Optimal. Leaf size=291 \[ -\frac {2 f \sqrt {c+d x} \cos \left (a+b (c+d x)^{3/2}\right )}{3 b d^2}-\frac {e^{i a} f \sqrt {c+d x} \Gamma \left (\frac {1}{3},-i b (c+d x)^{3/2}\right )}{9 b d^2 \sqrt [3]{-i b (c+d x)^{3/2}}}-\frac {e^{-i a} f \sqrt {c+d x} \Gamma \left (\frac {1}{3},i b (c+d x)^{3/2}\right )}{9 b d^2 \sqrt [3]{i b (c+d x)^{3/2}}}+\frac {i e^{i a} (d e-c f) (c+d x) \Gamma \left (\frac {2}{3},-i b (c+d x)^{3/2}\right )}{3 d^2 \left (-i b (c+d x)^{3/2}\right )^{2/3}}-\frac {i e^{-i a} (d e-c f) (c+d x) \Gamma \left (\frac {2}{3},i b (c+d x)^{3/2}\right )}{3 d^2 \left (i b (c+d x)^{3/2}\right )^{2/3}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.13, 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 = {3514, 3470,
2250, 3466, 3437, 2239} \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2239
Rule 2250
Rule 3437
Rule 3466
Rule 3470
Rule 3514
Rubi steps
\begin {align*} \int (e+f x) \sin \left (a+b (c+d x)^{3/2}\right ) \, dx &=\frac {2 \text {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) \text {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)) \text {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) \text {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)) \text {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)) \text {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 \text {Subst}\left (\int e^{-i a-i b x^3} \, dx,x,\sqrt {c+d x}\right )}{3 b d^2}+\frac {f \text {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*}
________________________________________________________________________________________
Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice
the leaf count of optimal. \(705\) vs. \(2(291)=582\).
time = 1.65, size = 705, normalized size = 2.42 \begin {gather*} -\frac {2 f \sqrt {c+d x} \cos (a) \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 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 {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 \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 ) \sin (a)}{6 b d^2}+\frac {e \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 ) \sin (a)}{2 d}-\frac {c f \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 ) \sin (a)}{2 d^2}+\frac {2 f \sqrt {c+d x} \sin (a) \sin \left (b (c+d x)^{3/2}\right )}{3 b d^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.02, 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.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.63, size = 377, normalized size = 1.30 \begin {gather*} \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 \left (a\right ) - {\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 \left (a\right )\right )} c f}{\sqrt {d x + c} b d} - \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 \left (a\right ) - {\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 \left (a\right )\right )} e}{\sqrt {d x + c} b} - \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 \left (a\right ) + {\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 \left (a\right )\right )}\right )} f}{\left ({\left (d x + c\right )}^{\frac {3}{2}} b\right )^{\frac {1}{3}} b d}}{18 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.13, size = 187, normalized size = 0.64 \begin {gather*} \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 c f - b d e\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 c f - b d e\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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (e + f x\right ) \sin {\left (a + b c \sqrt {c + d x} + b d x \sqrt {c + d x} \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \sin \left (a+b\,{\left (c+d\,x\right )}^{3/2}\right )\,\left (e+f\,x\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________