Optimal. Leaf size=298 \[ -\frac{2 \left (a+b \sec ^{-1}(c x)\right )}{3 e (d+e x)^{3/2}}-\frac{4 b e \left (1-c^2 x^2\right )}{3 c d x \sqrt{1-\frac{1}{c^2 x^2}} \left (c^2 d^2-e^2\right ) \sqrt{d+e x}}+\frac{4 b \sqrt{1-c^2 x^2} \sqrt{d+e x} E\left (\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )}{3 d x \sqrt{1-\frac{1}{c^2 x^2}} \left (c^2 d^2-e^2\right ) \sqrt{\frac{c (d+e x)}{c d+e}}}-\frac{4 b \sqrt{1-c^2 x^2} \sqrt{\frac{c (d+e x)}{c d+e}} \Pi \left (2;\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )}{3 c d e x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{d+e x}} \]
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Rubi [A] time = 0.405632, antiderivative size = 298, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 11, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.611, Rules used = {5226, 1574, 958, 745, 21, 719, 424, 933, 168, 538, 537} \[ -\frac{2 \left (a+b \sec ^{-1}(c x)\right )}{3 e (d+e x)^{3/2}}-\frac{4 b e \left (1-c^2 x^2\right )}{3 c d x \sqrt{1-\frac{1}{c^2 x^2}} \left (c^2 d^2-e^2\right ) \sqrt{d+e x}}+\frac{4 b \sqrt{1-c^2 x^2} \sqrt{d+e x} E\left (\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )}{3 d x \sqrt{1-\frac{1}{c^2 x^2}} \left (c^2 d^2-e^2\right ) \sqrt{\frac{c (d+e x)}{c d+e}}}-\frac{4 b \sqrt{1-c^2 x^2} \sqrt{\frac{c (d+e x)}{c d+e}} \Pi \left (2;\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )}{3 c d e x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{d+e x}} \]
Antiderivative was successfully verified.
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Rule 5226
Rule 1574
Rule 958
Rule 745
Rule 21
Rule 719
Rule 424
Rule 933
Rule 168
Rule 538
Rule 537
Rubi steps
\begin{align*} \int \frac{a+b \sec ^{-1}(c x)}{(d+e x)^{5/2}} \, dx &=-\frac{2 \left (a+b \sec ^{-1}(c x)\right )}{3 e (d+e x)^{3/2}}+\frac{(2 b) \int \frac{1}{\sqrt{1-\frac{1}{c^2 x^2}} x^2 (d+e x)^{3/2}} \, dx}{3 c e}\\ &=-\frac{2 \left (a+b \sec ^{-1}(c x)\right )}{3 e (d+e x)^{3/2}}+\frac{\left (2 b \sqrt{-\frac{1}{c^2}+x^2}\right ) \int \frac{1}{x (d+e x)^{3/2} \sqrt{-\frac{1}{c^2}+x^2}} \, dx}{3 c e \sqrt{1-\frac{1}{c^2 x^2}} x}\\ &=-\frac{2 \left (a+b \sec ^{-1}(c x)\right )}{3 e (d+e x)^{3/2}}+\frac{\left (2 b \sqrt{-\frac{1}{c^2}+x^2}\right ) \int \left (-\frac{e}{d (d+e x)^{3/2} \sqrt{-\frac{1}{c^2}+x^2}}+\frac{1}{d x \sqrt{d+e x} \sqrt{-\frac{1}{c^2}+x^2}}\right ) \, dx}{3 c e \sqrt{1-\frac{1}{c^2 x^2}} x}\\ &=-\frac{2 \left (a+b \sec ^{-1}(c x)\right )}{3 e (d+e x)^{3/2}}-\frac{\left (2 b \sqrt{-\frac{1}{c^2}+x^2}\right ) \int \frac{1}{(d+e x)^{3/2} \sqrt{-\frac{1}{c^2}+x^2}} \, dx}{3 c d \sqrt{1-\frac{1}{c^2 x^2}} x}+\frac{\left (2 b \sqrt{-\frac{1}{c^2}+x^2}\right ) \int \frac{1}{x \sqrt{d+e x} \sqrt{-\frac{1}{c^2}+x^2}} \, dx}{3 c d e \sqrt{1-\frac{1}{c^2 x^2}} x}\\ &=-\frac{4 b e \left (1-c^2 x^2\right )}{3 c d \left (c^2 d^2-e^2\right ) \sqrt{1-\frac{1}{c^2 x^2}} x \sqrt{d+e x}}-\frac{2 \left (a+b \sec ^{-1}(c x)\right )}{3 e (d+e x)^{3/2}}+\frac{\left (4 b \sqrt{-\frac{1}{c^2}+x^2}\right ) \int \frac{-\frac{d}{2}-\frac{e x}{2}}{\sqrt{d+e x} \sqrt{-\frac{1}{c^2}+x^2}} \, dx}{3 c d \left (d^2-\frac{e^2}{c^2}\right ) \sqrt{1-\frac{1}{c^2 x^2}} x}+\frac{\left (2 b \sqrt{1-c^2 x^2}\right ) \int \frac{1}{x \sqrt{1-c x} \sqrt{1+c x} \sqrt{d+e x}} \, dx}{3 c d e \sqrt{1-\frac{1}{c^2 x^2}} x}\\ &=-\frac{4 b e \left (1-c^2 x^2\right )}{3 c d \left (c^2 d^2-e^2\right ) \sqrt{1-\frac{1}{c^2 x^2}} x \sqrt{d+e x}}-\frac{2 \left (a+b \sec ^{-1}(c x)\right )}{3 e (d+e x)^{3/2}}-\frac{\left (2 b \sqrt{-\frac{1}{c^2}+x^2}\right ) \int \frac{\sqrt{d+e x}}{\sqrt{-\frac{1}{c^2}+x^2}} \, dx}{3 c d \left (d^2-\frac{e^2}{c^2}\right ) \sqrt{1-\frac{1}{c^2 x^2}} x}-\frac{\left (4 b \sqrt{1-c^2 x^2}\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 )}{3 c d e \sqrt{1-\frac{1}{c^2 x^2}} x}\\ &=-\frac{4 b e \left (1-c^2 x^2\right )}{3 c d \left (c^2 d^2-e^2\right ) \sqrt{1-\frac{1}{c^2 x^2}} x \sqrt{d+e x}}-\frac{2 \left (a+b \sec ^{-1}(c x)\right )}{3 e (d+e x)^{3/2}}-\frac{\left (4 b \sqrt{\frac{c (d+e x)}{c d+e}} \sqrt{1-c^2 x^2}\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 )}{3 c d e \sqrt{1-\frac{1}{c^2 x^2}} x \sqrt{d+e x}}+\frac{\left (4 b \sqrt{d+e x} \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1-\frac{2 e x^2}{c \left (d+\frac{e}{c}\right )}}}{\sqrt{1-x^2}} \, dx,x,\frac{\sqrt{1-c x}}{\sqrt{2}}\right )}{3 c^2 d \left (d^2-\frac{e^2}{c^2}\right ) \sqrt{1-\frac{1}{c^2 x^2}} x \sqrt{\frac{d+e x}{d+\frac{e}{c}}}}\\ &=-\frac{4 b e \left (1-c^2 x^2\right )}{3 c d \left (c^2 d^2-e^2\right ) \sqrt{1-\frac{1}{c^2 x^2}} x \sqrt{d+e x}}-\frac{2 \left (a+b \sec ^{-1}(c x)\right )}{3 e (d+e x)^{3/2}}+\frac{4 b \sqrt{d+e x} \sqrt{1-c^2 x^2} E\left (\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )}{3 d \left (c^2 d^2-e^2\right ) \sqrt{1-\frac{1}{c^2 x^2}} x \sqrt{\frac{c (d+e x)}{c d+e}}}-\frac{4 b \sqrt{\frac{c (d+e x)}{c d+e}} \sqrt{1-c^2 x^2} \Pi \left (2;\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )}{3 c d e \sqrt{1-\frac{1}{c^2 x^2}} x \sqrt{d+e x}}\\ \end{align*}
Mathematica [C] time = 6.17543, size = 326, normalized size = 1.09 \[ \frac{2 \left (-\frac{2 i b \sqrt{\frac{e (c x+1)}{e-c d}} \sqrt{\frac{e-c e x}{c d+e}} \left (c d \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{-\frac{c}{c d+e}} \sqrt{d+e x}\right ),\frac{c d+e}{c d-e}\right )-c d E\left (i \sinh ^{-1}\left (\sqrt{-\frac{c}{c d+e}} \sqrt{d+e x}\right )|\frac{c d+e}{c d-e}\right )+(c d+e) \Pi \left (\frac{e}{c d}+1;i \sinh ^{-1}\left (\sqrt{-\frac{c}{c d+e}} \sqrt{d+e x}\right )|\frac{c d+e}{c d-e}\right )\right )}{d^2 x \sqrt{1-\frac{1}{c^2 x^2}} \left (-\frac{c}{c d+e}\right )^{3/2} (c d+e)^2}-\frac{a}{(d+e x)^{3/2}}+\frac{2 b c e^2 x \sqrt{1-\frac{1}{c^2 x^2}}}{\left (c^2 d^3-d e^2\right ) \sqrt{d+e x}}-\frac{b \sec ^{-1}(c x)}{(d+e x)^{3/2}}\right )}{3 e} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.267, size = 886, normalized size = 3. \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] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{e x + d}{\left (b \operatorname{arcsec}\left (c x\right ) + a\right )}}{e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}}, x\right ) \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{arcsec}\left (c x\right ) + a}{{\left (e x + d\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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