Optimal. Leaf size=382 \[ \frac {2 f^2 \sin \left (a+b (c+d x)^{3/2}\right )}{3 b^2 d^3}-\frac {4 f \sqrt {c+d x} (d e-c f) \cos \left (a+b (c+d x)^{3/2}\right )}{3 b d^3}-\frac {2 e^{i a} f \sqrt {c+d x} (d e-c f) \Gamma \left (\frac {1}{3},-i b (c+d x)^{3/2}\right )}{9 b d^3 \sqrt [3]{-i b (c+d x)^{3/2}}}-\frac {2 e^{-i a} f \sqrt {c+d x} (d e-c f) \Gamma \left (\frac {1}{3},i b (c+d x)^{3/2}\right )}{9 b d^3 \sqrt [3]{i b (c+d x)^{3/2}}}+\frac {i e^{i a} (c+d x) (d e-c f)^2 \Gamma \left (\frac {2}{3},-i b (c+d x)^{3/2}\right )}{3 d^3 \left (-i b (c+d x)^{3/2}\right )^{2/3}}-\frac {i e^{-i a} (c+d x) (d e-c f)^2 \Gamma \left (\frac {2}{3},i b (c+d x)^{3/2}\right )}{3 d^3 \left (i b (c+d x)^{3/2}\right )^{2/3}}-\frac {2 f^2 (c+d x)^{3/2} \cos \left (a+b (c+d x)^{3/2}\right )}{3 b d^3} \]
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Rubi [A] time = 0.31, antiderivative size = 382, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {3433, 3389, 2218, 3385, 3356, 2208, 3379, 3296, 2637} \[ -\frac {2 e^{i a} f \sqrt {c+d x} (d e-c f) \text {Gamma}\left (\frac {1}{3},-i b (c+d x)^{3/2}\right )}{9 b d^3 \sqrt [3]{-i b (c+d x)^{3/2}}}-\frac {2 e^{-i a} f \sqrt {c+d x} (d e-c f) \text {Gamma}\left (\frac {1}{3},i b (c+d x)^{3/2}\right )}{9 b d^3 \sqrt [3]{i b (c+d x)^{3/2}}}+\frac {i e^{i a} (c+d x) (d e-c f)^2 \text {Gamma}\left (\frac {2}{3},-i b (c+d x)^{3/2}\right )}{3 d^3 \left (-i b (c+d x)^{3/2}\right )^{2/3}}-\frac {i e^{-i a} (c+d x) (d e-c f)^2 \text {Gamma}\left (\frac {2}{3},i b (c+d x)^{3/2}\right )}{3 d^3 \left (i b (c+d x)^{3/2}\right )^{2/3}}+\frac {2 f^2 \sin \left (a+b (c+d x)^{3/2}\right )}{3 b^2 d^3}-\frac {4 f \sqrt {c+d x} (d e-c f) \cos \left (a+b (c+d x)^{3/2}\right )}{3 b d^3}-\frac {2 f^2 (c+d x)^{3/2} \cos \left (a+b (c+d x)^{3/2}\right )}{3 b d^3} \]
Antiderivative was successfully verified.
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Rule 2208
Rule 2218
Rule 2637
Rule 3296
Rule 3356
Rule 3379
Rule 3385
Rule 3389
Rule 3433
Rubi steps
\begin {align*} \int (e+f x)^2 \sin \left (a+b (c+d x)^{3/2}\right ) \, dx &=\frac {2 \operatorname {Subst}\left (\int \left ((d e-c f)^2 x \sin \left (a+b x^3\right )-2 f (-d e+c f) x^3 \sin \left (a+b x^3\right )+f^2 x^5 \sin \left (a+b x^3\right )\right ) \, dx,x,\sqrt {c+d x}\right )}{d^3}\\ &=\frac {\left (2 f^2\right ) \operatorname {Subst}\left (\int x^5 \sin \left (a+b x^3\right ) \, dx,x,\sqrt {c+d x}\right )}{d^3}+\frac {(4 f (d e-c f)) \operatorname {Subst}\left (\int x^3 \sin \left (a+b x^3\right ) \, dx,x,\sqrt {c+d x}\right )}{d^3}+\frac {\left (2 (d e-c f)^2\right ) \operatorname {Subst}\left (\int x \sin \left (a+b x^3\right ) \, dx,x,\sqrt {c+d x}\right )}{d^3}\\ &=-\frac {4 f (d e-c f) \sqrt {c+d x} \cos \left (a+b (c+d x)^{3/2}\right )}{3 b d^3}+\frac {\left (2 f^2\right ) \operatorname {Subst}\left (\int x \sin (a+b x) \, dx,x,(c+d x)^{3/2}\right )}{3 d^3}+\frac {(4 f (d e-c f)) \operatorname {Subst}\left (\int \cos \left (a+b x^3\right ) \, dx,x,\sqrt {c+d x}\right )}{3 b d^3}+\frac {\left (i (d e-c f)^2\right ) \operatorname {Subst}\left (\int e^{-i a-i b x^3} x \, dx,x,\sqrt {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} x \, dx,x,\sqrt {c+d x}\right )}{d^3}\\ &=-\frac {4 f (d e-c f) \sqrt {c+d x} \cos \left (a+b (c+d x)^{3/2}\right )}{3 b d^3}-\frac {2 f^2 (c+d x)^{3/2} \cos \left (a+b (c+d x)^{3/2}\right )}{3 b d^3}+\frac {i e^{i a} (d e-c f)^2 (c+d x) \Gamma \left (\frac {2}{3},-i b (c+d x)^{3/2}\right )}{3 d^3 \left (-i b (c+d x)^{3/2}\right )^{2/3}}-\frac {i e^{-i a} (d e-c f)^2 (c+d x) \Gamma \left (\frac {2}{3},i b (c+d x)^{3/2}\right )}{3 d^3 \left (i b (c+d x)^{3/2}\right )^{2/3}}+\frac {\left (2 f^2\right ) \operatorname {Subst}\left (\int \cos (a+b x) \, dx,x,(c+d x)^{3/2}\right )}{3 b d^3}+\frac {(2 f (d e-c f)) \operatorname {Subst}\left (\int e^{-i a-i b x^3} \, dx,x,\sqrt {c+d x}\right )}{3 b d^3}+\frac {(2 f (d e-c f)) \operatorname {Subst}\left (\int e^{i a+i b x^3} \, dx,x,\sqrt {c+d x}\right )}{3 b d^3}\\ &=-\frac {4 f (d e-c f) \sqrt {c+d x} \cos \left (a+b (c+d x)^{3/2}\right )}{3 b d^3}-\frac {2 f^2 (c+d x)^{3/2} \cos \left (a+b (c+d x)^{3/2}\right )}{3 b d^3}-\frac {2 e^{i a} f (d e-c f) \sqrt {c+d x} \Gamma \left (\frac {1}{3},-i b (c+d x)^{3/2}\right )}{9 b d^3 \sqrt [3]{-i b (c+d x)^{3/2}}}-\frac {2 e^{-i a} f (d e-c f) \sqrt {c+d x} \Gamma \left (\frac {1}{3},i b (c+d x)^{3/2}\right )}{9 b d^3 \sqrt [3]{i b (c+d x)^{3/2}}}+\frac {i e^{i a} (d e-c f)^2 (c+d x) \Gamma \left (\frac {2}{3},-i b (c+d x)^{3/2}\right )}{3 d^3 \left (-i b (c+d x)^{3/2}\right )^{2/3}}-\frac {i e^{-i a} (d e-c f)^2 (c+d x) \Gamma \left (\frac {2}{3},i b (c+d x)^{3/2}\right )}{3 d^3 \left (i b (c+d x)^{3/2}\right )^{2/3}}+\frac {2 f^2 \sin \left (a+b (c+d x)^{3/2}\right )}{3 b^2 d^3}\\ \end {align*}
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Mathematica [A] time = 3.25, size = 419, normalized size = 1.10 \[ -\frac {i \left ((\cos (a)+i \sin (a)) \left (\frac {i f^2 \sin \left (b (c+d x)^{3/2}\right )}{b^2}+\frac {f^2 \cos \left (b (c+d x)^{3/2}\right )}{b^2}-\frac {2 f (c+d x)^2 (d e-c f) \Gamma \left (\frac {4}{3},-i b (c+d x)^{3/2}\right )}{\left (-i b (c+d x)^{3/2}\right )^{4/3}}-\frac {(c+d x) (d e-c f)^2 \Gamma \left (\frac {2}{3},-i b (c+d x)^{3/2}\right )}{\left (-i b (c+d x)^{3/2}\right )^{2/3}}+\frac {f^2 (c+d x)^{3/2} \left (\sin \left (b (c+d x)^{3/2}\right )-i \cos \left (b (c+d x)^{3/2}\right )\right )}{b}\right )-(\cos (a)-i \sin (a)) \left (-\frac {i f^2 \sin \left (b (c+d x)^{3/2}\right )}{b^2}+\frac {f^2 \cos \left (b (c+d x)^{3/2}\right )}{b^2}-\frac {2 f (c+d x)^2 (d e-c f) \Gamma \left (\frac {4}{3},i b (c+d x)^{3/2}\right )}{\left (i b (c+d x)^{3/2}\right )^{4/3}}-\frac {(c+d x) (d e-c f)^2 \Gamma \left (\frac {2}{3},i b (c+d x)^{3/2}\right )}{\left (i b (c+d x)^{3/2}\right )^{2/3}}+\frac {f^2 (c+d x)^{3/2} \left (\sin \left (b (c+d x)^{3/2}\right )+i \cos \left (b (c+d x)^{3/2}\right )\right )}{b}\right )\right )}{3 d^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.69, size = 276, normalized size = 0.72 \[ \frac {{\left (2 i \, d e f - 2 i \, c f^{2}\right )} \left (i \, b\right )^{\frac {2}{3}} e^{\left (-i \, a\right )} \Gamma \left (\frac {1}{3}, {\left (i \, b d x + i \, b c\right )} \sqrt {d x + c}\right ) + {\left (-2 i \, d e f + 2 i \, c f^{2}\right )} \left (-i \, b\right )^{\frac {2}{3}} e^{\left (i \, a\right )} \Gamma \left (\frac {1}{3}, {\left (-i \, b d x - i \, b c\right )} \sqrt {d x + c}\right ) - 3 \, {\left (b d^{2} e^{2} - 2 \, b c d e f + b c^{2} f^{2}\right )} \left (i \, b\right )^{\frac {1}{3}} e^{\left (-i \, a\right )} \Gamma \left (\frac {2}{3}, {\left (i \, b d x + i \, b c\right )} \sqrt {d x + c}\right ) - 3 \, {\left (b d^{2} e^{2} - 2 \, b c d e f + b c^{2} f^{2}\right )} \left (-i \, b\right )^{\frac {1}{3}} e^{\left (i \, a\right )} \Gamma \left (\frac {2}{3}, {\left (-i \, b d x - i \, b c\right )} \sqrt {d x + c}\right ) + 6 \, f^{2} \sin \left ({\left (b d x + b c\right )} \sqrt {d x + c} + a\right ) - 6 \, {\left (b d f^{2} x + 2 \, b d e f - b c f^{2}\right )} \sqrt {d x + c} \cos \left ({\left (b d x + b c\right )} \sqrt {d x + c} + a\right )}{9 \, b^{2} d^{3}} \]
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 \[ \int {\left (f x + e\right )}^{2} \sin \left ({\left (d x + c\right )}^{\frac {3}{2}} b + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.07, size = 0, normalized size = 0.00 \[ \int \left (f x +e \right )^{2} \sin \left (a +b \left (d x +c \right )^{\frac {3}{2}}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.20, size = 694, normalized size = 1.82 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \sin \left (a+b\,{\left (c+d\,x\right )}^{3/2}\right )\,{\left (e+f\,x\right )}^2 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (e + f x\right )^{2} \sin {\left (a + b c \sqrt {c + d x} + b d x \sqrt {c + d x} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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