Optimal. Leaf size=195 \[ -\frac {3 b^3 e (c+d x) \sqrt {(c+d x)^2+1} \left (a+b \sinh ^{-1}(c+d x)\right )}{2 d}+\frac {3 b^2 e (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )^2}{2 d}+\frac {3 b^2 e \left (a+b \sinh ^{-1}(c+d x)\right )^2}{4 d}-\frac {b e (c+d x) \sqrt {(c+d x)^2+1} \left (a+b \sinh ^{-1}(c+d x)\right )^3}{d}+\frac {e (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )^4}{2 d}+\frac {e \left (a+b \sinh ^{-1}(c+d x)\right )^4}{4 d}+\frac {3 b^4 e (c+d x)^2}{4 d} \]
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Rubi [A] time = 0.32, antiderivative size = 195, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {5865, 12, 5661, 5758, 5675, 30} \[ -\frac {3 b^3 e (c+d x) \sqrt {(c+d x)^2+1} \left (a+b \sinh ^{-1}(c+d x)\right )}{2 d}+\frac {3 b^2 e (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )^2}{2 d}+\frac {3 b^2 e \left (a+b \sinh ^{-1}(c+d x)\right )^2}{4 d}-\frac {b e (c+d x) \sqrt {(c+d x)^2+1} \left (a+b \sinh ^{-1}(c+d x)\right )^3}{d}+\frac {e (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )^4}{2 d}+\frac {e \left (a+b \sinh ^{-1}(c+d x)\right )^4}{4 d}+\frac {3 b^4 e (c+d x)^2}{4 d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 30
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 )^4 \, dx &=\frac {\operatorname {Subst}\left (\int e x \left (a+b \sinh ^{-1}(x)\right )^4 \, dx,x,c+d x\right )}{d}\\ &=\frac {e \operatorname {Subst}\left (\int x \left (a+b \sinh ^{-1}(x)\right )^4 \, dx,x,c+d x\right )}{d}\\ &=\frac {e (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )^4}{2 d}-\frac {(2 b e) \operatorname {Subst}\left (\int \frac {x^2 \left (a+b \sinh ^{-1}(x)\right )^3}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{d}\\ &=-\frac {b e (c+d x) \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^3}{d}+\frac {e (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )^4}{2 d}+\frac {(b e) \operatorname {Subst}\left (\int \frac {\left (a+b \sinh ^{-1}(x)\right )^3}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{d}+\frac {\left (3 b^2 e\right ) \operatorname {Subst}\left (\int x \left (a+b \sinh ^{-1}(x)\right )^2 \, dx,x,c+d x\right )}{d}\\ &=\frac {3 b^2 e (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )^2}{2 d}-\frac {b e (c+d x) \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^3}{d}+\frac {e \left (a+b \sinh ^{-1}(c+d x)\right )^4}{4 d}+\frac {e (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )^4}{2 d}-\frac {\left (3 b^3 e\right ) \operatorname {Subst}\left (\int \frac {x^2 \left (a+b \sinh ^{-1}(x)\right )}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{d}\\ &=-\frac {3 b^3 e (c+d x) \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )}{2 d}+\frac {3 b^2 e (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )^2}{2 d}-\frac {b e (c+d x) \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^3}{d}+\frac {e \left (a+b \sinh ^{-1}(c+d x)\right )^4}{4 d}+\frac {e (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )^4}{2 d}+\frac {\left (3 b^3 e\right ) \operatorname {Subst}\left (\int \frac {a+b \sinh ^{-1}(x)}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{2 d}+\frac {\left (3 b^4 e\right ) \operatorname {Subst}(\int x \, dx,x,c+d x)}{2 d}\\ &=\frac {3 b^4 e (c+d x)^2}{4 d}-\frac {3 b^3 e (c+d x) \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )}{2 d}+\frac {3 b^2 e \left (a+b \sinh ^{-1}(c+d x)\right )^2}{4 d}+\frac {3 b^2 e (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )^2}{2 d}-\frac {b e (c+d x) \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^3}{d}+\frac {e \left (a+b \sinh ^{-1}(c+d x)\right )^4}{4 d}+\frac {e (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )^4}{2 d}\\ \end {align*}
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Mathematica [A] time = 0.32, size = 300, normalized size = 1.54 \[ \frac {e \left (-2 a b \left (2 a^2+3 b^2\right ) (c+d x) \sqrt {(c+d x)^2+1}+3 b^2 \sinh ^{-1}(c+d x)^2 \left (4 a^2 (c+d x)^2+2 a^2-4 a b (c+d x) \sqrt {(c+d x)^2+1}+2 b^2 (c+d x)^2+b^2\right )+2 a b \left (2 a^2+3 b^2\right ) \sinh ^{-1}(c+d x)+\left (2 a^4+6 a^2 b^2+3 b^4\right ) (c+d x)^2-2 b (c+d x) \sinh ^{-1}(c+d x) \left (-4 a^3 (c+d x)+6 a^2 b \sqrt {(c+d x)^2+1}-6 a b^2 (c+d x)+3 b^3 \sqrt {(c+d x)^2+1}\right )+4 b^3 \sinh ^{-1}(c+d x)^3 \left (2 a (c+d x)^2+a-b \sqrt {(c+d x)^2+1} (c+d x)\right )+b^4 \left (2 (c+d x)^2+1\right ) \sinh ^{-1}(c+d x)^4\right )}{4 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.60, size = 574, normalized size = 2.94 \[ \frac {{\left (2 \, a^{4} + 6 \, a^{2} b^{2} + 3 \, b^{4}\right )} d^{2} e x^{2} + 2 \, {\left (2 \, a^{4} + 6 \, a^{2} b^{2} + 3 \, b^{4}\right )} c d e x + {\left (2 \, b^{4} d^{2} e x^{2} + 4 \, b^{4} c d e x + {\left (2 \, b^{4} c^{2} + b^{4}\right )} e\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )^{4} + 4 \, {\left (2 \, a b^{3} d^{2} e x^{2} + 4 \, a b^{3} c d e x + {\left (2 \, a b^{3} c^{2} + a b^{3}\right )} e - {\left (b^{4} d e x + b^{4} c e\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )^{3} + 3 \, {\left (2 \, {\left (2 \, a^{2} b^{2} + b^{4}\right )} d^{2} e x^{2} + 4 \, {\left (2 \, a^{2} b^{2} + b^{4}\right )} c d e x + {\left (2 \, a^{2} b^{2} + b^{4} + 2 \, {\left (2 \, a^{2} b^{2} + b^{4}\right )} c^{2}\right )} e - 4 \, {\left (a b^{3} d e x + a b^{3} c e\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )^{2} + 2 \, {\left (2 \, {\left (2 \, a^{3} b + 3 \, a b^{3}\right )} d^{2} e x^{2} + 4 \, {\left (2 \, a^{3} b + 3 \, a b^{3}\right )} c d e x + {\left (2 \, a^{3} b + 3 \, a b^{3} + 2 \, {\left (2 \, a^{3} b + 3 \, a b^{3}\right )} c^{2}\right )} e - 3 \, {\left ({\left (2 \, a^{2} b^{2} + b^{4}\right )} d e x + {\left (2 \, a^{2} b^{2} + b^{4}\right )} c e\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right ) - 2 \, {\left ({\left (2 \, a^{3} b + 3 \, a b^{3}\right )} d e x + {\left (2 \, a^{3} b + 3 \, a b^{3}\right )} c e\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}}{4 \, d} \]
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 \[ \int {\left (d e x + c e\right )} {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{4}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 371, normalized size = 1.90 \[ \frac {\frac {\left (d x +c \right )^{2} e \,a^{4}}{2}+e \,b^{4} \left (\frac {\left (1+\left (d x +c \right )^{2}\right ) \arcsinh \left (d x +c \right )^{4}}{2}-\arcsinh \left (d x +c \right )^{3} \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}-\frac {\arcsinh \left (d x +c \right )^{4}}{4}+\frac {3 \arcsinh \left (d x +c \right )^{2} \left (1+\left (d x +c \right )^{2}\right )}{2}-\frac {3 \arcsinh \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}\, \left (d x +c \right )}{2}-\frac {3 \arcsinh \left (d x +c \right )^{2}}{4}+\frac {3 \left (d x +c \right )^{2}}{4}+\frac {3}{4}\right )+4 e a \,b^{3} \left (\frac {\arcsinh \left (d x +c \right )^{3} \left (1+\left (d x +c \right )^{2}\right )}{2}-\frac {3 \arcsinh \left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}\, \left (d x +c \right )}{4}-\frac {\arcsinh \left (d x +c \right )^{3}}{4}+\frac {3 \arcsinh \left (d x +c \right ) \left (1+\left (d x +c \right )^{2}\right )}{4}-\frac {3 \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}}{8}-\frac {3 \arcsinh \left (d x +c \right )}{8}\right )+6 e \,a^{2} b^{2} \left (\frac {\arcsinh \left (d x +c \right )^{2} \left (1+\left (d x +c \right )^{2}\right )}{2}-\frac {\arcsinh \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}\, \left (d x +c \right )}{2}-\frac {\arcsinh \left (d x +c \right )^{2}}{4}+\frac {\left (d x +c \right )^{2}}{4}+\frac {1}{4}\right )+4 e \,a^{3} b \left (\frac {\left (d x +c \right )^{2} \arcsinh \left (d x +c \right )}{2}-\frac {\left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}}{4}+\frac {\arcsinh \left (d x +c \right )}{4}\right )}{d} \]
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^{4} d e x^{2} + {\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^{3} b d e + a^{4} c e x + \frac {4 \, {\left ({\left (d x + c\right )} \operatorname {arsinh}\left (d x + c\right ) - \sqrt {{\left (d x + c\right )}^{2} + 1}\right )} a^{3} b c e}{d} + \frac {1}{2} \, {\left (b^{4} d e x^{2} + 2 \, b^{4} c e x\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )^{4} + \int \frac {2 \, {\left ({\left (2 \, {\left (c^{4} e + c^{2} e\right )} a b^{3} + {\left (2 \, a b^{3} d^{4} e - b^{4} d^{4} e\right )} x^{4} + 4 \, {\left (2 \, a b^{3} c d^{3} e - b^{4} c d^{3} e\right )} x^{3} + {\left (2 \, {\left (6 \, c^{2} d^{2} e + d^{2} e\right )} a b^{3} - {\left (5 \, c^{2} d^{2} e + d^{2} e\right )} b^{4}\right )} x^{2} + 2 \, {\left (2 \, {\left (2 \, c^{3} d e + c d e\right )} a b^{3} - {\left (c^{3} d e + c d e\right )} b^{4}\right )} x + {\left (2 \, {\left (c^{3} e + c e\right )} a b^{3} + {\left (2 \, a b^{3} d^{3} e - b^{4} d^{3} e\right )} x^{3} + 3 \, {\left (2 \, a b^{3} c d^{2} e - b^{4} c d^{2} e\right )} x^{2} - 2 \, {\left (b^{4} c^{2} d e - {\left (3 \, c^{2} d e + d e\right )} a b^{3}\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 )^{3} + 3 \, {\left (a^{2} b^{2} d^{4} e x^{4} + 4 \, a^{2} b^{2} c d^{3} e x^{3} + {\left (6 \, c^{2} d^{2} e + d^{2} e\right )} a^{2} b^{2} x^{2} + 2 \, {\left (2 \, c^{3} d e + c d e\right )} a^{2} b^{2} x + {\left (c^{4} e + c^{2} e\right )} a^{2} b^{2} + {\left (a^{2} b^{2} d^{3} e x^{3} + 3 \, a^{2} b^{2} c d^{2} e x^{2} + {\left (3 \, c^{2} d e + d e\right )} a^{2} b^{2} x + {\left (c^{3} e + c e\right )} a^{2} b^{2}\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}\right )}}{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}\,{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 \[ \int \left (c\,e+d\,e\,x\right )\,{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^4 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.85, size = 1027, normalized size = 5.27 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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