Optimal. Leaf size=291 \[ \frac {i e^{i a} (c+d x) (d e-c f) \Gamma \left (\frac {2}{3},-i b (c+d x)^{3/2}\right )}{3 d^2 \left (-i b (c+d x)^{3/2}\right )^{2/3}}-\frac {i e^{-i a} (c+d x) (d e-c f) \Gamma \left (\frac {2}{3},i b (c+d x)^{3/2}\right )}{3 d^2 \left (i b (c+d x)^{3/2}\right )^{2/3}}-\frac {2 f \sqrt {c+d x} \cos \left (a+b (c+d x)^{3/2}\right )}{3 b d^2}-\frac {e^{i a} f \sqrt {c+d x} \Gamma \left (\frac {1}{3},-i b (c+d x)^{3/2}\right )}{9 b d^2 \sqrt [3]{-i b (c+d x)^{3/2}}}-\frac {e^{-i a} f \sqrt {c+d x} \Gamma \left (\frac {1}{3},i b (c+d x)^{3/2}\right )}{9 b d^2 \sqrt [3]{i b (c+d x)^{3/2}}} \]
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Rubi [A] time = 0.20, antiderivative size = 291, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3433, 3389, 2218, 3385, 3356, 2208} \[ \frac {i e^{i a} (c+d x) (d e-c f) \text {Gamma}\left (\frac {2}{3},-i b (c+d x)^{3/2}\right )}{3 d^2 \left (-i b (c+d x)^{3/2}\right )^{2/3}}-\frac {i e^{-i a} (c+d x) (d e-c f) \text {Gamma}\left (\frac {2}{3},i b (c+d x)^{3/2}\right )}{3 d^2 \left (i b (c+d x)^{3/2}\right )^{2/3}}-\frac {e^{i a} f \sqrt {c+d x} \text {Gamma}\left (\frac {1}{3},-i b (c+d x)^{3/2}\right )}{9 b d^2 \sqrt [3]{-i b (c+d x)^{3/2}}}-\frac {e^{-i a} f \sqrt {c+d x} \text {Gamma}\left (\frac {1}{3},i b (c+d x)^{3/2}\right )}{9 b d^2 \sqrt [3]{i b (c+d x)^{3/2}}}-\frac {2 f \sqrt {c+d x} \cos \left (a+b (c+d x)^{3/2}\right )}{3 b d^2} \]
Antiderivative was successfully verified.
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Rule 2208
Rule 2218
Rule 3356
Rule 3385
Rule 3389
Rule 3433
Rubi steps
\begin {align*} \int (e+f x) \sin \left (a+b (c+d x)^{3/2}\right ) \, dx &=\frac {2 \operatorname {Subst}\left (\int \left ((d e-c f) x \sin \left (a+b x^3\right )+f x^3 \sin \left (a+b x^3\right )\right ) \, dx,x,\sqrt {c+d x}\right )}{d^2}\\ &=\frac {(2 f) \operatorname {Subst}\left (\int x^3 \sin \left (a+b x^3\right ) \, dx,x,\sqrt {c+d x}\right )}{d^2}+\frac {(2 (d e-c f)) \operatorname {Subst}\left (\int x \sin \left (a+b x^3\right ) \, dx,x,\sqrt {c+d x}\right )}{d^2}\\ &=-\frac {2 f \sqrt {c+d x} \cos \left (a+b (c+d x)^{3/2}\right )}{3 b d^2}+\frac {(2 f) \operatorname {Subst}\left (\int \cos \left (a+b x^3\right ) \, dx,x,\sqrt {c+d x}\right )}{3 b d^2}+\frac {(i (d e-c f)) \operatorname {Subst}\left (\int e^{-i a-i b x^3} x \, dx,x,\sqrt {c+d x}\right )}{d^2}-\frac {(i (d e-c f)) \operatorname {Subst}\left (\int e^{i a+i b x^3} x \, dx,x,\sqrt {c+d x}\right )}{d^2}\\ &=-\frac {2 f \sqrt {c+d x} \cos \left (a+b (c+d x)^{3/2}\right )}{3 b d^2}+\frac {i e^{i a} (d e-c f) (c+d x) \Gamma \left (\frac {2}{3},-i b (c+d x)^{3/2}\right )}{3 d^2 \left (-i b (c+d x)^{3/2}\right )^{2/3}}-\frac {i e^{-i a} (d e-c f) (c+d x) \Gamma \left (\frac {2}{3},i b (c+d x)^{3/2}\right )}{3 d^2 \left (i b (c+d x)^{3/2}\right )^{2/3}}+\frac {f \operatorname {Subst}\left (\int e^{-i a-i b x^3} \, dx,x,\sqrt {c+d x}\right )}{3 b d^2}+\frac {f \operatorname {Subst}\left (\int e^{i a+i b x^3} \, dx,x,\sqrt {c+d x}\right )}{3 b d^2}\\ &=-\frac {2 f \sqrt {c+d x} \cos \left (a+b (c+d x)^{3/2}\right )}{3 b d^2}-\frac {e^{i a} f \sqrt {c+d x} \Gamma \left (\frac {1}{3},-i b (c+d x)^{3/2}\right )}{9 b d^2 \sqrt [3]{-i b (c+d x)^{3/2}}}-\frac {e^{-i a} f \sqrt {c+d x} \Gamma \left (\frac {1}{3},i b (c+d x)^{3/2}\right )}{9 b d^2 \sqrt [3]{i b (c+d x)^{3/2}}}+\frac {i e^{i a} (d e-c f) (c+d x) \Gamma \left (\frac {2}{3},-i b (c+d x)^{3/2}\right )}{3 d^2 \left (-i b (c+d x)^{3/2}\right )^{2/3}}-\frac {i e^{-i a} (d e-c f) (c+d x) \Gamma \left (\frac {2}{3},i b (c+d x)^{3/2}\right )}{3 d^2 \left (i b (c+d x)^{3/2}\right )^{2/3}}\\ \end {align*}
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Mathematica [B] time = 2.63, size = 705, normalized size = 2.42 \[ \frac {2 f \sin (a) \sqrt {c+d x} \sin \left (b (c+d x)^{3/2}\right )}{3 b d^2}-\frac {2 f \cos (a) \sqrt {c+d x} \cos \left (b (c+d x)^{3/2}\right )}{3 b d^2}+\frac {f \cos (a) \left (-\frac {2 \sqrt {c+d x} \Gamma \left (\frac {1}{3},-i b (c+d x)^{3/2}\right )}{3 \sqrt [3]{-i b (c+d x)^{3/2}}}-\frac {2 \sqrt {c+d x} \Gamma \left (\frac {1}{3},i b (c+d x)^{3/2}\right )}{3 \sqrt [3]{i b (c+d x)^{3/2}}}\right )}{6 b d^2}+\frac {i c f \cos (a) \left (\frac {2 (c+d x) \Gamma \left (\frac {2}{3},i b (c+d x)^{3/2}\right )}{3 \left (i b (c+d x)^{3/2}\right )^{2/3}}-\frac {2 (c+d x) \Gamma \left (\frac {2}{3},-i b (c+d x)^{3/2}\right )}{3 \left (-i b (c+d x)^{3/2}\right )^{2/3}}\right )}{2 d^2}+\frac {i f \sin (a) \left (\frac {2 \sqrt {c+d x} \Gamma \left (\frac {1}{3},i b (c+d x)^{3/2}\right )}{3 \sqrt [3]{i b (c+d x)^{3/2}}}-\frac {2 \sqrt {c+d x} \Gamma \left (\frac {1}{3},-i b (c+d x)^{3/2}\right )}{3 \sqrt [3]{-i b (c+d x)^{3/2}}}\right )}{6 b d^2}-\frac {c f \sin (a) \left (-\frac {2 (c+d x) \Gamma \left (\frac {2}{3},-i b (c+d x)^{3/2}\right )}{3 \left (-i b (c+d x)^{3/2}\right )^{2/3}}-\frac {2 (c+d x) \Gamma \left (\frac {2}{3},i b (c+d x)^{3/2}\right )}{3 \left (i b (c+d x)^{3/2}\right )^{2/3}}\right )}{2 d^2}-\frac {i e \cos (a) \left (\frac {2 (c+d x) \Gamma \left (\frac {2}{3},i b (c+d x)^{3/2}\right )}{3 \left (i b (c+d x)^{3/2}\right )^{2/3}}-\frac {2 (c+d x) \Gamma \left (\frac {2}{3},-i b (c+d x)^{3/2}\right )}{3 \left (-i b (c+d x)^{3/2}\right )^{2/3}}\right )}{2 d}+\frac {e \sin (a) \left (-\frac {2 (c+d x) \Gamma \left (\frac {2}{3},-i b (c+d x)^{3/2}\right )}{3 \left (-i b (c+d x)^{3/2}\right )^{2/3}}-\frac {2 (c+d x) \Gamma \left (\frac {2}{3},i b (c+d x)^{3/2}\right )}{3 \left (i b (c+d x)^{3/2}\right )^{2/3}}\right )}{2 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.90, size = 185, normalized size = 0.64 \[ \frac {i \, \left (i \, b\right )^{\frac {2}{3}} f e^{\left (-i \, a\right )} \Gamma \left (\frac {1}{3}, {\left (i \, b d x + i \, b c\right )} \sqrt {d x + c}\right ) - i \, \left (-i \, b\right )^{\frac {2}{3}} f e^{\left (i \, a\right )} \Gamma \left (\frac {1}{3}, {\left (-i \, b d x - i \, b c\right )} \sqrt {d x + c}\right ) - 6 \, \sqrt {d x + c} b f \cos \left ({\left (b d x + b c\right )} \sqrt {d x + c} + a\right ) - 3 \, {\left (b d e - b c f\right )} \left (i \, b\right )^{\frac {1}{3}} e^{\left (-i \, a\right )} \Gamma \left (\frac {2}{3}, {\left (i \, b d x + i \, b c\right )} \sqrt {d x + c}\right ) - 3 \, {\left (b d e - b c f\right )} \left (-i \, b\right )^{\frac {1}{3}} e^{\left (i \, a\right )} \Gamma \left (\frac {2}{3}, {\left (-i \, b d x - i \, b c\right )} \sqrt {d x + c}\right )}{9 \, b^{2} d^{2}} \]
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 \[ \int {\left (f x + e\right )} \sin \left ({\left (d x + c\right )}^{\frac {3}{2}} b + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.04, size = 0, normalized size = 0.00 \[ \int \left (f x +e \right ) \sin \left (a +b \left (d x +c \right )^{\frac {3}{2}}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.98, size = 375, normalized size = 1.29 \[ -\frac {\frac {3 \, \left ({\left (d x + c\right )}^{\frac {3}{2}} b\right )^{\frac {1}{3}} {\left ({\left ({\left (\sqrt {3} + i\right )} \Gamma \left (\frac {2}{3}, i \, {\left (d x + c\right )}^{\frac {3}{2}} b\right ) + {\left (\sqrt {3} - i\right )} \Gamma \left (\frac {2}{3}, -i \, {\left (d x + c\right )}^{\frac {3}{2}} b\right )\right )} \cos \relax (a) - {\left ({\left (i \, \sqrt {3} - 1\right )} \Gamma \left (\frac {2}{3}, i \, {\left (d x + c\right )}^{\frac {3}{2}} b\right ) + {\left (-i \, \sqrt {3} - 1\right )} \Gamma \left (\frac {2}{3}, -i \, {\left (d x + c\right )}^{\frac {3}{2}} b\right )\right )} \sin \relax (a)\right )} e}{\sqrt {d x + c} b} - \frac {3 \, \left ({\left (d x + c\right )}^{\frac {3}{2}} b\right )^{\frac {1}{3}} {\left ({\left ({\left (\sqrt {3} + i\right )} \Gamma \left (\frac {2}{3}, i \, {\left (d x + c\right )}^{\frac {3}{2}} b\right ) + {\left (\sqrt {3} - i\right )} \Gamma \left (\frac {2}{3}, -i \, {\left (d x + c\right )}^{\frac {3}{2}} b\right )\right )} \cos \relax (a) - {\left ({\left (i \, \sqrt {3} - 1\right )} \Gamma \left (\frac {2}{3}, i \, {\left (d x + c\right )}^{\frac {3}{2}} b\right ) + {\left (-i \, \sqrt {3} - 1\right )} \Gamma \left (\frac {2}{3}, -i \, {\left (d x + c\right )}^{\frac {3}{2}} b\right )\right )} \sin \relax (a)\right )} c f}{\sqrt {d x + c} b d} + \frac {{\left (12 \, \left ({\left (d x + c\right )}^{\frac {3}{2}} b\right )^{\frac {1}{3}} \sqrt {d x + c} \cos \left ({\left (d x + c\right )}^{\frac {3}{2}} b + a\right ) + \sqrt {d x + c} {\left ({\left ({\left (\sqrt {3} - i\right )} \Gamma \left (\frac {1}{3}, i \, {\left (d x + c\right )}^{\frac {3}{2}} b\right ) + {\left (\sqrt {3} + i\right )} \Gamma \left (\frac {1}{3}, -i \, {\left (d x + c\right )}^{\frac {3}{2}} b\right )\right )} \cos \relax (a) + {\left ({\left (-i \, \sqrt {3} - 1\right )} \Gamma \left (\frac {1}{3}, i \, {\left (d x + c\right )}^{\frac {3}{2}} b\right ) + {\left (i \, \sqrt {3} - 1\right )} \Gamma \left (\frac {1}{3}, -i \, {\left (d x + c\right )}^{\frac {3}{2}} b\right )\right )} \sin \relax (a)\right )}\right )} f}{\left ({\left (d x + c\right )}^{\frac {3}{2}} b\right )^{\frac {1}{3}} b d}}{18 \, d} \]
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 \[ \int \sin \left (a+b\,{\left (c+d\,x\right )}^{3/2}\right )\,\left (e+f\,x\right ) \,d x \]
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 \[ \int \left (e + f x\right ) \sin {\left (a + b c \sqrt {c + d x} + b d x \sqrt {c + d x} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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