Optimal. Leaf size=169 \[ -\frac {a+b \text {csch}^{-1}(c x)}{e^2 \sqrt {d+e x^2}}+\frac {d \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^2 \left (d+e x^2\right )^{3/2}}-\frac {2 b c x \tan ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {-c^2 x^2-1}}\right )}{3 \sqrt {d} e^2 \sqrt {-c^2 x^2}}-\frac {b c x \sqrt {-c^2 x^2-1}}{3 e \sqrt {-c^2 x^2} \left (c^2 d-e\right ) \sqrt {d+e x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.27, antiderivative size = 169, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 8, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {266, 43, 6302, 12, 573, 152, 93, 204} \[ -\frac {a+b \text {csch}^{-1}(c x)}{e^2 \sqrt {d+e x^2}}+\frac {d \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^2 \left (d+e x^2\right )^{3/2}}-\frac {2 b c x \tan ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {-c^2 x^2-1}}\right )}{3 \sqrt {d} e^2 \sqrt {-c^2 x^2}}-\frac {b c x \sqrt {-c^2 x^2-1}}{3 e \sqrt {-c^2 x^2} \left (c^2 d-e\right ) \sqrt {d+e x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 43
Rule 93
Rule 152
Rule 204
Rule 266
Rule 573
Rule 6302
Rubi steps
\begin {align*} \int \frac {x^3 \left (a+b \text {csch}^{-1}(c x)\right )}{\left (d+e x^2\right )^{5/2}} \, dx &=\frac {d \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^2 \left (d+e x^2\right )^{3/2}}-\frac {a+b \text {csch}^{-1}(c x)}{e^2 \sqrt {d+e x^2}}-\frac {(b c x) \int \frac {-2 d-3 e x^2}{3 e^2 x \sqrt {-1-c^2 x^2} \left (d+e x^2\right )^{3/2}} \, dx}{\sqrt {-c^2 x^2}}\\ &=\frac {d \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^2 \left (d+e x^2\right )^{3/2}}-\frac {a+b \text {csch}^{-1}(c x)}{e^2 \sqrt {d+e x^2}}-\frac {(b c x) \int \frac {-2 d-3 e x^2}{x \sqrt {-1-c^2 x^2} \left (d+e x^2\right )^{3/2}} \, dx}{3 e^2 \sqrt {-c^2 x^2}}\\ &=\frac {d \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^2 \left (d+e x^2\right )^{3/2}}-\frac {a+b \text {csch}^{-1}(c x)}{e^2 \sqrt {d+e x^2}}-\frac {(b c x) \operatorname {Subst}\left (\int \frac {-2 d-3 e x}{x \sqrt {-1-c^2 x} (d+e x)^{3/2}} \, dx,x,x^2\right )}{6 e^2 \sqrt {-c^2 x^2}}\\ &=-\frac {b c x \sqrt {-1-c^2 x^2}}{3 \left (c^2 d-e\right ) e \sqrt {-c^2 x^2} \sqrt {d+e x^2}}+\frac {d \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^2 \left (d+e x^2\right )^{3/2}}-\frac {a+b \text {csch}^{-1}(c x)}{e^2 \sqrt {d+e x^2}}+\frac {(b c x) \operatorname {Subst}\left (\int \frac {d \left (c^2 d-e\right )}{x \sqrt {-1-c^2 x} \sqrt {d+e x}} \, dx,x,x^2\right )}{3 d \left (c^2 d-e\right ) e^2 \sqrt {-c^2 x^2}}\\ &=-\frac {b c x \sqrt {-1-c^2 x^2}}{3 \left (c^2 d-e\right ) e \sqrt {-c^2 x^2} \sqrt {d+e x^2}}+\frac {d \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^2 \left (d+e x^2\right )^{3/2}}-\frac {a+b \text {csch}^{-1}(c x)}{e^2 \sqrt {d+e x^2}}+\frac {(b c x) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {-1-c^2 x} \sqrt {d+e x}} \, dx,x,x^2\right )}{3 e^2 \sqrt {-c^2 x^2}}\\ &=-\frac {b c x \sqrt {-1-c^2 x^2}}{3 \left (c^2 d-e\right ) e \sqrt {-c^2 x^2} \sqrt {d+e x^2}}+\frac {d \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^2 \left (d+e x^2\right )^{3/2}}-\frac {a+b \text {csch}^{-1}(c x)}{e^2 \sqrt {d+e x^2}}+\frac {(2 b c 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^2 \sqrt {-c^2 x^2}}\\ &=-\frac {b c x \sqrt {-1-c^2 x^2}}{3 \left (c^2 d-e\right ) e \sqrt {-c^2 x^2} \sqrt {d+e x^2}}+\frac {d \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^2 \left (d+e x^2\right )^{3/2}}-\frac {a+b \text {csch}^{-1}(c x)}{e^2 \sqrt {d+e x^2}}-\frac {2 b c x \tan ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {-1-c^2 x^2}}\right )}{3 \sqrt {d} e^2 \sqrt {-c^2 x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.30, size = 201, normalized size = 1.19 \[ \frac {a \left (c^2 d-e\right ) \left (2 d+3 e x^2\right )+b c e x \sqrt {\frac {1}{c^2 x^2}+1} \left (d+e x^2\right )+b \left (c^2 d-e\right ) \text {csch}^{-1}(c x) \left (2 d+3 e x^2\right )}{3 e^2 \left (e-c^2 d\right ) \left (d+e x^2\right )^{3/2}}+\frac {2 b c x \sqrt {\frac {1}{c^2 x^2}+1} \sqrt {-d-e x^2} \tan ^{-1}\left (\frac {\sqrt {d} \sqrt {c^2 x^2+1}}{\sqrt {-d-e x^2}}\right )}{3 \sqrt {d} e^2 \sqrt {c^2 x^2+1} \sqrt {d+e x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.94, size = 786, normalized size = 4.65 \[ \left [-\frac {2 \, {\left (2 \, b c^{2} d^{3} - 2 \, b d^{2} e + 3 \, {\left (b c^{2} d^{2} e - b d e^{2}\right )} x^{2}\right )} \sqrt {e x^{2} + d} \log \left (\frac {c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right ) - {\left (b c^{2} d^{3} + {\left (b c^{2} d e^{2} - b e^{3}\right )} x^{4} - b d^{2} e + 2 \, {\left (b c^{2} d^{2} e - b d e^{2}\right )} x^{2}\right )} \sqrt {d} \log \left (\frac {{\left (c^{4} d^{2} + 6 \, c^{2} d e + e^{2}\right )} x^{4} + 8 \, {\left (c^{2} d^{2} + d e\right )} x^{2} + 4 \, {\left ({\left (c^{3} d + c e\right )} x^{3} + 2 \, c d x\right )} \sqrt {e x^{2} + d} \sqrt {d} \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 8 \, d^{2}}{x^{4}}\right ) + 2 \, {\left (2 \, a c^{2} d^{3} - 2 \, a d^{2} e + 3 \, {\left (a c^{2} d^{2} e - a d e^{2}\right )} x^{2} + {\left (b c d e^{2} x^{3} + b c d^{2} e x\right )} \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}}\right )} \sqrt {e x^{2} + d}}{6 \, {\left (c^{2} d^{4} e^{2} - d^{3} e^{3} + {\left (c^{2} d^{2} e^{4} - d e^{5}\right )} x^{4} + 2 \, {\left (c^{2} d^{3} e^{3} - d^{2} e^{4}\right )} x^{2}\right )}}, -\frac {{\left (b c^{2} d^{3} + {\left (b c^{2} d e^{2} - b e^{3}\right )} x^{4} - b d^{2} e + 2 \, {\left (b c^{2} d^{2} e - b d e^{2}\right )} x^{2}\right )} \sqrt {-d} \arctan \left (\frac {{\left ({\left (c^{3} d + c e\right )} x^{3} + 2 \, c d x\right )} \sqrt {e x^{2} + d} \sqrt {-d} \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}}}{2 \, {\left (c^{2} d e x^{4} + {\left (c^{2} d^{2} + d e\right )} x^{2} + d^{2}\right )}}\right ) + {\left (2 \, b c^{2} d^{3} - 2 \, b d^{2} e + 3 \, {\left (b c^{2} d^{2} e - b d e^{2}\right )} x^{2}\right )} \sqrt {e x^{2} + d} \log \left (\frac {c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right ) + {\left (2 \, a c^{2} d^{3} - 2 \, a d^{2} e + 3 \, {\left (a c^{2} d^{2} e - a d e^{2}\right )} x^{2} + {\left (b c d e^{2} x^{3} + b c d^{2} e x\right )} \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}}\right )} \sqrt {e x^{2} + d}}{3 \, {\left (c^{2} d^{4} e^{2} - d^{3} e^{3} + {\left (c^{2} d^{2} e^{4} - d e^{5}\right )} x^{4} + 2 \, {\left (c^{2} d^{3} e^{3} - d^{2} e^{4}\right )} x^{2}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \operatorname {arcsch}\left (c x\right ) + a\right )} x^{3}}{{\left (e x^{2} + d\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.45, size = 0, normalized size = 0.00 \[ \int \frac {x^{3} \left (a +b \,\mathrm {arccsch}\left (c x \right )\right )}{\left (e \,x^{2}+d \right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {1}{3} \, a {\left (\frac {3 \, x^{2}}{{\left (e x^{2} + d\right )}^{\frac {3}{2}} e} + \frac {2 \, d}{{\left (e x^{2} + d\right )}^{\frac {3}{2}} e^{2}}\right )} + b \int \frac {x^{3} \log \left (\sqrt {\frac {1}{c^{2} x^{2}} + 1} + \frac {1}{c x}\right )}{{\left (e x^{2} + d\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^3\,\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right )}{{\left (e\,x^2+d\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[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]
________________________________________________________________________________________