Optimal. Leaf size=609 \[ \frac{4 b c \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{\frac{c (d+e x)}{c d+e}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right ),\frac{2 e}{c d+e}\right )}{15 d \left (c^2 d^2-e^2\right ) \sqrt{d+e x}}-\frac{2 \left (a+b \text{sech}^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}+\frac{16 b c^2 e \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2}}{15 \left (c^2 d^2-e^2\right )^2 \sqrt{d+e x}}+\frac{4 b e \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2}}{5 d^2 \left (c^2 d^2-e^2\right ) \sqrt{d+e x}}+\frac{4 b e \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2}}{15 d \left (c^2 d^2-e^2\right ) (d+e x)^{3/2}}-\frac{16 b c^3 \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{d+e x} E\left (\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )}{15 \left (c^2 d^2-e^2\right )^2 \sqrt{\frac{c (d+e x)}{c d+e}}}-\frac{4 b c \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{d+e x} E\left (\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )}{5 d^2 \left (c^2 d^2-e^2\right ) \sqrt{\frac{c (d+e x)}{c d+e}}}+\frac{4 b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{\frac{c (d+e x)}{c d+e}} \Pi \left (2;\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )}{5 d^2 e \sqrt{d+e x}} \]
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Rubi [A] time = 0.594117, antiderivative size = 609, normalized size of antiderivative = 1., 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 6288
Rule 958
Rule 745
Rule 835
Rule 844
Rule 719
Rule 424
Rule 419
Rule 21
Rule 932
Rule 168
Rule 538
Rule 537
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*}
Mathematica [C] time = 12.9241, size = 8675, normalized size = 14.24 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.352, size = 1632, normalized size = 2.7 \begin{align*} \text{result too large to display} \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 [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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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 \frac{b \operatorname{arsech}\left (c x\right ) + a}{{\left (e x + d\right )}^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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