Optimal. Leaf size=331 \[ -\frac {a^4 \sinh ^{-1}(a+b x)^2}{4 b^4}+\frac {2 a^3 \sqrt {(a+b x)^2+1} \sinh ^{-1}(a+b x)}{b^4}-\frac {2 a^3 x}{b^3}+\frac {3 a^2 (a+b x)^2}{4 b^4}+\frac {3 a^2 \sinh ^{-1}(a+b x)^2}{4 b^4}-\frac {3 a^2 (a+b x) \sqrt {(a+b x)^2+1} \sinh ^{-1}(a+b x)}{2 b^4}-\frac {2 a (a+b x)^3}{9 b^4}+\frac {(a+b x)^4}{32 b^4}-\frac {3 (a+b x)^2}{32 b^4}+\frac {2 a (a+b x)^2 \sqrt {(a+b x)^2+1} \sinh ^{-1}(a+b x)}{3 b^4}-\frac {4 a \sqrt {(a+b x)^2+1} \sinh ^{-1}(a+b x)}{3 b^4}-\frac {3 \sinh ^{-1}(a+b x)^2}{32 b^4}-\frac {(a+b x)^3 \sqrt {(a+b x)^2+1} \sinh ^{-1}(a+b x)}{8 b^4}+\frac {3 (a+b x) \sqrt {(a+b x)^2+1} \sinh ^{-1}(a+b x)}{16 b^4}+\frac {4 a x}{3 b^3}+\frac {1}{4} x^4 \sinh ^{-1}(a+b x)^2 \]
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Rubi [A] time = 0.55, antiderivative size = 331, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 8, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {5865, 5801, 5821, 5675, 5717, 8, 5758, 30} \[ -\frac {2 a^3 x}{b^3}+\frac {3 a^2 (a+b x)^2}{4 b^4}-\frac {a^4 \sinh ^{-1}(a+b x)^2}{4 b^4}+\frac {2 a^3 \sqrt {(a+b x)^2+1} \sinh ^{-1}(a+b x)}{b^4}+\frac {3 a^2 \sinh ^{-1}(a+b x)^2}{4 b^4}-\frac {3 a^2 (a+b x) \sqrt {(a+b x)^2+1} \sinh ^{-1}(a+b x)}{2 b^4}-\frac {2 a (a+b x)^3}{9 b^4}+\frac {4 a x}{3 b^3}+\frac {(a+b x)^4}{32 b^4}-\frac {3 (a+b x)^2}{32 b^4}+\frac {2 a (a+b x)^2 \sqrt {(a+b x)^2+1} \sinh ^{-1}(a+b x)}{3 b^4}-\frac {4 a \sqrt {(a+b x)^2+1} \sinh ^{-1}(a+b x)}{3 b^4}-\frac {3 \sinh ^{-1}(a+b x)^2}{32 b^4}-\frac {(a+b x)^3 \sqrt {(a+b x)^2+1} \sinh ^{-1}(a+b x)}{8 b^4}+\frac {3 (a+b x) \sqrt {(a+b x)^2+1} \sinh ^{-1}(a+b x)}{16 b^4}+\frac {1}{4} x^4 \sinh ^{-1}(a+b x)^2 \]
Antiderivative was successfully verified.
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Rule 8
Rule 30
Rule 5675
Rule 5717
Rule 5758
Rule 5801
Rule 5821
Rule 5865
Rubi steps
\begin {align*} \int x^3 \sinh ^{-1}(a+b x)^2 \, dx &=\frac {\operatorname {Subst}\left (\int \left (-\frac {a}{b}+\frac {x}{b}\right )^3 \sinh ^{-1}(x)^2 \, dx,x,a+b x\right )}{b}\\ &=\frac {1}{4} x^4 \sinh ^{-1}(a+b x)^2-\frac {1}{2} \operatorname {Subst}\left (\int \frac {\left (-\frac {a}{b}+\frac {x}{b}\right )^4 \sinh ^{-1}(x)}{\sqrt {1+x^2}} \, dx,x,a+b x\right )\\ &=\frac {1}{4} x^4 \sinh ^{-1}(a+b x)^2-\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {a^4 \sinh ^{-1}(x)}{b^4 \sqrt {1+x^2}}-\frac {4 a^3 x \sinh ^{-1}(x)}{b^4 \sqrt {1+x^2}}+\frac {6 a^2 x^2 \sinh ^{-1}(x)}{b^4 \sqrt {1+x^2}}-\frac {4 a x^3 \sinh ^{-1}(x)}{b^4 \sqrt {1+x^2}}+\frac {x^4 \sinh ^{-1}(x)}{b^4 \sqrt {1+x^2}}\right ) \, dx,x,a+b x\right )\\ &=\frac {1}{4} x^4 \sinh ^{-1}(a+b x)^2-\frac {\operatorname {Subst}\left (\int \frac {x^4 \sinh ^{-1}(x)}{\sqrt {1+x^2}} \, dx,x,a+b x\right )}{2 b^4}+\frac {(2 a) \operatorname {Subst}\left (\int \frac {x^3 \sinh ^{-1}(x)}{\sqrt {1+x^2}} \, dx,x,a+b x\right )}{b^4}-\frac {\left (3 a^2\right ) \operatorname {Subst}\left (\int \frac {x^2 \sinh ^{-1}(x)}{\sqrt {1+x^2}} \, dx,x,a+b x\right )}{b^4}+\frac {\left (2 a^3\right ) \operatorname {Subst}\left (\int \frac {x \sinh ^{-1}(x)}{\sqrt {1+x^2}} \, dx,x,a+b x\right )}{b^4}-\frac {a^4 \operatorname {Subst}\left (\int \frac {\sinh ^{-1}(x)}{\sqrt {1+x^2}} \, dx,x,a+b x\right )}{2 b^4}\\ &=\frac {2 a^3 \sqrt {1+(a+b x)^2} \sinh ^{-1}(a+b x)}{b^4}-\frac {3 a^2 (a+b x) \sqrt {1+(a+b x)^2} \sinh ^{-1}(a+b x)}{2 b^4}+\frac {2 a (a+b x)^2 \sqrt {1+(a+b x)^2} \sinh ^{-1}(a+b x)}{3 b^4}-\frac {(a+b x)^3 \sqrt {1+(a+b x)^2} \sinh ^{-1}(a+b x)}{8 b^4}-\frac {a^4 \sinh ^{-1}(a+b x)^2}{4 b^4}+\frac {1}{4} x^4 \sinh ^{-1}(a+b x)^2+\frac {\operatorname {Subst}\left (\int x^3 \, dx,x,a+b x\right )}{8 b^4}+\frac {3 \operatorname {Subst}\left (\int \frac {x^2 \sinh ^{-1}(x)}{\sqrt {1+x^2}} \, dx,x,a+b x\right )}{8 b^4}-\frac {(2 a) \operatorname {Subst}\left (\int x^2 \, dx,x,a+b x\right )}{3 b^4}-\frac {(4 a) \operatorname {Subst}\left (\int \frac {x \sinh ^{-1}(x)}{\sqrt {1+x^2}} \, dx,x,a+b x\right )}{3 b^4}+\frac {\left (3 a^2\right ) \operatorname {Subst}(\int x \, dx,x,a+b x)}{2 b^4}+\frac {\left (3 a^2\right ) \operatorname {Subst}\left (\int \frac {\sinh ^{-1}(x)}{\sqrt {1+x^2}} \, dx,x,a+b x\right )}{2 b^4}-\frac {\left (2 a^3\right ) \operatorname {Subst}(\int 1 \, dx,x,a+b x)}{b^4}\\ &=-\frac {2 a^3 x}{b^3}+\frac {3 a^2 (a+b x)^2}{4 b^4}-\frac {2 a (a+b x)^3}{9 b^4}+\frac {(a+b x)^4}{32 b^4}-\frac {4 a \sqrt {1+(a+b x)^2} \sinh ^{-1}(a+b x)}{3 b^4}+\frac {2 a^3 \sqrt {1+(a+b x)^2} \sinh ^{-1}(a+b x)}{b^4}+\frac {3 (a+b x) \sqrt {1+(a+b x)^2} \sinh ^{-1}(a+b x)}{16 b^4}-\frac {3 a^2 (a+b x) \sqrt {1+(a+b x)^2} \sinh ^{-1}(a+b x)}{2 b^4}+\frac {2 a (a+b x)^2 \sqrt {1+(a+b x)^2} \sinh ^{-1}(a+b x)}{3 b^4}-\frac {(a+b x)^3 \sqrt {1+(a+b x)^2} \sinh ^{-1}(a+b x)}{8 b^4}+\frac {3 a^2 \sinh ^{-1}(a+b x)^2}{4 b^4}-\frac {a^4 \sinh ^{-1}(a+b x)^2}{4 b^4}+\frac {1}{4} x^4 \sinh ^{-1}(a+b x)^2-\frac {3 \operatorname {Subst}(\int x \, dx,x,a+b x)}{16 b^4}-\frac {3 \operatorname {Subst}\left (\int \frac {\sinh ^{-1}(x)}{\sqrt {1+x^2}} \, dx,x,a+b x\right )}{16 b^4}+\frac {(4 a) \operatorname {Subst}(\int 1 \, dx,x,a+b x)}{3 b^4}\\ &=\frac {4 a x}{3 b^3}-\frac {2 a^3 x}{b^3}-\frac {3 (a+b x)^2}{32 b^4}+\frac {3 a^2 (a+b x)^2}{4 b^4}-\frac {2 a (a+b x)^3}{9 b^4}+\frac {(a+b x)^4}{32 b^4}-\frac {4 a \sqrt {1+(a+b x)^2} \sinh ^{-1}(a+b x)}{3 b^4}+\frac {2 a^3 \sqrt {1+(a+b x)^2} \sinh ^{-1}(a+b x)}{b^4}+\frac {3 (a+b x) \sqrt {1+(a+b x)^2} \sinh ^{-1}(a+b x)}{16 b^4}-\frac {3 a^2 (a+b x) \sqrt {1+(a+b x)^2} \sinh ^{-1}(a+b x)}{2 b^4}+\frac {2 a (a+b x)^2 \sqrt {1+(a+b x)^2} \sinh ^{-1}(a+b x)}{3 b^4}-\frac {(a+b x)^3 \sqrt {1+(a+b x)^2} \sinh ^{-1}(a+b x)}{8 b^4}-\frac {3 \sinh ^{-1}(a+b x)^2}{32 b^4}+\frac {3 a^2 \sinh ^{-1}(a+b x)^2}{4 b^4}-\frac {a^4 \sinh ^{-1}(a+b x)^2}{4 b^4}+\frac {1}{4} x^4 \sinh ^{-1}(a+b x)^2\\ \end {align*}
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Mathematica [A] time = 0.19, size = 145, normalized size = 0.44 \[ \frac {-9 \left (8 a^4-24 a^2-8 b^4 x^4+3\right ) \sinh ^{-1}(a+b x)^2+b x \left (-300 a^3+78 a^2 b x+a \left (330-28 b^2 x^2\right )+9 b x \left (b^2 x^2-3\right )\right )+6 \sqrt {a^2+2 a b x+b^2 x^2+1} \left (50 a^3-26 a^2 b x+a \left (14 b^2 x^2-55\right )-6 b^3 x^3+9 b x\right ) \sinh ^{-1}(a+b x)}{288 b^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.54, size = 182, normalized size = 0.55 \[ \frac {9 \, b^{4} x^{4} - 28 \, a b^{3} x^{3} + 3 \, {\left (26 \, a^{2} - 9\right )} b^{2} x^{2} - 30 \, {\left (10 \, a^{3} - 11 \, a\right )} b x + 9 \, {\left (8 \, b^{4} x^{4} - 8 \, a^{4} + 24 \, a^{2} - 3\right )} \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )^{2} - 6 \, {\left (6 \, b^{3} x^{3} - 14 \, a b^{2} x^{2} - 50 \, a^{3} + {\left (26 \, a^{2} - 9\right )} b x + 55 \, a\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )}{288 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{3} \operatorname {arsinh}\left (b x + a\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.11, size = 377, normalized size = 1.14 \[ \frac {-a^{3} \left (\arcsinh \left (b x +a \right )^{2} \left (b x +a \right )-2 \arcsinh \left (b x +a \right ) \sqrt {1+\left (b x +a \right )^{2}}+2 b x +2 a \right )+\frac {3 a^{2} \left (2 \arcsinh \left (b x +a \right )^{2} \left (b x +a \right )^{2}-2 \arcsinh \left (b x +a \right ) \sqrt {1+\left (b x +a \right )^{2}}\, \left (b x +a \right )+\arcsinh \left (b x +a \right )^{2}+\left (b x +a \right )^{2}+1\right )}{4}-\frac {a \left (9 \arcsinh \left (b x +a \right )^{2} \left (b x +a \right )^{3}-6 \arcsinh \left (b x +a \right ) \sqrt {1+\left (b x +a \right )^{2}}\, \left (b x +a \right )^{2}+27 \arcsinh \left (b x +a \right )^{2} \left (b x +a \right )+2 \left (b x +a \right )^{3}-42 \arcsinh \left (b x +a \right ) \sqrt {1+\left (b x +a \right )^{2}}+42 b x +42 a \right )}{9}+\frac {\arcsinh \left (b x +a \right )^{2} \left (1+\left (b x +a \right )^{2}\right )^{2}}{4}-\frac {\arcsinh \left (b x +a \right ) \left (b x +a \right ) \left (1+\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{8}+\frac {5 \arcsinh \left (b x +a \right ) \sqrt {1+\left (b x +a \right )^{2}}\, \left (b x +a \right )}{16}+\frac {5 \arcsinh \left (b x +a \right )^{2}}{32}+\frac {\left (1+\left (b x +a \right )^{2}\right )^{2}}{32}-\frac {5 \left (b x +a \right )^{2}}{32}-\frac {5}{32}+3 a \left (\arcsinh \left (b x +a \right )^{2} \left (b x +a \right )-2 \arcsinh \left (b x +a \right ) \sqrt {1+\left (b x +a \right )^{2}}+2 b x +2 a \right )-\frac {\arcsinh \left (b x +a \right )^{2} \left (1+\left (b x +a \right )^{2}\right )}{2}}{b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{4} \, x^{4} \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )^{2} - \int \frac {{\left (b^{3} x^{6} + 2 \, a b^{2} x^{5} + {\left (a^{2} b + b\right )} x^{4} + {\left (b^{2} x^{5} + a b x^{4}\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )} \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )}{2 \, {\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + a^{3} + {\left (3 \, a^{2} b + b\right )} x + {\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {3}{2}} + a\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int x^3\,{\mathrm {asinh}\left (a+b\,x\right )}^2 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.11, size = 366, normalized size = 1.11 \[ \begin {cases} - \frac {a^{4} \operatorname {asinh}^{2}{\left (a + b x \right )}}{4 b^{4}} - \frac {25 a^{3} x}{24 b^{3}} + \frac {25 a^{3} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} \operatorname {asinh}{\left (a + b x \right )}}{24 b^{4}} + \frac {13 a^{2} x^{2}}{48 b^{2}} - \frac {13 a^{2} x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} \operatorname {asinh}{\left (a + b x \right )}}{24 b^{3}} + \frac {3 a^{2} \operatorname {asinh}^{2}{\left (a + b x \right )}}{4 b^{4}} - \frac {7 a x^{3}}{72 b} + \frac {7 a x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} \operatorname {asinh}{\left (a + b x \right )}}{24 b^{2}} + \frac {55 a x}{48 b^{3}} - \frac {55 a \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} \operatorname {asinh}{\left (a + b x \right )}}{48 b^{4}} + \frac {x^{4} \operatorname {asinh}^{2}{\left (a + b x \right )}}{4} + \frac {x^{4}}{32} - \frac {x^{3} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} \operatorname {asinh}{\left (a + b x \right )}}{8 b} - \frac {3 x^{2}}{32 b^{2}} + \frac {3 x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} \operatorname {asinh}{\left (a + b x \right )}}{16 b^{3}} - \frac {3 \operatorname {asinh}^{2}{\left (a + b x \right )}}{32 b^{4}} & \text {for}\: b \neq 0 \\\frac {x^{4} \operatorname {asinh}^{2}{\relax (a )}}{4} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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