Optimal. Leaf size=67 \[ \frac {a (c+d x)^3}{3 d}-\frac {2 b d (c+d x) \sinh (e+f x)}{f^2}+\frac {b (c+d x)^2 \cosh (e+f x)}{f}+\frac {2 b d^2 \cosh (e+f x)}{f^3} \]
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Rubi [A] time = 0.10, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3317, 3296, 2638} \[ \frac {a (c+d x)^3}{3 d}-\frac {2 b d (c+d x) \sinh (e+f x)}{f^2}+\frac {b (c+d x)^2 \cosh (e+f x)}{f}+\frac {2 b d^2 \cosh (e+f x)}{f^3} \]
Antiderivative was successfully verified.
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Rule 2638
Rule 3296
Rule 3317
Rubi steps
\begin {align*} \int (c+d x)^2 (a+b \sinh (e+f x)) \, dx &=\int \left (a (c+d x)^2+b (c+d x)^2 \sinh (e+f x)\right ) \, dx\\ &=\frac {a (c+d x)^3}{3 d}+b \int (c+d x)^2 \sinh (e+f x) \, dx\\ &=\frac {a (c+d x)^3}{3 d}+\frac {b (c+d x)^2 \cosh (e+f x)}{f}-\frac {(2 b d) \int (c+d x) \cosh (e+f x) \, dx}{f}\\ &=\frac {a (c+d x)^3}{3 d}+\frac {b (c+d x)^2 \cosh (e+f x)}{f}-\frac {2 b d (c+d x) \sinh (e+f x)}{f^2}+\frac {\left (2 b d^2\right ) \int \sinh (e+f x) \, dx}{f^2}\\ &=\frac {a (c+d x)^3}{3 d}+\frac {2 b d^2 \cosh (e+f x)}{f^3}+\frac {b (c+d x)^2 \cosh (e+f x)}{f}-\frac {2 b d (c+d x) \sinh (e+f x)}{f^2}\\ \end {align*}
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Mathematica [A] time = 0.32, size = 83, normalized size = 1.24 \[ \frac {1}{3} a x \left (3 c^2+3 c d x+d^2 x^2\right )+\frac {b \left (c^2 f^2+2 c d f^2 x+d^2 \left (f^2 x^2+2\right )\right ) \cosh (e+f x)}{f^3}-\frac {2 b d (c+d x) \sinh (e+f x)}{f^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.06, size = 102, normalized size = 1.52 \[ \frac {a d^{2} f^{3} x^{3} + 3 \, a c d f^{3} x^{2} + 3 \, a c^{2} f^{3} x + 3 \, {\left (b d^{2} f^{2} x^{2} + 2 \, b c d f^{2} x + b c^{2} f^{2} + 2 \, b d^{2}\right )} \cosh \left (f x + e\right ) - 6 \, {\left (b d^{2} f x + b c d f\right )} \sinh \left (f x + e\right )}{3 \, f^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.22, size = 148, normalized size = 2.21 \[ \frac {1}{3} \, a d^{2} x^{3} + a c d x^{2} + a c^{2} x + \frac {{\left (b d^{2} f^{2} x^{2} + 2 \, b c d f^{2} x + b c^{2} f^{2} - 2 \, b d^{2} f x - 2 \, b c d f + 2 \, b d^{2}\right )} e^{\left (f x + e\right )}}{2 \, f^{3}} + \frac {{\left (b d^{2} f^{2} x^{2} + 2 \, b c d f^{2} x + b c^{2} f^{2} + 2 \, b d^{2} f x + 2 \, b c d f + 2 \, b d^{2}\right )} e^{\left (-f x - e\right )}}{2 \, f^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 240, normalized size = 3.58 \[ \frac {\frac {d^{2} a \left (f x +e \right )^{3}}{3 f^{2}}+\frac {d^{2} b \left (\left (f x +e \right )^{2} \cosh \left (f x +e \right )-2 \left (f x +e \right ) \sinh \left (f x +e \right )+2 \cosh \left (f x +e \right )\right )}{f^{2}}-\frac {d^{2} e a \left (f x +e \right )^{2}}{f^{2}}-\frac {2 d^{2} e b \left (\left (f x +e \right ) \cosh \left (f x +e \right )-\sinh \left (f x +e \right )\right )}{f^{2}}+\frac {d^{2} e^{2} a \left (f x +e \right )}{f^{2}}+\frac {d^{2} e^{2} b \cosh \left (f x +e \right )}{f^{2}}+\frac {d c a \left (f x +e \right )^{2}}{f}+\frac {2 c d b \left (\left (f x +e \right ) \cosh \left (f x +e \right )-\sinh \left (f x +e \right )\right )}{f}-\frac {2 d e c a \left (f x +e \right )}{f}-\frac {2 c d e b \cosh \left (f x +e \right )}{f}+c^{2} a \left (f x +e \right )+b \,c^{2} \cosh \left (f x +e \right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.33, size = 139, normalized size = 2.07 \[ \frac {1}{3} \, a d^{2} x^{3} + a c d x^{2} + a c^{2} x + b c d {\left (\frac {{\left (f x e^{e} - e^{e}\right )} e^{\left (f x\right )}}{f^{2}} + \frac {{\left (f x + 1\right )} e^{\left (-f x - e\right )}}{f^{2}}\right )} + \frac {1}{2} \, b d^{2} {\left (\frac {{\left (f^{2} x^{2} e^{e} - 2 \, f x e^{e} + 2 \, e^{e}\right )} e^{\left (f x\right )}}{f^{3}} + \frac {{\left (f^{2} x^{2} + 2 \, f x + 2\right )} e^{\left (-f x - e\right )}}{f^{3}}\right )} + \frac {b c^{2} \cosh \left (f x + e\right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.11, size = 110, normalized size = 1.64 \[ \frac {a\,d^2\,x^3}{3}+\frac {\mathrm {cosh}\left (e+f\,x\right )\,\left (b\,c^2\,f^2+2\,b\,d^2\right )}{f^3}+a\,c^2\,x+a\,c\,d\,x^2-\frac {2\,b\,d^2\,x\,\mathrm {sinh}\left (e+f\,x\right )}{f^2}+\frac {b\,d^2\,x^2\,\mathrm {cosh}\left (e+f\,x\right )}{f}-\frac {2\,b\,c\,d\,\mathrm {sinh}\left (e+f\,x\right )}{f^2}+\frac {2\,b\,c\,d\,x\,\mathrm {cosh}\left (e+f\,x\right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.62, size = 151, normalized size = 2.25 \[ \begin {cases} a c^{2} x + a c d x^{2} + \frac {a d^{2} x^{3}}{3} + \frac {b c^{2} \cosh {\left (e + f x \right )}}{f} + \frac {2 b c d x \cosh {\left (e + f x \right )}}{f} - \frac {2 b c d \sinh {\left (e + f x \right )}}{f^{2}} + \frac {b d^{2} x^{2} \cosh {\left (e + f x \right )}}{f} - \frac {2 b d^{2} x \sinh {\left (e + f x \right )}}{f^{2}} + \frac {2 b d^{2} \cosh {\left (e + f x \right )}}{f^{3}} & \text {for}\: f \neq 0 \\\left (a + b \sinh {\relax (e )}\right ) \left (c^{2} x + c d x^{2} + \frac {d^{2} x^{3}}{3}\right ) & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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