {result too large to display} \]
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 \[ \int \frac {a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )}{{\left (d+e\,x\right )}^{7/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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