Optimal. Leaf size=298 \[ -\frac{2 \left (a+b \csc ^{-1}(c x)\right )}{3 e (d+e x)^{3/2}}+\frac{4 b e \left (1-c^2 x^2\right )}{3 c d x \sqrt{1-\frac{1}{c^2 x^2}} \left (c^2 d^2-e^2\right ) \sqrt{d+e x}}-\frac{4 b \sqrt{1-c^2 x^2} \sqrt{d+e x} E\left (\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )}{3 d x \sqrt{1-\frac{1}{c^2 x^2}} \left (c^2 d^2-e^2\right ) \sqrt{\frac{c (d+e x)}{c d+e}}}+\frac{4 b \sqrt{1-c^2 x^2} \sqrt{\frac{c (d+e x)}{c d+e}} \Pi \left (2;\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )}{3 c d e x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{d+e x}} \]
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Rubi [A] time = 0.408926, 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 = {5227, 1574, 958, 745, 21, 719, 424, 933, 168, 538, 537} \[ -\frac{2 \left (a+b \csc ^{-1}(c x)\right )}{3 e (d+e x)^{3/2}}+\frac{4 b e \left (1-c^2 x^2\right )}{3 c d x \sqrt{1-\frac{1}{c^2 x^2}} \left (c^2 d^2-e^2\right ) \sqrt{d+e x}}-\frac{4 b \sqrt{1-c^2 x^2} \sqrt{d+e x} E\left (\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )}{3 d x \sqrt{1-\frac{1}{c^2 x^2}} \left (c^2 d^2-e^2\right ) \sqrt{\frac{c (d+e x)}{c d+e}}}+\frac{4 b \sqrt{1-c^2 x^2} \sqrt{\frac{c (d+e x)}{c d+e}} \Pi \left (2;\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )}{3 c d e x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{d+e x}} \]
Antiderivative was successfully verified.
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Rule 5227
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 \csc ^{-1}(c x)}{(d+e x)^{5/2}} \, dx &=-\frac{2 \left (a+b \csc ^{-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 \csc ^{-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 \csc ^{-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 \csc ^{-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 \csc ^{-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 \csc ^{-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 \csc ^{-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 \csc ^{-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 [B] time = 13.5131, size = 725, normalized size = 2.43 \[ \frac{b \left (\frac{2 (c x)^{5/2} \left (\frac{d}{x}+e\right )^{5/2} \left (\frac{2 c d \sqrt{1-c^2 x^2} \sqrt{\frac{c d+c e x}{c d+e}} \Pi \left (2;\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )}{\sqrt{1-\frac{1}{c^2 x^2}} (c x)^{3/2} \sqrt{\frac{d}{x}+e}}-\frac{2 e \cos \left (2 \csc ^{-1}(c x)\right ) \left (c^2 d x \sqrt{1-c^2 x^2} \sqrt{\frac{c d+c e x}{c d+e}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right ),\frac{2 e}{c d+e}\right )-\frac{c x (c x+1) \sqrt{\frac{e-c e x}{c d+e}} \sqrt{\frac{c d+c e x}{c d-e}} \left ((c d+e) E\left (\sin ^{-1}\left (\sqrt{\frac{c d+c e x}{c d-e}}\right )|\frac{c d-e}{c d+e}\right )-e \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{c d+c e x}{c d-e}}\right ),\frac{c d-e}{c d+e}\right )\right )}{\sqrt{\frac{e (c x+1)}{e-c d}}}+\left (c^2 x^2-1\right ) (c d+c e x)+c e x \sqrt{1-c^2 x^2} \sqrt{\frac{c d+c e x}{c d+e}} \Pi \left (2;\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )\right )}{c d \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{c x} \left (c^2 x^2-2\right ) \sqrt{\frac{d}{x}+e}}\right )}{3 e (c d-e) (c d+e) (d+e x)^{5/2}}-\frac{c^3 x^3 \left (\frac{d}{x}+e\right )^3 \left (-\frac{4 \sqrt{1-\frac{1}{c^2 x^2}}}{3 c d \left (c^2 d^2-e^2\right )}-\frac{4 \left (c^2 d^2 \csc ^{-1}(c x)-c d e \sqrt{1-\frac{1}{c^2 x^2}}-e^2 \csc ^{-1}(c x)\right )}{3 c^2 d^2 \left (c^2 d^2-e^2\right ) \left (\frac{d}{x}+e\right )}+\frac{2 \csc ^{-1}(c x)}{3 c^2 d^2 e}+\frac{2 e \csc ^{-1}(c x)}{3 c^2 d^2 \left (\frac{d}{x}+e\right )^2}\right )}{(d+e x)^{5/2}}\right )}{c}-\frac{2 a}{3 e (d+e x)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.279, 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(-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{arccsc}\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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