Optimal. Leaf size=176 \[ -\frac {7 b d (d+e x)^2 \sqrt {1+c^2 x^2}}{48 c}-\frac {b (d+e x)^3 \sqrt {1+c^2 x^2}}{16 c}-\frac {b \left (4 d \left (19 c^2 d^2-16 e^2\right )+e \left (26 c^2 d^2-9 e^2\right ) x\right ) \sqrt {1+c^2 x^2}}{96 c^3}-\frac {b \left (8 c^4 d^4-24 c^2 d^2 e^2+3 e^4\right ) \sinh ^{-1}(c x)}{32 c^4 e}+\frac {(d+e x)^4 \left (a+b \sinh ^{-1}(c x)\right )}{4 e} \]
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Rubi [A]
time = 0.12, antiderivative size = 176, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {5828, 757, 847,
794, 221} \begin {gather*} \frac {(d+e x)^4 \left (a+b \sinh ^{-1}(c x)\right )}{4 e}-\frac {b \sqrt {c^2 x^2+1} (d+e x)^3}{16 c}-\frac {7 b d \sqrt {c^2 x^2+1} (d+e x)^2}{48 c}-\frac {b \left (8 c^4 d^4-24 c^2 d^2 e^2+3 e^4\right ) \sinh ^{-1}(c x)}{32 c^4 e}-\frac {b \sqrt {c^2 x^2+1} \left (e x \left (26 c^2 d^2-9 e^2\right )+4 d \left (19 c^2 d^2-16 e^2\right )\right )}{96 c^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 221
Rule 757
Rule 794
Rule 847
Rule 5828
Rubi steps
\begin {align*} \int (d+e x)^3 \left (a+b \sinh ^{-1}(c x)\right ) \, dx &=\frac {(d+e x)^4 \left (a+b \sinh ^{-1}(c x)\right )}{4 e}-\frac {(b c) \int \frac {(d+e x)^4}{\sqrt {1+c^2 x^2}} \, dx}{4 e}\\ &=-\frac {b (d+e x)^3 \sqrt {1+c^2 x^2}}{16 c}+\frac {(d+e x)^4 \left (a+b \sinh ^{-1}(c x)\right )}{4 e}-\frac {b \int \frac {(d+e x)^2 \left (4 c^2 d^2-3 e^2+7 c^2 d e x\right )}{\sqrt {1+c^2 x^2}} \, dx}{16 c e}\\ &=-\frac {7 b d (d+e x)^2 \sqrt {1+c^2 x^2}}{48 c}-\frac {b (d+e x)^3 \sqrt {1+c^2 x^2}}{16 c}+\frac {(d+e x)^4 \left (a+b \sinh ^{-1}(c x)\right )}{4 e}-\frac {b \int \frac {(d+e x) \left (c^2 d \left (12 c^2 d^2-23 e^2\right )+c^2 e \left (26 c^2 d^2-9 e^2\right ) x\right )}{\sqrt {1+c^2 x^2}} \, dx}{48 c^3 e}\\ &=-\frac {7 b d (d+e x)^2 \sqrt {1+c^2 x^2}}{48 c}-\frac {b (d+e x)^3 \sqrt {1+c^2 x^2}}{16 c}-\frac {b \left (4 d \left (19 c^2 d^2-16 e^2\right )+e \left (26 c^2 d^2-9 e^2\right ) x\right ) \sqrt {1+c^2 x^2}}{96 c^3}+\frac {(d+e x)^4 \left (a+b \sinh ^{-1}(c x)\right )}{4 e}-\frac {\left (b \left (8 c^4 d^4-24 c^2 d^2 e^2+3 e^4\right )\right ) \int \frac {1}{\sqrt {1+c^2 x^2}} \, dx}{32 c^3 e}\\ &=-\frac {7 b d (d+e x)^2 \sqrt {1+c^2 x^2}}{48 c}-\frac {b (d+e x)^3 \sqrt {1+c^2 x^2}}{16 c}-\frac {b \left (4 d \left (19 c^2 d^2-16 e^2\right )+e \left (26 c^2 d^2-9 e^2\right ) x\right ) \sqrt {1+c^2 x^2}}{96 c^3}-\frac {b \left (8 c^4 d^4-24 c^2 d^2 e^2+3 e^4\right ) \sinh ^{-1}(c x)}{32 c^4 e}+\frac {(d+e x)^4 \left (a+b \sinh ^{-1}(c x)\right )}{4 e}\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 166, normalized size = 0.94 \begin {gather*} \frac {24 a c^4 x \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right )-b c \sqrt {1+c^2 x^2} \left (-e^2 (64 d+9 e x)+c^2 \left (96 d^3+72 d^2 e x+32 d e^2 x^2+6 e^3 x^3\right )\right )+3 b \left (24 c^2 d^2 e-3 e^3+8 c^4 x \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right )\right ) \sinh ^{-1}(c x)}{96 c^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.68, size = 259, normalized size = 1.47
method | result | size |
derivativedivides | \(\frac {\frac {\left (c e x +c d \right )^{4} a}{4 c^{3} e}+\frac {b \left (\frac {\arcsinh \left (c x \right ) c^{4} d^{4}}{4 e}+\arcsinh \left (c x \right ) c^{4} d^{3} x +\frac {3 e \arcsinh \left (c x \right ) c^{4} d^{2} x^{2}}{2}+e^{2} \arcsinh \left (c x \right ) c^{4} d \,x^{3}+\frac {e^{3} \arcsinh \left (c x \right ) c^{4} x^{4}}{4}-\frac {c^{4} d^{4} \arcsinh \left (c x \right )+4 d^{3} c^{3} e \sqrt {c^{2} x^{2}+1}+6 d^{2} c^{2} e^{2} \left (\frac {\sqrt {c^{2} x^{2}+1}\, c x}{2}-\frac {\arcsinh \left (c x \right )}{2}\right )+4 d c \,e^{3} \left (\frac {c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {c^{2} x^{2}+1}}{3}\right )+e^{4} \left (\frac {\sqrt {c^{2} x^{2}+1}\, x^{3} c^{3}}{4}-\frac {3 \sqrt {c^{2} x^{2}+1}\, c x}{8}+\frac {3 \arcsinh \left (c x \right )}{8}\right )}{4 e}\right )}{c^{3}}}{c}\) | \(259\) |
default | \(\frac {\frac {\left (c e x +c d \right )^{4} a}{4 c^{3} e}+\frac {b \left (\frac {\arcsinh \left (c x \right ) c^{4} d^{4}}{4 e}+\arcsinh \left (c x \right ) c^{4} d^{3} x +\frac {3 e \arcsinh \left (c x \right ) c^{4} d^{2} x^{2}}{2}+e^{2} \arcsinh \left (c x \right ) c^{4} d \,x^{3}+\frac {e^{3} \arcsinh \left (c x \right ) c^{4} x^{4}}{4}-\frac {c^{4} d^{4} \arcsinh \left (c x \right )+4 d^{3} c^{3} e \sqrt {c^{2} x^{2}+1}+6 d^{2} c^{2} e^{2} \left (\frac {\sqrt {c^{2} x^{2}+1}\, c x}{2}-\frac {\arcsinh \left (c x \right )}{2}\right )+4 d c \,e^{3} \left (\frac {c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {c^{2} x^{2}+1}}{3}\right )+e^{4} \left (\frac {\sqrt {c^{2} x^{2}+1}\, x^{3} c^{3}}{4}-\frac {3 \sqrt {c^{2} x^{2}+1}\, c x}{8}+\frac {3 \arcsinh \left (c x \right )}{8}\right )}{4 e}\right )}{c^{3}}}{c}\) | \(259\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 228, normalized size = 1.30 \begin {gather*} \frac {1}{4} \, a x^{4} e^{3} + a d x^{3} e^{2} + \frac {3}{2} \, a d^{2} x^{2} e + a d^{3} x + \frac {3}{4} \, {\left (2 \, x^{2} \operatorname {arsinh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} + 1} x}{c^{2}} - \frac {\operatorname {arsinh}\left (c x\right )}{c^{3}}\right )}\right )} b d^{2} e + \frac {{\left (c x \operatorname {arsinh}\left (c x\right ) - \sqrt {c^{2} x^{2} + 1}\right )} b d^{3}}{c} + \frac {1}{3} \, {\left (3 \, x^{3} \operatorname {arsinh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} + 1} x^{2}}{c^{2}} - \frac {2 \, \sqrt {c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} b d e^{2} + \frac {1}{32} \, {\left (8 \, x^{4} \operatorname {arsinh}\left (c x\right ) - {\left (\frac {2 \, \sqrt {c^{2} x^{2} + 1} x^{3}}{c^{2}} - \frac {3 \, \sqrt {c^{2} x^{2} + 1} x}{c^{4}} + \frac {3 \, \operatorname {arsinh}\left (c x\right )}{c^{5}}\right )} c\right )} b e^{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 525 vs.
\(2 (158) = 316\).
time = 0.38, size = 525, normalized size = 2.98 \begin {gather*} \frac {24 \, a c^{4} x^{4} \cosh \left (1\right )^{3} + 24 \, a c^{4} x^{4} \sinh \left (1\right )^{3} + 96 \, a c^{4} d x^{3} \cosh \left (1\right )^{2} + 144 \, a c^{4} d^{2} x^{2} \cosh \left (1\right ) + 96 \, a c^{4} d^{3} x + 24 \, {\left (3 \, a c^{4} x^{4} \cosh \left (1\right ) + 4 \, a c^{4} d x^{3}\right )} \sinh \left (1\right )^{2} + 3 \, {\left (32 \, b c^{4} d x^{3} \cosh \left (1\right )^{2} + 32 \, b c^{4} d^{3} x + {\left (8 \, b c^{4} x^{4} - 3 \, b\right )} \cosh \left (1\right )^{3} + {\left (8 \, b c^{4} x^{4} - 3 \, b\right )} \sinh \left (1\right )^{3} + {\left (32 \, b c^{4} d x^{3} + 3 \, {\left (8 \, b c^{4} x^{4} - 3 \, b\right )} \cosh \left (1\right )\right )} \sinh \left (1\right )^{2} + 24 \, {\left (2 \, b c^{4} d^{2} x^{2} + b c^{2} d^{2}\right )} \cosh \left (1\right ) + {\left (64 \, b c^{4} d x^{3} \cosh \left (1\right ) + 48 \, b c^{4} d^{2} x^{2} + 24 \, b c^{2} d^{2} + 3 \, {\left (8 \, b c^{4} x^{4} - 3 \, b\right )} \cosh \left (1\right )^{2}\right )} \sinh \left (1\right )\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + 24 \, {\left (3 \, a c^{4} x^{4} \cosh \left (1\right )^{2} + 8 \, a c^{4} d x^{3} \cosh \left (1\right ) + 6 \, a c^{4} d^{2} x^{2}\right )} \sinh \left (1\right ) - {\left (72 \, b c^{3} d^{2} x \cosh \left (1\right ) + 96 \, b c^{3} d^{3} + 3 \, {\left (2 \, b c^{3} x^{3} - 3 \, b c x\right )} \cosh \left (1\right )^{3} + 3 \, {\left (2 \, b c^{3} x^{3} - 3 \, b c x\right )} \sinh \left (1\right )^{3} + 32 \, {\left (b c^{3} d x^{2} - 2 \, b c d\right )} \cosh \left (1\right )^{2} + {\left (32 \, b c^{3} d x^{2} - 64 \, b c d + 9 \, {\left (2 \, b c^{3} x^{3} - 3 \, b c x\right )} \cosh \left (1\right )\right )} \sinh \left (1\right )^{2} + {\left (72 \, b c^{3} d^{2} x + 9 \, {\left (2 \, b c^{3} x^{3} - 3 \, b c x\right )} \cosh \left (1\right )^{2} + 64 \, {\left (b c^{3} d x^{2} - 2 \, b c d\right )} \cosh \left (1\right )\right )} \sinh \left (1\right )\right )} \sqrt {c^{2} x^{2} + 1}}{96 \, c^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.28, size = 316, normalized size = 1.80 \begin {gather*} \begin {cases} a d^{3} x + \frac {3 a d^{2} e x^{2}}{2} + a d e^{2} x^{3} + \frac {a e^{3} x^{4}}{4} + b d^{3} x \operatorname {asinh}{\left (c x \right )} + \frac {3 b d^{2} e x^{2} \operatorname {asinh}{\left (c x \right )}}{2} + b d e^{2} x^{3} \operatorname {asinh}{\left (c x \right )} + \frac {b e^{3} x^{4} \operatorname {asinh}{\left (c x \right )}}{4} - \frac {b d^{3} \sqrt {c^{2} x^{2} + 1}}{c} - \frac {3 b d^{2} e x \sqrt {c^{2} x^{2} + 1}}{4 c} - \frac {b d e^{2} x^{2} \sqrt {c^{2} x^{2} + 1}}{3 c} - \frac {b e^{3} x^{3} \sqrt {c^{2} x^{2} + 1}}{16 c} + \frac {3 b d^{2} e \operatorname {asinh}{\left (c x \right )}}{4 c^{2}} + \frac {2 b d e^{2} \sqrt {c^{2} x^{2} + 1}}{3 c^{3}} + \frac {3 b e^{3} x \sqrt {c^{2} x^{2} + 1}}{32 c^{3}} - \frac {3 b e^{3} \operatorname {asinh}{\left (c x \right )}}{32 c^{4}} & \text {for}\: c \neq 0 \\a \left (d^{3} x + \frac {3 d^{2} e x^{2}}{2} + d e^{2} x^{3} + \frac {e^{3} x^{4}}{4}\right ) & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
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 \begin {gather*} \int \left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (d+e\,x\right )}^3 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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