Optimal. Leaf size=280 \[ \frac{i e^{i a} f (c+d x)^2 (d e-c f) \text{Gamma}\left (\frac{2}{3},-i b (c+d x)^3\right )}{3 d^3 \left (-i b (c+d x)^3\right )^{2/3}}-\frac{i e^{-i a} f (c+d x)^2 (d e-c f) \text{Gamma}\left (\frac{2}{3},i b (c+d x)^3\right )}{3 d^3 \left (i b (c+d x)^3\right )^{2/3}}+\frac{i e^{i a} (c+d x) (d e-c f)^2 \text{Gamma}\left (\frac{1}{3},-i b (c+d x)^3\right )}{6 d^3 \sqrt [3]{-i b (c+d x)^3}}-\frac{i e^{-i a} (c+d x) (d e-c f)^2 \text{Gamma}\left (\frac{1}{3},i b (c+d x)^3\right )}{6 d^3 \sqrt [3]{i b (c+d x)^3}}-\frac{f^2 \cos \left (a+b (c+d x)^3\right )}{3 b d^3} \]
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Rubi [A] time = 0.276133, antiderivative size = 280, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35, Rules used = {3433, 3355, 2208, 3389, 2218, 3379, 2638} \[ \frac{i e^{i a} f (c+d x)^2 (d e-c f) \text{Gamma}\left (\frac{2}{3},-i b (c+d x)^3\right )}{3 d^3 \left (-i b (c+d x)^3\right )^{2/3}}-\frac{i e^{-i a} f (c+d x)^2 (d e-c f) \text{Gamma}\left (\frac{2}{3},i b (c+d x)^3\right )}{3 d^3 \left (i b (c+d x)^3\right )^{2/3}}+\frac{i e^{i a} (c+d x) (d e-c f)^2 \text{Gamma}\left (\frac{1}{3},-i b (c+d x)^3\right )}{6 d^3 \sqrt [3]{-i b (c+d x)^3}}-\frac{i e^{-i a} (c+d x) (d e-c f)^2 \text{Gamma}\left (\frac{1}{3},i b (c+d x)^3\right )}{6 d^3 \sqrt [3]{i b (c+d x)^3}}-\frac{f^2 \cos \left (a+b (c+d x)^3\right )}{3 b d^3} \]
Antiderivative was successfully verified.
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Rule 3433
Rule 3355
Rule 2208
Rule 3389
Rule 2218
Rule 3379
Rule 2638
Rubi steps
\begin{align*} \int (e+f x)^2 \sin \left (a+b (c+d x)^3\right ) \, dx &=\frac{\operatorname{Subst}\left (\int \left (d^2 e^2 \left (1+\frac{c f (-2 d e+c f)}{d^2 e^2}\right ) \sin \left (a+b x^3\right )+2 d e f \left (1-\frac{c f}{d e}\right ) x \sin \left (a+b x^3\right )+f^2 x^2 \sin \left (a+b x^3\right )\right ) \, dx,x,c+d x\right )}{d^3}\\ &=\frac{f^2 \operatorname{Subst}\left (\int x^2 \sin \left (a+b x^3\right ) \, dx,x,c+d x\right )}{d^3}+\frac{(2 f (d e-c f)) \operatorname{Subst}\left (\int x \sin \left (a+b x^3\right ) \, dx,x,c+d x\right )}{d^3}+\frac{(d e-c f)^2 \operatorname{Subst}\left (\int \sin \left (a+b x^3\right ) \, dx,x,c+d x\right )}{d^3}\\ &=\frac{f^2 \operatorname{Subst}\left (\int \sin (a+b x) \, dx,x,(c+d x)^3\right )}{3 d^3}+\frac{(i f (d e-c f)) \operatorname{Subst}\left (\int e^{-i a-i b x^3} x \, dx,x,c+d x\right )}{d^3}-\frac{(i f (d e-c f)) \operatorname{Subst}\left (\int e^{i a+i b x^3} x \, dx,x,c+d x\right )}{d^3}+\frac{\left (i (d e-c f)^2\right ) \operatorname{Subst}\left (\int e^{-i a-i b x^3} \, dx,x,c+d x\right )}{2 d^3}-\frac{\left (i (d e-c f)^2\right ) \operatorname{Subst}\left (\int e^{i a+i b x^3} \, dx,x,c+d x\right )}{2 d^3}\\ &=-\frac{f^2 \cos \left (a+b (c+d x)^3\right )}{3 b d^3}+\frac{i e^{i a} (d e-c f)^2 (c+d x) \Gamma \left (\frac{1}{3},-i b (c+d x)^3\right )}{6 d^3 \sqrt [3]{-i b (c+d x)^3}}-\frac{i e^{-i a} (d e-c f)^2 (c+d x) \Gamma \left (\frac{1}{3},i b (c+d x)^3\right )}{6 d^3 \sqrt [3]{i b (c+d x)^3}}+\frac{i e^{i a} f (d e-c f) (c+d x)^2 \Gamma \left (\frac{2}{3},-i b (c+d x)^3\right )}{3 d^3 \left (-i b (c+d x)^3\right )^{2/3}}-\frac{i e^{-i a} f (d e-c f) (c+d x)^2 \Gamma \left (\frac{2}{3},i b (c+d x)^3\right )}{3 d^3 \left (i b (c+d x)^3\right )^{2/3}}\\ \end{align*}
Mathematica [F] time = 40.1766, size = 0, normalized size = 0. \[ \int (e+f x)^2 \sin \left (a+b (c+d x)^3\right ) \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.106, size = 0, normalized size = 0. \begin{align*} \int \left ( fx+e \right ) ^{2}\sin \left ( a+ \left ( dx+c \right ) ^{3}b \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (f x + e\right )}^{2} \sin \left ({\left (d x + c\right )}^{3} b + a\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.87395, size = 774, normalized size = 2.76 \begin{align*} -\frac{2 \, d^{2} f^{2} \cos \left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right ) + \left (i \, b d^{3}\right )^{\frac{2}{3}}{\left (d^{2} e^{2} - 2 \, c d e f + c^{2} f^{2}\right )} e^{\left (-i \, a\right )} \Gamma \left (\frac{1}{3}, i \, b d^{3} x^{3} + 3 i \, b c d^{2} x^{2} + 3 i \, b c^{2} d x + i \, b c^{3}\right ) + \left (-i \, b d^{3}\right )^{\frac{2}{3}}{\left (d^{2} e^{2} - 2 \, c d e f + c^{2} f^{2}\right )} e^{\left (i \, a\right )} \Gamma \left (\frac{1}{3}, -i \, b d^{3} x^{3} - 3 i \, b c d^{2} x^{2} - 3 i \, b c^{2} d x - i \, b c^{3}\right ) + 2 \, \left (i \, b d^{3}\right )^{\frac{1}{3}}{\left (d^{2} e f - c d f^{2}\right )} e^{\left (-i \, a\right )} \Gamma \left (\frac{2}{3}, i \, b d^{3} x^{3} + 3 i \, b c d^{2} x^{2} + 3 i \, b c^{2} d x + i \, b c^{3}\right ) + 2 \, \left (-i \, b d^{3}\right )^{\frac{1}{3}}{\left (d^{2} e f - c d f^{2}\right )} e^{\left (i \, a\right )} \Gamma \left (\frac{2}{3}, -i \, b d^{3} x^{3} - 3 i \, b c d^{2} x^{2} - 3 i \, b c^{2} d x - i \, b c^{3}\right )}{6 \, b d^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (e + f x\right )^{2} \sin{\left (a + b c^{3} + 3 b c^{2} d x + 3 b c d^{2} x^{2} + b d^{3} x^{3} \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (f x + e\right )}^{2} \sin \left ({\left (d x + c\right )}^{3} b + a\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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