Optimal. Leaf size=161 \[ \frac {3 b^2 e (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )}{4 d}-\frac {3 b e (c+d x) \sqrt {(c+d x)^2+1} \left (a+b \sinh ^{-1}(c+d x)\right )^2}{4 d}+\frac {e (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )^3}{2 d}+\frac {e \left (a+b \sinh ^{-1}(c+d x)\right )^3}{4 d}-\frac {3 b^3 e (c+d x) \sqrt {(c+d x)^2+1}}{8 d}+\frac {3 b^3 e \sinh ^{-1}(c+d x)}{8 d} \]
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Rubi [A] time = 0.21, antiderivative size = 161, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5865, 12, 5661, 5758, 5675, 321, 215} \[ \frac {3 b^2 e (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )}{4 d}-\frac {3 b e (c+d x) \sqrt {(c+d x)^2+1} \left (a+b \sinh ^{-1}(c+d x)\right )^2}{4 d}+\frac {e (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )^3}{2 d}+\frac {e \left (a+b \sinh ^{-1}(c+d x)\right )^3}{4 d}-\frac {3 b^3 e (c+d x) \sqrt {(c+d x)^2+1}}{8 d}+\frac {3 b^3 e \sinh ^{-1}(c+d x)}{8 d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 215
Rule 321
Rule 5661
Rule 5675
Rule 5758
Rule 5865
Rubi steps
\begin {align*} \int (c e+d e x) \left (a+b \sinh ^{-1}(c+d x)\right )^3 \, dx &=\frac {\operatorname {Subst}\left (\int e x \left (a+b \sinh ^{-1}(x)\right )^3 \, dx,x,c+d x\right )}{d}\\ &=\frac {e \operatorname {Subst}\left (\int x \left (a+b \sinh ^{-1}(x)\right )^3 \, dx,x,c+d x\right )}{d}\\ &=\frac {e (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )^3}{2 d}-\frac {(3 b e) \operatorname {Subst}\left (\int \frac {x^2 \left (a+b \sinh ^{-1}(x)\right )^2}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{2 d}\\ &=-\frac {3 b e (c+d x) \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^2}{4 d}+\frac {e (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )^3}{2 d}+\frac {(3 b e) \operatorname {Subst}\left (\int \frac {\left (a+b \sinh ^{-1}(x)\right )^2}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{4 d}+\frac {\left (3 b^2 e\right ) \operatorname {Subst}\left (\int x \left (a+b \sinh ^{-1}(x)\right ) \, dx,x,c+d x\right )}{2 d}\\ &=\frac {3 b^2 e (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )}{4 d}-\frac {3 b e (c+d x) \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^2}{4 d}+\frac {e \left (a+b \sinh ^{-1}(c+d x)\right )^3}{4 d}+\frac {e (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )^3}{2 d}-\frac {\left (3 b^3 e\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{4 d}\\ &=-\frac {3 b^3 e (c+d x) \sqrt {1+(c+d x)^2}}{8 d}+\frac {3 b^2 e (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )}{4 d}-\frac {3 b e (c+d x) \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^2}{4 d}+\frac {e \left (a+b \sinh ^{-1}(c+d x)\right )^3}{4 d}+\frac {e (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )^3}{2 d}+\frac {\left (3 b^3 e\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{8 d}\\ &=-\frac {3 b^3 e (c+d x) \sqrt {1+(c+d x)^2}}{8 d}+\frac {3 b^3 e \sinh ^{-1}(c+d x)}{8 d}+\frac {3 b^2 e (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )}{4 d}-\frac {3 b e (c+d x) \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^2}{4 d}+\frac {e \left (a+b \sinh ^{-1}(c+d x)\right )^3}{4 d}+\frac {e (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )^3}{2 d}\\ \end {align*}
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Mathematica [A] time = 0.21, size = 200, normalized size = 1.24 \[ \frac {e \left (2 a \left (2 a^2+3 b^2\right ) (c+d x)^2-3 b \left (2 a^2+b^2\right ) (c+d x) \sqrt {(c+d x)^2+1}+3 b \left (2 a^2+b^2\right ) \sinh ^{-1}(c+d x)-6 b (c+d x) \sinh ^{-1}(c+d x) \left (-2 a^2 (c+d x)+2 a b \sqrt {(c+d x)^2+1}-b^2 (c+d x)\right )+6 b^2 \sinh ^{-1}(c+d x)^2 \left (2 a (c+d x)^2+a-b \sqrt {(c+d x)^2+1} (c+d x)\right )+2 b^3 \left (2 (c+d x)^2+1\right ) \sinh ^{-1}(c+d x)^3\right )}{8 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.54, size = 391, normalized size = 2.43 \[ \frac {2 \, {\left (2 \, a^{3} + 3 \, a b^{2}\right )} d^{2} e x^{2} + 4 \, {\left (2 \, a^{3} + 3 \, a b^{2}\right )} c d e x + 2 \, {\left (2 \, b^{3} d^{2} e x^{2} + 4 \, b^{3} c d e x + {\left (2 \, b^{3} c^{2} + b^{3}\right )} e\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )^{3} + 6 \, {\left (2 \, a b^{2} d^{2} e x^{2} + 4 \, a b^{2} c d e x + {\left (2 \, a b^{2} c^{2} + a b^{2}\right )} e - {\left (b^{3} d e x + b^{3} c e\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )^{2} + 3 \, {\left (2 \, {\left (2 \, a^{2} b + b^{3}\right )} d^{2} e x^{2} + 4 \, {\left (2 \, a^{2} b + b^{3}\right )} c d e x + {\left (2 \, a^{2} b + b^{3} + 2 \, {\left (2 \, a^{2} b + b^{3}\right )} c^{2}\right )} e - 4 \, {\left (a b^{2} d e x + a b^{2} c e\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right ) - 3 \, {\left ({\left (2 \, a^{2} b + b^{3}\right )} d e x + {\left (2 \, a^{2} b + b^{3}\right )} c e\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}}{8 \, d} \]
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 \[ \int {\left (d e x + c e\right )} {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 243, normalized size = 1.51 \[ \frac {\frac {\left (d x +c \right )^{2} e \,a^{3}}{2}+e \,b^{3} \left (\frac {\arcsinh \left (d x +c \right )^{3} \left (1+\left (d x +c \right )^{2}\right )}{2}-\frac {3 \arcsinh \left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}\, \left (d x +c \right )}{4}-\frac {\arcsinh \left (d x +c \right )^{3}}{4}+\frac {3 \arcsinh \left (d x +c \right ) \left (1+\left (d x +c \right )^{2}\right )}{4}-\frac {3 \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}}{8}-\frac {3 \arcsinh \left (d x +c \right )}{8}\right )+3 e a \,b^{2} \left (\frac {\arcsinh \left (d x +c \right )^{2} \left (1+\left (d x +c \right )^{2}\right )}{2}-\frac {\arcsinh \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}\, \left (d x +c \right )}{2}-\frac {\arcsinh \left (d x +c \right )^{2}}{4}+\frac {\left (d x +c \right )^{2}}{4}+\frac {1}{4}\right )+3 e \,a^{2} b \left (\frac {\left (d x +c \right )^{2} \arcsinh \left (d x +c \right )}{2}-\frac {\left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}}{4}+\frac {\arcsinh \left (d x +c \right )}{4}\right )}{d} \]
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}{2} \, a^{3} d e x^{2} + \frac {3}{4} \, {\left (2 \, x^{2} \operatorname {arsinh}\left (d x + c\right ) - d {\left (\frac {3 \, c^{2} \operatorname {arsinh}\left (\frac {2 \, {\left (d^{2} x + c d\right )}}{\sqrt {-4 \, c^{2} d^{2} + 4 \, {\left (c^{2} + 1\right )} d^{2}}}\right )}{d^{3}} + \frac {\sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1} x}{d^{2}} - \frac {{\left (c^{2} + 1\right )} \operatorname {arsinh}\left (\frac {2 \, {\left (d^{2} x + c d\right )}}{\sqrt {-4 \, c^{2} d^{2} + 4 \, {\left (c^{2} + 1\right )} d^{2}}}\right )}{d^{3}} - \frac {3 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1} c}{d^{3}}\right )}\right )} a^{2} b d e + a^{3} c e x + \frac {3 \, {\left ({\left (d x + c\right )} \operatorname {arsinh}\left (d x + c\right ) - \sqrt {{\left (d x + c\right )}^{2} + 1}\right )} a^{2} b c e}{d} + \frac {1}{2} \, {\left (b^{3} d e x^{2} + 2 \, b^{3} c e x\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )^{3} + \int \frac {3 \, {\left ({\left (2 \, a b^{2} d^{4} e - b^{3} d^{4} e\right )} x^{4} + 2 \, {\left (c^{4} e + c^{2} e\right )} a b^{2} + 4 \, {\left (2 \, a b^{2} c d^{3} e - b^{3} c d^{3} e\right )} x^{3} + {\left (2 \, {\left (6 \, c^{2} d^{2} e + d^{2} e\right )} a b^{2} - {\left (5 \, c^{2} d^{2} e + d^{2} e\right )} b^{3}\right )} x^{2} + 2 \, {\left (2 \, {\left (2 \, c^{3} d e + c d e\right )} a b^{2} - {\left (c^{3} d e + c d e\right )} b^{3}\right )} x + {\left (2 \, {\left (c^{3} e + c e\right )} a b^{2} + {\left (2 \, a b^{2} d^{3} e - b^{3} d^{3} e\right )} x^{3} + 3 \, {\left (2 \, a b^{2} c d^{2} e - b^{3} c d^{2} e\right )} x^{2} - 2 \, {\left (b^{3} c^{2} d e - {\left (3 \, c^{2} d e + d e\right )} a b^{2}\right )} x\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )^{2}}{2 \, {\left (d^{3} x^{3} + 3 \, c d^{2} x^{2} + c^{3} + {\left (3 \, c^{2} d + d\right )} x + {\left (d^{2} x^{2} + 2 \, c d x + c^{2} + 1\right )}^{\frac {3}{2}} + c\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 = -0.01 \[ \int \left (c\,e+d\,e\,x\right )\,{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^3 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.08, size = 685, normalized size = 4.25 \[ \begin {cases} a^{3} c e x + \frac {a^{3} d e x^{2}}{2} + \frac {3 a^{2} b c^{2} e \operatorname {asinh}{\left (c + d x \right )}}{2 d} + 3 a^{2} b c e x \operatorname {asinh}{\left (c + d x \right )} - \frac {3 a^{2} b c e \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1}}{4 d} + \frac {3 a^{2} b d e x^{2} \operatorname {asinh}{\left (c + d x \right )}}{2} - \frac {3 a^{2} b e x \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1}}{4} + \frac {3 a^{2} b e \operatorname {asinh}{\left (c + d x \right )}}{4 d} + \frac {3 a b^{2} c^{2} e \operatorname {asinh}^{2}{\left (c + d x \right )}}{2 d} + 3 a b^{2} c e x \operatorname {asinh}^{2}{\left (c + d x \right )} + \frac {3 a b^{2} c e x}{2} - \frac {3 a b^{2} c e \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1} \operatorname {asinh}{\left (c + d x \right )}}{2 d} + \frac {3 a b^{2} d e x^{2} \operatorname {asinh}^{2}{\left (c + d x \right )}}{2} + \frac {3 a b^{2} d e x^{2}}{4} - \frac {3 a b^{2} e x \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1} \operatorname {asinh}{\left (c + d x \right )}}{2} + \frac {3 a b^{2} e \operatorname {asinh}^{2}{\left (c + d x \right )}}{4 d} + \frac {b^{3} c^{2} e \operatorname {asinh}^{3}{\left (c + d x \right )}}{2 d} + \frac {3 b^{3} c^{2} e \operatorname {asinh}{\left (c + d x \right )}}{4 d} + b^{3} c e x \operatorname {asinh}^{3}{\left (c + d x \right )} + \frac {3 b^{3} c e x \operatorname {asinh}{\left (c + d x \right )}}{2} - \frac {3 b^{3} c e \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1} \operatorname {asinh}^{2}{\left (c + d x \right )}}{4 d} - \frac {3 b^{3} c e \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1}}{8 d} + \frac {b^{3} d e x^{2} \operatorname {asinh}^{3}{\left (c + d x \right )}}{2} + \frac {3 b^{3} d e x^{2} \operatorname {asinh}{\left (c + d x \right )}}{4} - \frac {3 b^{3} e x \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1} \operatorname {asinh}^{2}{\left (c + d x \right )}}{4} - \frac {3 b^{3} e x \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1}}{8} + \frac {b^{3} e \operatorname {asinh}^{3}{\left (c + d x \right )}}{4 d} + \frac {3 b^{3} e \operatorname {asinh}{\left (c + d x \right )}}{8 d} & \text {for}\: d \neq 0 \\c e x \left (a + b \operatorname {asinh}{\relax (c )}\right )^{3} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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