Optimal. Leaf size=270 \[ \frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e}+\frac {b c d^{5/2} x \tan ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {-c^2 x^2-1}}\right )}{5 e \sqrt {-c^2 x^2}}+\frac {b x \sqrt {-c^2 x^2-1} \left (d+e x^2\right )^{3/2}}{20 c \sqrt {-c^2 x^2}}+\frac {b x \left (15 c^4 d^2-10 c^2 d e+3 e^2\right ) \tan ^{-1}\left (\frac {\sqrt {e} \sqrt {-c^2 x^2-1}}{c \sqrt {d+e x^2}}\right )}{40 c^4 \sqrt {e} \sqrt {-c^2 x^2}}+\frac {b x \sqrt {-c^2 x^2-1} \left (7 c^2 d-3 e\right ) \sqrt {d+e x^2}}{40 c^3 \sqrt {-c^2 x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.28, antiderivative size = 270, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 10, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {6300, 446, 102, 154, 157, 63, 217, 203, 93, 204} \[ \frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e}+\frac {b x \left (15 c^4 d^2-10 c^2 d e+3 e^2\right ) \tan ^{-1}\left (\frac {\sqrt {e} \sqrt {-c^2 x^2-1}}{c \sqrt {d+e x^2}}\right )}{40 c^4 \sqrt {e} \sqrt {-c^2 x^2}}+\frac {b c d^{5/2} x \tan ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {-c^2 x^2-1}}\right )}{5 e \sqrt {-c^2 x^2}}+\frac {b x \sqrt {-c^2 x^2-1} \left (d+e x^2\right )^{3/2}}{20 c \sqrt {-c^2 x^2}}+\frac {b x \sqrt {-c^2 x^2-1} \left (7 c^2 d-3 e\right ) \sqrt {d+e x^2}}{40 c^3 \sqrt {-c^2 x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 63
Rule 93
Rule 102
Rule 154
Rule 157
Rule 203
Rule 204
Rule 217
Rule 446
Rule 6300
Rubi steps
\begin {align*} \int x \left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right ) \, dx &=\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e}-\frac {(b c x) \int \frac {\left (d+e x^2\right )^{5/2}}{x \sqrt {-1-c^2 x^2}} \, dx}{5 e \sqrt {-c^2 x^2}}\\ &=\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e}-\frac {(b c x) \operatorname {Subst}\left (\int \frac {(d+e x)^{5/2}}{x \sqrt {-1-c^2 x}} \, dx,x,x^2\right )}{10 e \sqrt {-c^2 x^2}}\\ &=\frac {b x \sqrt {-1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{20 c \sqrt {-c^2 x^2}}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e}+\frac {(b x) \operatorname {Subst}\left (\int \frac {\sqrt {d+e x} \left (-2 c^2 d^2-\frac {1}{2} \left (7 c^2 d-3 e\right ) e x\right )}{x \sqrt {-1-c^2 x}} \, dx,x,x^2\right )}{20 c e \sqrt {-c^2 x^2}}\\ &=\frac {b \left (7 c^2 d-3 e\right ) x \sqrt {-1-c^2 x^2} \sqrt {d+e x^2}}{40 c^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 \sqrt {-c^2 x^2}}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e}-\frac {(b x) \operatorname {Subst}\left (\int \frac {2 c^4 d^3+\frac {1}{4} e \left (15 c^4 d^2-10 c^2 d e+3 e^2\right ) x}{x \sqrt {-1-c^2 x} \sqrt {d+e x}} \, dx,x,x^2\right )}{20 c^3 e \sqrt {-c^2 x^2}}\\ &=\frac {b \left (7 c^2 d-3 e\right ) x \sqrt {-1-c^2 x^2} \sqrt {d+e x^2}}{40 c^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 \sqrt {-c^2 x^2}}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e}-\frac {\left (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 )}{10 e \sqrt {-c^2 x^2}}-\frac {\left (b \left (15 c^4 d^2-10 c^2 d e+3 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 )}{80 c^3 \sqrt {-c^2 x^2}}\\ &=\frac {b \left (7 c^2 d-3 e\right ) x \sqrt {-1-c^2 x^2} \sqrt {d+e x^2}}{40 c^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 \sqrt {-c^2 x^2}}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e}-\frac {\left (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 )}{5 e \sqrt {-c^2 x^2}}+\frac {\left (b \left (15 c^4 d^2-10 c^2 d e+3 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 )}{40 c^5 \sqrt {-c^2 x^2}}\\ &=\frac {b \left (7 c^2 d-3 e\right ) x \sqrt {-1-c^2 x^2} \sqrt {d+e x^2}}{40 c^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 \sqrt {-c^2 x^2}}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e}+\frac {b c d^{5/2} x \tan ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {-1-c^2 x^2}}\right )}{5 e \sqrt {-c^2 x^2}}+\frac {\left (b \left (15 c^4 d^2-10 c^2 d e+3 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 )}{40 c^5 \sqrt {-c^2 x^2}}\\ &=\frac {b \left (7 c^2 d-3 e\right ) x \sqrt {-1-c^2 x^2} \sqrt {d+e x^2}}{40 c^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 \sqrt {-c^2 x^2}}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e}+\frac {b \left (15 c^4 d^2-10 c^2 d e+3 e^2\right ) x \tan ^{-1}\left (\frac {\sqrt {e} \sqrt {-1-c^2 x^2}}{c \sqrt {d+e x^2}}\right )}{40 c^4 \sqrt {e} \sqrt {-c^2 x^2}}+\frac {b c d^{5/2} x \tan ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {-1-c^2 x^2}}\right )}{5 e \sqrt {-c^2 x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.80, size = 314, normalized size = 1.16 \[ \frac {\sqrt {d+e x^2} \left (8 a c^3 \left (d+e x^2\right )^2+8 b c^3 \text {csch}^{-1}(c x) \left (d+e x^2\right )^2+b e x \sqrt {\frac {1}{c^2 x^2}+1} \left (c^2 \left (9 d+2 e x^2\right )-3 e\right )\right )}{40 c^3 e}-\frac {b x \sqrt {\frac {1}{c^2 x^2}+1} \left (8 c^7 d^{5/2} \sqrt {-d-e x^2} \tan ^{-1}\left (\frac {\sqrt {d} \sqrt {c^2 x^2+1}}{\sqrt {-d-e x^2}}\right )+\sqrt {c^2} \sqrt {e} \sqrt {c^2 d-e} \left (-15 c^4 d^2+10 c^2 d e-3 e^2\right ) \sqrt {\frac {c^2 \left (d+e x^2\right )}{c^2 d-e}} \sinh ^{-1}\left (\frac {c \sqrt {e} \sqrt {c^2 x^2+1}}{\sqrt {c^2} \sqrt {c^2 d-e}}\right )\right )}{40 c^6 e \sqrt {c^2 x^2+1} \sqrt {d+e x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 2.01, size = 1625, normalized size = 6.02 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (e x^{2} + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arcsch}\left (c x\right ) + a\right )} x\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.43, size = 0, normalized size = 0.00 \[ \int x \left (e \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \,\mathrm {arccsch}\left (c x \right )\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {{\left (e x^{2} + d\right )}^{\frac {5}{2}} a}{5 \, e} + \frac {1}{5} \, {\left (\frac {{\left (e^{2} x^{4} + 2 \, d e x^{2} + d^{2}\right )} \sqrt {e x^{2} + d} \log \left (\sqrt {c^{2} x^{2} + 1} + 1\right )}{e} + 5 \, \int \frac {{\left (c^{2} e^{2} x^{5} + 2 \, c^{2} d e x^{3} + c^{2} d^{2} x\right )} \sqrt {e x^{2} + d}}{5 \, {\left (c^{2} e x^{2} + {\left (c^{2} e x^{2} + e\right )} \sqrt {c^{2} x^{2} + 1} + e\right )}}\,{d x} - 5 \, \int \frac {{\left ({\left (5 \, e^{2} \log \relax (c) + e^{2}\right )} c^{2} x^{5} + {\left ({\left (5 \, d e \log \relax (c) + 2 \, d e\right )} c^{2} + 5 \, e^{2} \log \relax (c)\right )} x^{3} + {\left (c^{2} d^{2} + 5 \, d e \log \relax (c)\right )} x + 5 \, {\left (c^{2} e^{2} x^{5} + {\left (c^{2} d e + e^{2}\right )} x^{3} + d e x\right )} \log \relax (x)\right )} \sqrt {e x^{2} + d}}{5 \, {\left (c^{2} e x^{2} + e\right )}}\,{d x}\right )} b \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int x\,{\left (e\,x^2+d\right )}^{3/2}\,\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right ) \,d x \]
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]
________________________________________________________________________________________