Optimal. Leaf size=321 \[ \frac {d^2 \sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{e^3}-\frac {2 d \left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^3}-\frac {8 b c d^{5/2} x \tan ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {c^2 x^2-1}}\right )}{15 e^3 \sqrt {c^2 x^2}}+\frac {b x \sqrt {c^2 x^2-1} \left (d+e x^2\right )^{3/2}}{20 c e^2 \sqrt {c^2 x^2}}+\frac {b x \left (45 c^4 d^2-10 c^2 d e+9 e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {c^2 x^2-1}}{c \sqrt {d+e x^2}}\right )}{120 c^4 e^{5/2} \sqrt {c^2 x^2}}-\frac {b x \sqrt {c^2 x^2-1} \left (19 c^2 d-9 e\right ) \sqrt {d+e x^2}}{120 c^3 e^2 \sqrt {c^2 x^2}} \]
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Rubi [A] time = 1.05, antiderivative size = 321, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 12, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.522, Rules used = {266, 43, 5239, 12, 1615, 154, 157, 63, 217, 206, 93, 204} \[ \frac {d^2 \sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{e^3}-\frac {2 d \left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^3}-\frac {8 b c d^{5/2} x \tan ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {c^2 x^2-1}}\right )}{15 e^3 \sqrt {c^2 x^2}}+\frac {b x \left (45 c^4 d^2-10 c^2 d e+9 e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {c^2 x^2-1}}{c \sqrt {d+e x^2}}\right )}{120 c^4 e^{5/2} \sqrt {c^2 x^2}}+\frac {b x \sqrt {c^2 x^2-1} \left (d+e x^2\right )^{3/2}}{20 c e^2 \sqrt {c^2 x^2}}-\frac {b x \sqrt {c^2 x^2-1} \left (19 c^2 d-9 e\right ) \sqrt {d+e x^2}}{120 c^3 e^2 \sqrt {c^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 43
Rule 63
Rule 93
Rule 154
Rule 157
Rule 204
Rule 206
Rule 217
Rule 266
Rule 1615
Rule 5239
Rubi steps
\begin {align*} \int \frac {x^5 \left (a+b \csc ^{-1}(c x)\right )}{\sqrt {d+e x^2}} \, dx &=\frac {d^2 \sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{e^3}-\frac {2 d \left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^3}+\frac {(b c x) \int \frac {\sqrt {d+e x^2} \left (8 d^2-4 d e x^2+3 e^2 x^4\right )}{15 e^3 x \sqrt {-1+c^2 x^2}} \, dx}{\sqrt {c^2 x^2}}\\ &=\frac {d^2 \sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{e^3}-\frac {2 d \left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^3}+\frac {(b c x) \int \frac {\sqrt {d+e x^2} \left (8 d^2-4 d e x^2+3 e^2 x^4\right )}{x \sqrt {-1+c^2 x^2}} \, dx}{15 e^3 \sqrt {c^2 x^2}}\\ &=\frac {d^2 \sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{e^3}-\frac {2 d \left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^3}+\frac {(b c x) \operatorname {Subst}\left (\int \frac {\sqrt {d+e x} \left (8 d^2-4 d e x+3 e^2 x^2\right )}{x \sqrt {-1+c^2 x}} \, dx,x,x^2\right )}{30 e^3 \sqrt {c^2 x^2}}\\ &=\frac {b x \sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{3/2}}{20 c e^2 \sqrt {c^2 x^2}}+\frac {d^2 \sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{e^3}-\frac {2 d \left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^3}+\frac {(b x) \operatorname {Subst}\left (\int \frac {\sqrt {d+e x} \left (16 c^2 d^2 e-\frac {1}{2} \left (19 c^2 d-9 e\right ) e^2 x\right )}{x \sqrt {-1+c^2 x}} \, dx,x,x^2\right )}{60 c e^4 \sqrt {c^2 x^2}}\\ &=-\frac {b \left (19 c^2 d-9 e\right ) x \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{120 c^3 e^2 \sqrt {c^2 x^2}}+\frac {b x \sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{3/2}}{20 c e^2 \sqrt {c^2 x^2}}+\frac {d^2 \sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{e^3}-\frac {2 d \left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^3}+\frac {(b x) \operatorname {Subst}\left (\int \frac {16 c^4 d^3 e+\frac {1}{4} e^2 \left (45 c^4 d^2-10 c^2 d e+9 e^2\right ) x}{x \sqrt {-1+c^2 x} \sqrt {d+e x}} \, dx,x,x^2\right )}{60 c^3 e^4 \sqrt {c^2 x^2}}\\ &=-\frac {b \left (19 c^2 d-9 e\right ) x \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{120 c^3 e^2 \sqrt {c^2 x^2}}+\frac {b x \sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{3/2}}{20 c e^2 \sqrt {c^2 x^2}}+\frac {d^2 \sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{e^3}-\frac {2 d \left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^3}+\frac {\left (4 b c d^3 x\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {-1+c^2 x} \sqrt {d+e x}} \, dx,x,x^2\right )}{15 e^3 \sqrt {c^2 x^2}}+\frac {\left (b \left (45 c^4 d^2-10 c^2 d e+9 e^2\right ) x\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+c^2 x} \sqrt {d+e x}} \, dx,x,x^2\right )}{240 c^3 e^2 \sqrt {c^2 x^2}}\\ &=-\frac {b \left (19 c^2 d-9 e\right ) x \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{120 c^3 e^2 \sqrt {c^2 x^2}}+\frac {b x \sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{3/2}}{20 c e^2 \sqrt {c^2 x^2}}+\frac {d^2 \sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{e^3}-\frac {2 d \left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^3}+\frac {\left (8 b c d^3 x\right ) \operatorname {Subst}\left (\int \frac {1}{-d-x^2} \, dx,x,\frac {\sqrt {d+e x^2}}{\sqrt {-1+c^2 x^2}}\right )}{15 e^3 \sqrt {c^2 x^2}}+\frac {\left (b \left (45 c^4 d^2-10 c^2 d e+9 e^2\right ) x\right ) \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 )}{120 c^5 e^2 \sqrt {c^2 x^2}}\\ &=-\frac {b \left (19 c^2 d-9 e\right ) x \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{120 c^3 e^2 \sqrt {c^2 x^2}}+\frac {b x \sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{3/2}}{20 c e^2 \sqrt {c^2 x^2}}+\frac {d^2 \sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{e^3}-\frac {2 d \left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^3}-\frac {8 b c d^{5/2} x \tan ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {-1+c^2 x^2}}\right )}{15 e^3 \sqrt {c^2 x^2}}+\frac {\left (b \left (45 c^4 d^2-10 c^2 d e+9 e^2\right ) x\right ) \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 )}{120 c^5 e^2 \sqrt {c^2 x^2}}\\ &=-\frac {b \left (19 c^2 d-9 e\right ) x \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{120 c^3 e^2 \sqrt {c^2 x^2}}+\frac {b x \sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{3/2}}{20 c e^2 \sqrt {c^2 x^2}}+\frac {d^2 \sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{e^3}-\frac {2 d \left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^3}-\frac {8 b c d^{5/2} x \tan ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {-1+c^2 x^2}}\right )}{15 e^3 \sqrt {c^2 x^2}}+\frac {b \left (45 c^4 d^2-10 c^2 d e+9 e^2\right ) x \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {-1+c^2 x^2}}{c \sqrt {d+e x^2}}\right )}{120 c^4 e^{5/2} \sqrt {c^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.75, size = 328, normalized size = 1.02 \[ \frac {\sqrt {d+e x^2} \left (8 a c^3 \left (8 d^2-4 d e x^2+3 e^2 x^4\right )+8 b c^3 \csc ^{-1}(c x) \left (8 d^2-4 d e x^2+3 e^2 x^4\right )+b e x \sqrt {1-\frac {1}{c^2 x^2}} \left (c^2 \left (6 e x^2-13 d\right )+9 e\right )\right )}{120 c^3 e^3}+\frac {b x \sqrt {1-\frac {1}{c^2 x^2}} \left (64 c^7 d^{5/2} \sqrt {d+e x^2} \tan ^{-1}\left (\frac {\sqrt {d} \sqrt {c^2 x^2-1}}{\sqrt {d+e x^2}}\right )+\sqrt {c^2} \sqrt {e} \sqrt {c^2 d+e} \left (45 c^4 d^2-10 c^2 d e+9 e^2\right ) \sqrt {\frac {c^2 \left (d+e x^2\right )}{c^2 d+e}} \sinh ^{-1}\left (\frac {c \sqrt {e} \sqrt {c^2 x^2-1}}{\sqrt {c^2} \sqrt {c^2 d+e}}\right )\right )}{120 c^6 e^3 \sqrt {c^2 x^2-1} \sqrt {d+e x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 4.80, size = 1383, normalized size = 4.31 \[ \text {result too large to display} \]
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 \[ \int \frac {{\left (b \operatorname {arccsc}\left (c x\right ) + a\right )} x^{5}}{\sqrt {e x^{2} + d}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 7.87, size = 0, normalized size = 0.00 \[ \int \frac {x^{5} \left (a +b \,\mathrm {arccsc}\left (c x \right )\right )}{\sqrt {e \,x^{2}+d}}\, dx \]
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 \[ \frac {1}{15} \, {\left (\frac {3 \, \sqrt {e x^{2} + d} x^{4}}{e} - \frac {4 \, \sqrt {e x^{2} + d} d x^{2}}{e^{2}} + \frac {8 \, \sqrt {e x^{2} + d} d^{2}}{e^{3}}\right )} a + \frac {{\left (e^{3} \int \frac {{\left (3 \, c^{2} e^{3} x^{7} - c^{2} d e^{2} x^{5} + 4 \, c^{2} d^{2} e x^{3} + 8 \, c^{2} d^{3} x\right )} e^{\left (-\frac {1}{2} \, \log \left (e x^{2} + d\right ) + \frac {1}{2} \, \log \left (c x + 1\right ) + \frac {1}{2} \, \log \left (c x - 1\right )\right )}}{c^{2} e^{3} x^{2} + {\left (c^{2} e^{3} x^{2} - e^{3}\right )} {\left (c x + 1\right )} {\left (c x - 1\right )} - e^{3}}\,{d x} + {\left (3 \, e^{2} x^{4} \arctan \left (1, \sqrt {c x + 1} \sqrt {c x - 1}\right ) - 4 \, d e x^{2} \arctan \left (1, \sqrt {c x + 1} \sqrt {c x - 1}\right ) + 8 \, d^{2} \arctan \left (1, \sqrt {c x + 1} \sqrt {c x - 1}\right )\right )} \sqrt {e x^{2} + d}\right )} b}{15 \, e^{3}} \]
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 \[ \int \frac {x^5\,\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )}{\sqrt {e\,x^2+d}} \,d x \]
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 \[ \int \frac {x^{5} \left (a + b \operatorname {acsc}{\left (c x \right )}\right )}{\sqrt {d + e x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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