Optimal. Leaf size=138 \[ \frac {b c x \sqrt {-1+c^2 x^2}}{3 d \left (c^2 d+e\right ) \sqrt {c^2 x^2} \sqrt {d+e x^2}}-\frac {a+b \csc ^{-1}(c x)}{3 e \left (d+e x^2\right )^{3/2}}+\frac {b c x \text {ArcTan}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {-1+c^2 x^2}}\right )}{3 d^{3/2} e \sqrt {c^2 x^2}} \]
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Rubi [A]
time = 0.10, antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {5345, 457, 98,
95, 210} \begin {gather*} -\frac {a+b \csc ^{-1}(c x)}{3 e \left (d+e x^2\right )^{3/2}}+\frac {b c x \text {ArcTan}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {c^2 x^2-1}}\right )}{3 d^{3/2} e \sqrt {c^2 x^2}}+\frac {b c x \sqrt {c^2 x^2-1}}{3 d \sqrt {c^2 x^2} \left (c^2 d+e\right ) \sqrt {d+e x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 95
Rule 98
Rule 210
Rule 457
Rule 5345
Rubi steps
\begin {align*} \int \frac {x \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^{5/2}} \, dx &=-\frac {a+b \csc ^{-1}(c x)}{3 e \left (d+e x^2\right )^{3/2}}-\frac {(b c x) \int \frac {1}{x \sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{3/2}} \, dx}{3 e \sqrt {c^2 x^2}}\\ &=-\frac {a+b \csc ^{-1}(c x)}{3 e \left (d+e x^2\right )^{3/2}}-\frac {(b c x) \text {Subst}\left (\int \frac {1}{x \sqrt {-1+c^2 x} (d+e x)^{3/2}} \, dx,x,x^2\right )}{6 e \sqrt {c^2 x^2}}\\ &=\frac {b c x \sqrt {-1+c^2 x^2}}{3 d \left (c^2 d+e\right ) \sqrt {c^2 x^2} \sqrt {d+e x^2}}-\frac {a+b \csc ^{-1}(c x)}{3 e \left (d+e x^2\right )^{3/2}}-\frac {(b c x) \text {Subst}\left (\int \frac {1}{x \sqrt {-1+c^2 x} \sqrt {d+e x}} \, dx,x,x^2\right )}{6 d e \sqrt {c^2 x^2}}\\ &=\frac {b c x \sqrt {-1+c^2 x^2}}{3 d \left (c^2 d+e\right ) \sqrt {c^2 x^2} \sqrt {d+e x^2}}-\frac {a+b \csc ^{-1}(c x)}{3 e \left (d+e x^2\right )^{3/2}}-\frac {(b c x) \text {Subst}\left (\int \frac {1}{-d-x^2} \, dx,x,\frac {\sqrt {d+e x^2}}{\sqrt {-1+c^2 x^2}}\right )}{3 d e \sqrt {c^2 x^2}}\\ &=\frac {b c x \sqrt {-1+c^2 x^2}}{3 d \left (c^2 d+e\right ) \sqrt {c^2 x^2} \sqrt {d+e x^2}}-\frac {a+b \csc ^{-1}(c x)}{3 e \left (d+e x^2\right )^{3/2}}+\frac {b c x \tan ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {-1+c^2 x^2}}\right )}{3 d^{3/2} e \sqrt {c^2 x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.13, size = 157, normalized size = 1.14 \begin {gather*} \frac {-a d \left (c^2 d+e\right )+b c e \sqrt {1-\frac {1}{c^2 x^2}} x \left (d+e x^2\right )-b d \left (c^2 d+e\right ) \csc ^{-1}(c x)}{3 d e \left (c^2 d+e\right ) \left (d+e x^2\right )^{3/2}}-\frac {b c \sqrt {1-\frac {1}{c^2 x^2}} x \text {ArcTan}\left (\frac {\sqrt {d} \sqrt {-1+c^2 x^2}}{\sqrt {d+e x^2}}\right )}{3 d^{3/2} e \sqrt {-1+c^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {x \left (a +b \,\mathrm {arccsc}\left (c x \right )\right )}{\left (e \,x^{2}+d \right )^{\frac {5}{2}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 285 vs.
\(2 (116) = 232\).
time = 0.47, size = 596, normalized size = 4.32 \begin {gather*} \left [-\frac {{\left (b c^{2} d^{3} + b x^{4} e^{3} + {\left (b c^{2} d x^{4} + 2 \, b d x^{2}\right )} e^{2} + {\left (2 \, b c^{2} d^{2} x^{2} + b d^{2}\right )} e\right )} \sqrt {-d} \log \left (\frac {c^{4} d^{2} x^{4} - 8 \, c^{2} d^{2} x^{2} + x^{4} e^{2} + 4 \, {\left (c^{2} d x^{2} - x^{2} e - 2 \, d\right )} \sqrt {c^{2} x^{2} - 1} \sqrt {x^{2} e + d} \sqrt {-d} + 8 \, d^{2} - 2 \, {\left (3 \, c^{2} d x^{4} - 4 \, d x^{2}\right )} e}{x^{4}}\right ) + 4 \, {\left (a c^{2} d^{3} + a d^{2} e + {\left (b c^{2} d^{3} + b d^{2} e\right )} \operatorname {arccsc}\left (c x\right ) - {\left (b d x^{2} e^{2} + b d^{2} e\right )} \sqrt {c^{2} x^{2} - 1}\right )} \sqrt {x^{2} e + d}}{12 \, {\left (c^{2} d^{5} e + d^{2} x^{4} e^{4} + {\left (c^{2} d^{3} x^{4} + 2 \, d^{3} x^{2}\right )} e^{3} + {\left (2 \, c^{2} d^{4} x^{2} + d^{4}\right )} e^{2}\right )}}, \frac {{\left (b c^{2} d^{3} + b x^{4} e^{3} + {\left (b c^{2} d x^{4} + 2 \, b d x^{2}\right )} e^{2} + {\left (2 \, b c^{2} d^{2} x^{2} + b d^{2}\right )} e\right )} \sqrt {d} \arctan \left (-\frac {{\left (c^{2} d x^{2} - x^{2} e - 2 \, d\right )} \sqrt {c^{2} x^{2} - 1} \sqrt {x^{2} e + d} \sqrt {d}}{2 \, {\left (c^{2} d^{2} x^{2} - d^{2} + {\left (c^{2} d x^{4} - d x^{2}\right )} e\right )}}\right ) - 2 \, {\left (a c^{2} d^{3} + a d^{2} e + {\left (b c^{2} d^{3} + b d^{2} e\right )} \operatorname {arccsc}\left (c x\right ) - {\left (b d x^{2} e^{2} + b d^{2} e\right )} \sqrt {c^{2} x^{2} - 1}\right )} \sqrt {x^{2} e + d}}{6 \, {\left (c^{2} d^{5} e + d^{2} x^{4} e^{4} + {\left (c^{2} d^{3} x^{4} + 2 \, d^{3} x^{2}\right )} e^{3} + {\left (2 \, c^{2} d^{4} x^{2} + d^{4}\right )} e^{2}\right )}}\right ] \end {gather*}
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 \begin {gather*} \int \frac {x \left (a + b \operatorname {acsc}{\left (c x \right )}\right )}{\left (d + e x^{2}\right )^{\frac {5}{2}}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {x\,\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )}{{\left (e\,x^2+d\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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