Optimal. Leaf size=243 \[ -\frac {d^2 \left (a+b \csc ^{-1}(c x)\right )}{3 e^3 \left (d+e x^2\right )^{3/2}}+\frac {2 d \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt {d+e x^2}}+\frac {\sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac {b x \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {c^2 x^2-1}}{c \sqrt {d+e x^2}}\right )}{e^{5/2} \sqrt {c^2 x^2}}-\frac {8 b c \sqrt {d} x \tan ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {c^2 x^2-1}}\right )}{3 e^3 \sqrt {c^2 x^2}}+\frac {b c d x \sqrt {c^2 x^2-1}}{3 e^2 \sqrt {c^2 x^2} \left (c^2 d+e\right ) \sqrt {d+e x^2}} \]
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Rubi [A] time = 1.06, antiderivative size = 243, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 11, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {266, 43, 5239, 12, 1614, 157, 63, 217, 206, 93, 204} \[ -\frac {d^2 \left (a+b \csc ^{-1}(c x)\right )}{3 e^3 \left (d+e x^2\right )^{3/2}}+\frac {2 d \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt {d+e x^2}}+\frac {\sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac {b c d x \sqrt {c^2 x^2-1}}{3 e^2 \sqrt {c^2 x^2} \left (c^2 d+e\right ) \sqrt {d+e x^2}}-\frac {8 b c \sqrt {d} x \tan ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {c^2 x^2-1}}\right )}{3 e^3 \sqrt {c^2 x^2}}+\frac {b x \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {c^2 x^2-1}}{c \sqrt {d+e x^2}}\right )}{e^{5/2} \sqrt {c^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 43
Rule 63
Rule 93
Rule 157
Rule 204
Rule 206
Rule 217
Rule 266
Rule 1614
Rule 5239
Rubi steps
\begin {align*} \int \frac {x^5 \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^{5/2}} \, dx &=-\frac {d^2 \left (a+b \csc ^{-1}(c x)\right )}{3 e^3 \left (d+e x^2\right )^{3/2}}+\frac {2 d \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt {d+e x^2}}+\frac {\sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac {(b c x) \int \frac {8 d^2+12 d e x^2+3 e^2 x^4}{3 e^3 x \sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{3/2}} \, dx}{\sqrt {c^2 x^2}}\\ &=-\frac {d^2 \left (a+b \csc ^{-1}(c x)\right )}{3 e^3 \left (d+e x^2\right )^{3/2}}+\frac {2 d \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt {d+e x^2}}+\frac {\sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac {(b c x) \int \frac {8 d^2+12 d e x^2+3 e^2 x^4}{x \sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{3/2}} \, dx}{3 e^3 \sqrt {c^2 x^2}}\\ &=-\frac {d^2 \left (a+b \csc ^{-1}(c x)\right )}{3 e^3 \left (d+e x^2\right )^{3/2}}+\frac {2 d \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt {d+e x^2}}+\frac {\sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac {(b c x) \operatorname {Subst}\left (\int \frac {8 d^2+12 d e x+3 e^2 x^2}{x \sqrt {-1+c^2 x} (d+e x)^{3/2}} \, dx,x,x^2\right )}{6 e^3 \sqrt {c^2 x^2}}\\ &=\frac {b c d x \sqrt {-1+c^2 x^2}}{3 e^2 \left (c^2 d+e\right ) \sqrt {c^2 x^2} \sqrt {d+e x^2}}-\frac {d^2 \left (a+b \csc ^{-1}(c x)\right )}{3 e^3 \left (d+e x^2\right )^{3/2}}+\frac {2 d \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt {d+e x^2}}+\frac {\sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{e^3}-\frac {(b c x) \operatorname {Subst}\left (\int \frac {-4 d^2 \left (c^2 d+e\right )-\frac {3}{2} d e \left (c^2 d+e\right ) x}{x \sqrt {-1+c^2 x} \sqrt {d+e x}} \, dx,x,x^2\right )}{3 d e^3 \left (c^2 d+e\right ) \sqrt {c^2 x^2}}\\ &=\frac {b c d x \sqrt {-1+c^2 x^2}}{3 e^2 \left (c^2 d+e\right ) \sqrt {c^2 x^2} \sqrt {d+e x^2}}-\frac {d^2 \left (a+b \csc ^{-1}(c x)\right )}{3 e^3 \left (d+e x^2\right )^{3/2}}+\frac {2 d \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt {d+e x^2}}+\frac {\sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac {(4 b c d x) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {-1+c^2 x} \sqrt {d+e x}} \, dx,x,x^2\right )}{3 e^3 \sqrt {c^2 x^2}}+\frac {(b c x) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+c^2 x} \sqrt {d+e x}} \, dx,x,x^2\right )}{2 e^2 \sqrt {c^2 x^2}}\\ &=\frac {b c d x \sqrt {-1+c^2 x^2}}{3 e^2 \left (c^2 d+e\right ) \sqrt {c^2 x^2} \sqrt {d+e x^2}}-\frac {d^2 \left (a+b \csc ^{-1}(c x)\right )}{3 e^3 \left (d+e x^2\right )^{3/2}}+\frac {2 d \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt {d+e x^2}}+\frac {\sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac {(8 b c d x) \operatorname {Subst}\left (\int \frac {1}{-d-x^2} \, dx,x,\frac {\sqrt {d+e x^2}}{\sqrt {-1+c^2 x^2}}\right )}{3 e^3 \sqrt {c^2 x^2}}+\frac {(b x) \operatorname {Subst}\left (\int \frac {1}{\sqrt {d+\frac {e}{c^2}+\frac {e x^2}{c^2}}} \, dx,x,\sqrt {-1+c^2 x^2}\right )}{c e^2 \sqrt {c^2 x^2}}\\ &=\frac {b c d x \sqrt {-1+c^2 x^2}}{3 e^2 \left (c^2 d+e\right ) \sqrt {c^2 x^2} \sqrt {d+e x^2}}-\frac {d^2 \left (a+b \csc ^{-1}(c x)\right )}{3 e^3 \left (d+e x^2\right )^{3/2}}+\frac {2 d \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt {d+e x^2}}+\frac {\sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{e^3}-\frac {8 b c \sqrt {d} x \tan ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {-1+c^2 x^2}}\right )}{3 e^3 \sqrt {c^2 x^2}}+\frac {(b x) \operatorname {Subst}\left (\int \frac {1}{1-\frac {e x^2}{c^2}} \, dx,x,\frac {\sqrt {-1+c^2 x^2}}{\sqrt {d+e x^2}}\right )}{c e^2 \sqrt {c^2 x^2}}\\ &=\frac {b c d x \sqrt {-1+c^2 x^2}}{3 e^2 \left (c^2 d+e\right ) \sqrt {c^2 x^2} \sqrt {d+e x^2}}-\frac {d^2 \left (a+b \csc ^{-1}(c x)\right )}{3 e^3 \left (d+e x^2\right )^{3/2}}+\frac {2 d \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt {d+e x^2}}+\frac {\sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{e^3}-\frac {8 b c \sqrt {d} x \tan ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {-1+c^2 x^2}}\right )}{3 e^3 \sqrt {c^2 x^2}}+\frac {b x \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {-1+c^2 x^2}}{c \sqrt {d+e x^2}}\right )}{e^{5/2} \sqrt {c^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.54, size = 311, normalized size = 1.28 \[ \frac {a \left (c^2 d+e\right ) \left (8 d^2+12 d e x^2+3 e^2 x^4\right )+b \left (c^2 d+e\right ) \csc ^{-1}(c x) \left (8 d^2+12 d e x^2+3 e^2 x^4\right )+b c d e x \sqrt {1-\frac {1}{c^2 x^2}} \left (d+e x^2\right )}{3 e^3 \left (c^2 d+e\right ) \left (d+e x^2\right )^{3/2}}+\frac {b x \sqrt {1-\frac {1}{c^2 x^2}} \left (3 \sqrt {c^2} \sqrt {e} \sqrt {c^2 d+e} \sqrt {\frac {c^2 \left (d+e x^2\right )}{c^2 d+e}} \sinh ^{-1}\left (\frac {c \sqrt {e} \sqrt {c^2 x^2-1}}{\sqrt {c^2} \sqrt {c^2 d+e}}\right )+8 c^3 \sqrt {d} \sqrt {d+e x^2} \tan ^{-1}\left (\frac {\sqrt {d} \sqrt {c^2 x^2-1}}{\sqrt {d+e x^2}}\right )\right )}{3 c^2 e^3 \sqrt {c^2 x^2-1} \sqrt {d+e x^2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.38, size = 2119, normalized size = 8.72 \[ \text {result too large to display} \]
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 \[ \int \frac {{\left (b \operatorname {arccsc}\left (c x\right ) + a\right )} x^{5}}{{\left (e x^{2} + d\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 7.34, size = 0, normalized size = 0.00 \[ \int \frac {x^{5} \left (a +b \,\mathrm {arccsc}\left (c x \right )\right )}{\left (e \,x^{2}+d \right )^{\frac {5}{2}}}\, dx \]
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 \[ \frac {1}{3} \, {\left (\frac {3 \, x^{4}}{{\left (e x^{2} + d\right )}^{\frac {3}{2}} e} + \frac {12 \, d x^{2}}{{\left (e x^{2} + d\right )}^{\frac {3}{2}} e^{2}} + \frac {8 \, d^{2}}{{\left (e x^{2} + d\right )}^{\frac {3}{2}} e^{3}}\right )} a + \frac {{\left (3 \, e^{2} x^{4} \arctan \left (1, \sqrt {c x + 1} \sqrt {c x - 1}\right ) + 12 \, d e x^{2} \arctan \left (1, \sqrt {c x + 1} \sqrt {c x - 1}\right ) + 8 \, d^{2} \arctan \left (1, \sqrt {c x + 1} \sqrt {c x - 1}\right ) + {\left (e^{4} x^{2} + d e^{3}\right )} \sqrt {e x^{2} + d} \int \frac {{\left (3 \, c^{2} e^{2} x^{5} + 12 \, c^{2} d e x^{3} + 8 \, c^{2} d^{2} x\right )} e^{\left (-\frac {1}{2} \, \log \left (e x^{2} + d\right ) + \frac {1}{2} \, \log \left (c x + 1\right ) + \frac {1}{2} \, \log \left (c x - 1\right )\right )}}{c^{2} e^{4} x^{4} - d e^{3} + {\left (c^{2} e^{4} x^{4} - d e^{3} + {\left (c^{2} d e^{3} - e^{4}\right )} x^{2}\right )} {\left (c x + 1\right )} {\left (c x - 1\right )} + {\left (c^{2} d e^{3} - e^{4}\right )} x^{2}}\,{d x}\right )} b}{3 \, {\left (e^{4} x^{2} + d e^{3}\right )} \sqrt {e x^{2} + d}} \]
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 \[ \int \frac {x^5\,\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )}{{\left (e\,x^2+d\right )}^{5/2}} \,d x \]
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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