Optimal. Leaf size=203 \[ -\frac {a^2 \sinh ^{-1}(a+b x)^3}{2 b^2}-\frac {3 (a+b x) \sqrt {(a+b x)^2+1}}{8 b^2}+\frac {6 a \sqrt {(a+b x)^2+1}}{b^2}+\frac {\sinh ^{-1}(a+b x)^3}{4 b^2}-\frac {3 (a+b x) \sqrt {(a+b x)^2+1} \sinh ^{-1}(a+b x)^2}{4 b^2}+\frac {3 a \sqrt {(a+b x)^2+1} \sinh ^{-1}(a+b x)^2}{b^2}+\frac {3 (a+b x)^2 \sinh ^{-1}(a+b x)}{4 b^2}-\frac {6 a (a+b x) \sinh ^{-1}(a+b x)}{b^2}+\frac {3 \sinh ^{-1}(a+b x)}{8 b^2}+\frac {1}{2} x^2 \sinh ^{-1}(a+b x)^3 \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.30, antiderivative size = 203, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 10, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {5865, 5801, 5831, 3317, 3296, 2638, 3311, 30, 2635, 8} \[ -\frac {a^2 \sinh ^{-1}(a+b x)^3}{2 b^2}-\frac {3 (a+b x) \sqrt {(a+b x)^2+1}}{8 b^2}+\frac {6 a \sqrt {(a+b x)^2+1}}{b^2}+\frac {\sinh ^{-1}(a+b x)^3}{4 b^2}-\frac {3 (a+b x) \sqrt {(a+b x)^2+1} \sinh ^{-1}(a+b x)^2}{4 b^2}+\frac {3 a \sqrt {(a+b x)^2+1} \sinh ^{-1}(a+b x)^2}{b^2}+\frac {3 (a+b x)^2 \sinh ^{-1}(a+b x)}{4 b^2}-\frac {6 a (a+b x) \sinh ^{-1}(a+b x)}{b^2}+\frac {3 \sinh ^{-1}(a+b x)}{8 b^2}+\frac {1}{2} x^2 \sinh ^{-1}(a+b x)^3 \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 30
Rule 2635
Rule 2638
Rule 3296
Rule 3311
Rule 3317
Rule 5801
Rule 5831
Rule 5865
Rubi steps
\begin {align*} \int x \sinh ^{-1}(a+b x)^3 \, dx &=\frac {\operatorname {Subst}\left (\int \left (-\frac {a}{b}+\frac {x}{b}\right ) \sinh ^{-1}(x)^3 \, dx,x,a+b x\right )}{b}\\ &=\frac {1}{2} x^2 \sinh ^{-1}(a+b x)^3-\frac {3}{2} \operatorname {Subst}\left (\int \frac {\left (-\frac {a}{b}+\frac {x}{b}\right )^2 \sinh ^{-1}(x)^2}{\sqrt {1+x^2}} \, dx,x,a+b x\right )\\ &=\frac {1}{2} x^2 \sinh ^{-1}(a+b x)^3-\frac {3}{2} \operatorname {Subst}\left (\int x^2 \left (-\frac {a}{b}+\frac {\sinh (x)}{b}\right )^2 \, dx,x,\sinh ^{-1}(a+b x)\right )\\ &=\frac {1}{2} x^2 \sinh ^{-1}(a+b x)^3-\frac {3}{2} \operatorname {Subst}\left (\int \left (\frac {a^2 x^2}{b^2}-\frac {2 a x^2 \sinh (x)}{b^2}+\frac {x^2 \sinh ^2(x)}{b^2}\right ) \, dx,x,\sinh ^{-1}(a+b x)\right )\\ &=-\frac {a^2 \sinh ^{-1}(a+b x)^3}{2 b^2}+\frac {1}{2} x^2 \sinh ^{-1}(a+b x)^3-\frac {3 \operatorname {Subst}\left (\int x^2 \sinh ^2(x) \, dx,x,\sinh ^{-1}(a+b x)\right )}{2 b^2}+\frac {(3 a) \operatorname {Subst}\left (\int x^2 \sinh (x) \, dx,x,\sinh ^{-1}(a+b x)\right )}{b^2}\\ &=\frac {3 (a+b x)^2 \sinh ^{-1}(a+b x)}{4 b^2}+\frac {3 a \sqrt {1+(a+b x)^2} \sinh ^{-1}(a+b x)^2}{b^2}-\frac {3 (a+b x) \sqrt {1+(a+b x)^2} \sinh ^{-1}(a+b x)^2}{4 b^2}-\frac {a^2 \sinh ^{-1}(a+b x)^3}{2 b^2}+\frac {1}{2} x^2 \sinh ^{-1}(a+b x)^3+\frac {3 \operatorname {Subst}\left (\int x^2 \, dx,x,\sinh ^{-1}(a+b x)\right )}{4 b^2}-\frac {3 \operatorname {Subst}\left (\int \sinh ^2(x) \, dx,x,\sinh ^{-1}(a+b x)\right )}{4 b^2}-\frac {(6 a) \operatorname {Subst}\left (\int x \cosh (x) \, dx,x,\sinh ^{-1}(a+b x)\right )}{b^2}\\ &=-\frac {3 (a+b x) \sqrt {1+(a+b x)^2}}{8 b^2}-\frac {6 a (a+b x) \sinh ^{-1}(a+b x)}{b^2}+\frac {3 (a+b x)^2 \sinh ^{-1}(a+b x)}{4 b^2}+\frac {3 a \sqrt {1+(a+b x)^2} \sinh ^{-1}(a+b x)^2}{b^2}-\frac {3 (a+b x) \sqrt {1+(a+b x)^2} \sinh ^{-1}(a+b x)^2}{4 b^2}+\frac {\sinh ^{-1}(a+b x)^3}{4 b^2}-\frac {a^2 \sinh ^{-1}(a+b x)^3}{2 b^2}+\frac {1}{2} x^2 \sinh ^{-1}(a+b x)^3+\frac {3 \operatorname {Subst}\left (\int 1 \, dx,x,\sinh ^{-1}(a+b x)\right )}{8 b^2}+\frac {(6 a) \operatorname {Subst}\left (\int \sinh (x) \, dx,x,\sinh ^{-1}(a+b x)\right )}{b^2}\\ &=\frac {6 a \sqrt {1+(a+b x)^2}}{b^2}-\frac {3 (a+b x) \sqrt {1+(a+b x)^2}}{8 b^2}+\frac {3 \sinh ^{-1}(a+b x)}{8 b^2}-\frac {6 a (a+b x) \sinh ^{-1}(a+b x)}{b^2}+\frac {3 (a+b x)^2 \sinh ^{-1}(a+b x)}{4 b^2}+\frac {3 a \sqrt {1+(a+b x)^2} \sinh ^{-1}(a+b x)^2}{b^2}-\frac {3 (a+b x) \sqrt {1+(a+b x)^2} \sinh ^{-1}(a+b x)^2}{4 b^2}+\frac {\sinh ^{-1}(a+b x)^3}{4 b^2}-\frac {a^2 \sinh ^{-1}(a+b x)^3}{2 b^2}+\frac {1}{2} x^2 \sinh ^{-1}(a+b x)^3\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.14, size = 129, normalized size = 0.64 \[ \frac {3 (15 a-b x) \sqrt {a^2+2 a b x+b^2 x^2+1}+\left (-4 a^2+4 b^2 x^2+2\right ) \sinh ^{-1}(a+b x)^3+6 (3 a-b x) \sqrt {a^2+2 a b x+b^2 x^2+1} \sinh ^{-1}(a+b x)^2+\left (-42 a^2-36 a b x+6 b^2 x^2+3\right ) \sinh ^{-1}(a+b x)}{8 b^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.50, size = 180, normalized size = 0.89 \[ \frac {2 \, {\left (2 \, b^{2} x^{2} - 2 \, a^{2} + 1\right )} \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )^{3} - 6 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (b x - 3 \, a\right )} \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )^{2} + 3 \, {\left (2 \, b^{2} x^{2} - 12 \, a b x - 14 \, a^{2} + 1\right )} \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right ) - 3 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (b x - 15 \, a\right )}}{8 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x \operatorname {arsinh}\left (b x + a\right )^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.08, size = 169, normalized size = 0.83 \[ \frac {\frac {\arcsinh \left (b x +a \right )^{3} \left (1+\left (b x +a \right )^{2}\right )}{2}-\frac {3 \arcsinh \left (b x +a \right )^{2} \sqrt {1+\left (b x +a \right )^{2}}\, \left (b x +a \right )}{4}-\frac {\arcsinh \left (b x +a \right )^{3}}{4}+\frac {3 \arcsinh \left (b x +a \right ) \left (1+\left (b x +a \right )^{2}\right )}{4}-\frac {3 \left (b x +a \right ) \sqrt {1+\left (b x +a \right )^{2}}}{8}-\frac {3 \arcsinh \left (b x +a \right )}{8}-a \left (\arcsinh \left (b x +a \right )^{3} \left (b x +a \right )-3 \arcsinh \left (b x +a \right )^{2} \sqrt {1+\left (b x +a \right )^{2}}+6 \left (b x +a \right ) \arcsinh \left (b x +a \right )-6 \sqrt {1+\left (b x +a \right )^{2}}\right )}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{2} \, x^{2} \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )^{3} - \int \frac {3 \, {\left (b^{3} x^{4} + 2 \, a b^{2} x^{3} + {\left (a^{2} b + b\right )} x^{2} + {\left (b^{2} x^{3} + a b x^{2}\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )} \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )^{2}}{2 \, {\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + a^{3} + {\left (3 \, a^{2} b + b\right )} x + {\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {3}{2}} + a\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int x\,{\mathrm {asinh}\left (a+b\,x\right )}^3 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 1.29, size = 248, normalized size = 1.22 \[ \begin {cases} - \frac {a^{2} \operatorname {asinh}^{3}{\left (a + b x \right )}}{2 b^{2}} - \frac {21 a^{2} \operatorname {asinh}{\left (a + b x \right )}}{4 b^{2}} - \frac {9 a x \operatorname {asinh}{\left (a + b x \right )}}{2 b} + \frac {9 a \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} \operatorname {asinh}^{2}{\left (a + b x \right )}}{4 b^{2}} + \frac {45 a \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}{8 b^{2}} + \frac {x^{2} \operatorname {asinh}^{3}{\left (a + b x \right )}}{2} + \frac {3 x^{2} \operatorname {asinh}{\left (a + b x \right )}}{4} - \frac {3 x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} \operatorname {asinh}^{2}{\left (a + b x \right )}}{4 b} - \frac {3 x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}{8 b} + \frac {\operatorname {asinh}^{3}{\left (a + b x \right )}}{4 b^{2}} + \frac {3 \operatorname {asinh}{\left (a + b x \right )}}{8 b^{2}} & \text {for}\: b \neq 0 \\\frac {x^{2} \operatorname {asinh}^{3}{\relax (a )}}{2} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________