Optimal. Leaf size=119 \[ \frac {x^3 \left (a+b \sinh ^{-1}(c x)\right )}{3 d \left (c^2 d x^2+d\right )^{3/2}}-\frac {b}{6 c^3 d^2 \sqrt {c^2 x^2+1} \sqrt {c^2 d x^2+d}}-\frac {b \sqrt {c^2 x^2+1} \log \left (c^2 x^2+1\right )}{6 c^3 d^2 \sqrt {c^2 d x^2+d}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.12, antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {5723, 266, 43} \[ \frac {x^3 \left (a+b \sinh ^{-1}(c x)\right )}{3 d \left (c^2 d x^2+d\right )^{3/2}}-\frac {b}{6 c^3 d^2 \sqrt {c^2 x^2+1} \sqrt {c^2 d x^2+d}}-\frac {b \sqrt {c^2 x^2+1} \log \left (c^2 x^2+1\right )}{6 c^3 d^2 \sqrt {c^2 d x^2+d}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 266
Rule 5723
Rubi steps
\begin {align*} \int \frac {x^2 \left (a+b \sinh ^{-1}(c x)\right )}{\left (d+c^2 d x^2\right )^{5/2}} \, dx &=\frac {x^3 \left (a+b \sinh ^{-1}(c x)\right )}{3 d \left (d+c^2 d x^2\right )^{3/2}}-\frac {\left (b c \sqrt {1+c^2 x^2}\right ) \int \frac {x^3}{\left (1+c^2 x^2\right )^2} \, dx}{3 d^2 \sqrt {d+c^2 d x^2}}\\ &=\frac {x^3 \left (a+b \sinh ^{-1}(c x)\right )}{3 d \left (d+c^2 d x^2\right )^{3/2}}-\frac {\left (b c \sqrt {1+c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {x}{\left (1+c^2 x\right )^2} \, dx,x,x^2\right )}{6 d^2 \sqrt {d+c^2 d x^2}}\\ &=\frac {x^3 \left (a+b \sinh ^{-1}(c x)\right )}{3 d \left (d+c^2 d x^2\right )^{3/2}}-\frac {\left (b c \sqrt {1+c^2 x^2}\right ) \operatorname {Subst}\left (\int \left (-\frac {1}{c^2 \left (1+c^2 x\right )^2}+\frac {1}{c^2 \left (1+c^2 x\right )}\right ) \, dx,x,x^2\right )}{6 d^2 \sqrt {d+c^2 d x^2}}\\ &=-\frac {b}{6 c^3 d^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}}+\frac {x^3 \left (a+b \sinh ^{-1}(c x)\right )}{3 d \left (d+c^2 d x^2\right )^{3/2}}-\frac {b \sqrt {1+c^2 x^2} \log \left (1+c^2 x^2\right )}{6 c^3 d^2 \sqrt {d+c^2 d x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.20, size = 118, normalized size = 0.99 \[ -\frac {\sqrt {c^2 d x^2+d} \left (-2 a c^3 x^3 \sqrt {c^2 x^2+1}+b c^2 x^2+b \left (c^2 x^2+1\right )^2 \log \left (c^2 x^2+1\right )-2 b c^3 x^3 \sqrt {c^2 x^2+1} \sinh ^{-1}(c x)+b\right )}{6 c^3 d^3 \left (c^2 x^2+1\right )^{5/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.57, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {c^{2} d x^{2} + d} {\left (b x^{2} \operatorname {arsinh}\left (c x\right ) + a x^{2}\right )}}{c^{6} d^{3} x^{6} + 3 \, c^{4} d^{3} x^{4} + 3 \, c^{2} d^{3} x^{2} + d^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x^{2}}{{\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.28, size = 1153, normalized size = 9.69 \[ -\frac {a x}{3 c^{2} d \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {a x}{3 c^{2} d^{2} \sqrt {c^{2} d \,x^{2}+d}}+\frac {2 b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right )}{3 \sqrt {c^{2} x^{2}+1}\, c^{3} d^{3}}+\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, c^{4} \arcsinh \left (c x \right ) x^{7}}{\left (3 c^{8} x^{8}+9 c^{6} x^{6}+10 c^{4} x^{4}+5 c^{2} x^{2}+1\right ) d^{3}}-\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, c^{3} \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{6}}{\left (3 c^{8} x^{8}+9 c^{6} x^{6}+10 c^{4} x^{4}+5 c^{2} x^{2}+1\right ) d^{3}}+\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, c^{4} x^{7}}{6 \left (3 c^{8} x^{8}+9 c^{6} x^{6}+10 c^{4} x^{4}+5 c^{2} x^{2}+1\right ) d^{3}}-\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, c^{2} \left (c^{2} x^{2}+1\right ) x^{5}}{6 \left (3 c^{8} x^{8}+9 c^{6} x^{6}+10 c^{4} x^{4}+5 c^{2} x^{2}+1\right ) d^{3}}+\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, c^{2} \arcsinh \left (c x \right ) x^{5}}{\left (3 c^{8} x^{8}+9 c^{6} x^{6}+10 c^{4} x^{4}+5 c^{2} x^{2}+1\right ) d^{3}}-\frac {2 b \sqrt {d \left (c^{2} x^{2}+1\right )}\, c \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{4}}{\left (3 c^{8} x^{8}+9 c^{6} x^{6}+10 c^{4} x^{4}+5 c^{2} x^{2}+1\right ) d^{3}}+\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, c^{2} x^{5}}{3 \left (3 c^{8} x^{8}+9 c^{6} x^{6}+10 c^{4} x^{4}+5 c^{2} x^{2}+1\right ) d^{3}}-\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, c \sqrt {c^{2} x^{2}+1}\, x^{4}}{2 \left (3 c^{8} x^{8}+9 c^{6} x^{6}+10 c^{4} x^{4}+5 c^{2} x^{2}+1\right ) d^{3}}-\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}+1\right ) x^{3}}{6 \left (3 c^{8} x^{8}+9 c^{6} x^{6}+10 c^{4} x^{4}+5 c^{2} x^{2}+1\right ) d^{3}}+\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right ) x^{3}}{3 \left (3 c^{8} x^{8}+9 c^{6} x^{6}+10 c^{4} x^{4}+5 c^{2} x^{2}+1\right ) d^{3}}-\frac {4 b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{2}}{3 \left (3 c^{8} x^{8}+9 c^{6} x^{6}+10 c^{4} x^{4}+5 c^{2} x^{2}+1\right ) c \,d^{3}}+\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, x^{3}}{6 \left (3 c^{8} x^{8}+9 c^{6} x^{6}+10 c^{4} x^{4}+5 c^{2} x^{2}+1\right ) d^{3}}-\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, x^{2} \sqrt {c^{2} x^{2}+1}}{2 \left (3 c^{8} x^{8}+9 c^{6} x^{6}+10 c^{4} x^{4}+5 c^{2} x^{2}+1\right ) c \,d^{3}}-\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}}{3 \left (3 c^{8} x^{8}+9 c^{6} x^{6}+10 c^{4} x^{4}+5 c^{2} x^{2}+1\right ) c^{3} d^{3}}-\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \sqrt {c^{2} x^{2}+1}}{6 \left (3 c^{8} x^{8}+9 c^{6} x^{6}+10 c^{4} x^{4}+5 c^{2} x^{2}+1\right ) c^{3} d^{3}}-\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{3 \sqrt {c^{2} x^{2}+1}\, c^{3} d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.39, size = 137, normalized size = 1.15 \[ -\frac {1}{6} \, b c {\left (\frac {1}{c^{6} d^{\frac {5}{2}} x^{2} + c^{4} d^{\frac {5}{2}}} + \frac {\log \left (c^{2} x^{2} + 1\right )}{c^{4} d^{\frac {5}{2}}}\right )} + \frac {1}{3} \, b {\left (\frac {x}{\sqrt {c^{2} d x^{2} + d} c^{2} d^{2}} - \frac {x}{{\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}} c^{2} d}\right )} \operatorname {arsinh}\left (c x\right ) + \frac {1}{3} \, a {\left (\frac {x}{\sqrt {c^{2} d x^{2} + d} c^{2} d^{2}} - \frac {x}{{\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}} c^{2} d}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^2\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}{{\left (d\,c^2\,x^2+d\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )}{\left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________