Optimal. Leaf size=252 \[ -\frac {d^2 \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt {d+e x^2}}-\frac {2 d \sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}+\frac {8 b c d^{3/2} x \tan ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {c^2 x^2-1}}\right )}{3 e^3 \sqrt {c^2 x^2}}-\frac {b x \left (9 c^2 d-e\right ) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {c^2 x^2-1}}{c \sqrt {d+e x^2}}\right )}{6 c^2 e^{5/2} \sqrt {c^2 x^2}}+\frac {b x \sqrt {c^2 x^2-1} \sqrt {d+e x^2}}{6 c e^2 \sqrt {c^2 x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 1.07, antiderivative size = 252, 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, 1615, 157, 63, 217, 206, 93, 204} \[ -\frac {d^2 \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt {d+e x^2}}-\frac {2 d \sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}+\frac {8 b c d^{3/2} x \tan ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {c^2 x^2-1}}\right )}{3 e^3 \sqrt {c^2 x^2}}+\frac {b x \sqrt {c^2 x^2-1} \sqrt {d+e x^2}}{6 c e^2 \sqrt {c^2 x^2}}-\frac {b x \left (9 c^2 d-e\right ) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {c^2 x^2-1}}{c \sqrt {d+e x^2}}\right )}{6 c^2 e^{5/2} \sqrt {c^2 x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 43
Rule 63
Rule 93
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 )}{\left (d+e x^2\right )^{3/2}} \, dx &=-\frac {d^2 \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt {d+e x^2}}-\frac {2 d \sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}+\frac {(b c x) \int \frac {-8 d^2-4 d e x^2+e^2 x^4}{3 e^3 x \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \, dx}{\sqrt {c^2 x^2}}\\ &=-\frac {d^2 \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt {d+e x^2}}-\frac {2 d \sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}+\frac {(b c x) \int \frac {-8 d^2-4 d e x^2+e^2 x^4}{x \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \, dx}{3 e^3 \sqrt {c^2 x^2}}\\ &=-\frac {d^2 \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt {d+e x^2}}-\frac {2 d \sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}+\frac {(b c x) \operatorname {Subst}\left (\int \frac {-8 d^2-4 d e x+e^2 x^2}{x \sqrt {-1+c^2 x} \sqrt {d+e x}} \, dx,x,x^2\right )}{6 e^3 \sqrt {c^2 x^2}}\\ &=\frac {b x \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{6 c e^2 \sqrt {c^2 x^2}}-\frac {d^2 \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt {d+e x^2}}-\frac {2 d \sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}+\frac {(b x) \operatorname {Subst}\left (\int \frac {-8 c^2 d^2 e-\frac {1}{2} \left (9 c^2 d-e\right ) e^2 x}{x \sqrt {-1+c^2 x} \sqrt {d+e x}} \, dx,x,x^2\right )}{6 c e^4 \sqrt {c^2 x^2}}\\ &=\frac {b x \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{6 c e^2 \sqrt {c^2 x^2}}-\frac {d^2 \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt {d+e x^2}}-\frac {2 d \sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}-\frac {\left (4 b c d^2 x\right ) \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 {\left (b \left (9 c^2 d-e\right ) x\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+c^2 x} \sqrt {d+e x}} \, dx,x,x^2\right )}{12 c e^2 \sqrt {c^2 x^2}}\\ &=\frac {b x \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{6 c e^2 \sqrt {c^2 x^2}}-\frac {d^2 \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt {d+e x^2}}-\frac {2 d \sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}-\frac {\left (8 b c d^2 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 )}{3 e^3 \sqrt {c^2 x^2}}-\frac {\left (b \left (9 c^2 d-e\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 )}{6 c^3 e^2 \sqrt {c^2 x^2}}\\ &=\frac {b x \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{6 c e^2 \sqrt {c^2 x^2}}-\frac {d^2 \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt {d+e x^2}}-\frac {2 d \sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}+\frac {8 b c d^{3/2} 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 {\left (b \left (9 c^2 d-e\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 )}{6 c^3 e^2 \sqrt {c^2 x^2}}\\ &=\frac {b x \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{6 c e^2 \sqrt {c^2 x^2}}-\frac {d^2 \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt {d+e x^2}}-\frac {2 d \sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}+\frac {8 b c d^{3/2} 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 \left (9 c^2 d-e\right ) x \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {-1+c^2 x^2}}{c \sqrt {d+e x^2}}\right )}{6 c^2 e^{5/2} \sqrt {c^2 x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.64, size = 302, normalized size = 1.20 \[ \frac {-2 a c \left (8 d^2+4 d e x^2-e^2 x^4\right )+b e x \sqrt {1-\frac {1}{c^2 x^2}} \left (d+e x^2\right )-2 b c \csc ^{-1}(c x) \left (8 d^2+4 d e x^2-e^2 x^4\right )}{6 c e^3 \sqrt {d+e x^2}}-\frac {b x \sqrt {1-\frac {1}{c^2 x^2}} \left (\sqrt {c^2} \sqrt {e} \left (9 c^2 d-e\right ) \sqrt {c^2 d+e} \sqrt {\frac {c^2 \left (d+e x^2\right )}{c^2 d+e}} \sinh ^{-1}\left (\frac {c \sqrt {e} \sqrt {c^2 x^2-1}}{\sqrt {c^2} \sqrt {c^2 d+e}}\right )+16 c^5 d^{3/2} \sqrt {d+e x^2} \tan ^{-1}\left (\frac {\sqrt {d} \sqrt {c^2 x^2-1}}{\sqrt {d+e x^2}}\right )\right )}{6 c^4 e^3 \sqrt {c^2 x^2-1} \sqrt {d+e x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.68, size = 1480, normalized size = 5.87 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \operatorname {arccsc}\left (c x\right ) + a\right )} x^{5}}{{\left (e x^{2} + d\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 7.40, size = 0, normalized size = 0.00 \[ \int \frac {x^{5} \left (a +b \,\mathrm {arccsc}\left (c x \right )\right )}{\left (e \,x^{2}+d \right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{3} \, {\left (\frac {x^{4}}{\sqrt {e x^{2} + d} e} - \frac {4 \, d x^{2}}{\sqrt {e x^{2} + d} e^{2}} - \frac {8 \, d^{2}}{\sqrt {e x^{2} + d} e^{3}}\right )} a + \frac {{\left (e^{2} x^{4} \arctan \left (1, \sqrt {c x + 1} \sqrt {c x - 1}\right ) - 4 \, d e x^{2} \arctan \left (1, \sqrt {c x + 1} \sqrt {c x - 1}\right ) + \sqrt {e x^{2} + d} e^{3} \int \frac {{\left (c^{2} e^{2} x^{5} - 4 \, c^{2} d e x^{3} - 8 \, c^{2} d^{2} x\right )} e^{\left (-\frac {1}{2} \, \log \left (e x^{2} + d\right ) + \frac {1}{2} \, \log \left (c x + 1\right ) + \frac {1}{2} \, \log \left (c x - 1\right )\right )}}{c^{2} e^{3} x^{2} + {\left (c^{2} e^{3} x^{2} - e^{3}\right )} {\left (c x + 1\right )} {\left (c x - 1\right )} - e^{3}}\,{d x} - 8 \, d^{2} \arctan \left (1, \sqrt {c x + 1} \sqrt {c x - 1}\right )\right )} b}{3 \, \sqrt {e x^{2} + d} e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^5\,\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )}{{\left (e\,x^2+d\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________