Optimal. Leaf size=251 \[ -\frac {i e^{i a} f \left (-\frac {i b}{(c+d x)^{3/2}}\right )^{4/3} (c+d x)^2 \Gamma \left (-\frac {4}{3},-\frac {i b}{(c+d x)^{3/2}}\right )}{3 d^2}+\frac {i e^{-i a} f \left (\frac {i b}{(c+d x)^{3/2}}\right )^{4/3} (c+d x)^2 \Gamma \left (-\frac {4}{3},\frac {i b}{(c+d x)^{3/2}}\right )}{3 d^2}-\frac {i e^{i a} (d e-c f) \left (-\frac {i b}{(c+d x)^{3/2}}\right )^{2/3} (c+d x) \Gamma \left (-\frac {2}{3},-\frac {i b}{(c+d x)^{3/2}}\right )}{3 d^2}+\frac {i e^{-i a} (d e-c f) \left (\frac {i b}{(c+d x)^{3/2}}\right )^{2/3} (c+d x) \Gamma \left (-\frac {2}{3},\frac {i b}{(c+d x)^{3/2}}\right )}{3 d^2} \]
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Rubi [A]
time = 0.15, antiderivative size = 251, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {3514, 3504,
2250} \begin {gather*} -\frac {i e^{i a} (c+d x) \left (-\frac {i b}{(c+d x)^{3/2}}\right )^{2/3} (d e-c f) \text {Gamma}\left (-\frac {2}{3},-\frac {i b}{(c+d x)^{3/2}}\right )}{3 d^2}+\frac {i e^{-i a} (c+d x) \left (\frac {i b}{(c+d x)^{3/2}}\right )^{2/3} (d e-c f) \text {Gamma}\left (-\frac {2}{3},\frac {i b}{(c+d x)^{3/2}}\right )}{3 d^2}-\frac {i e^{i a} f (c+d x)^2 \left (-\frac {i b}{(c+d x)^{3/2}}\right )^{4/3} \text {Gamma}\left (-\frac {4}{3},-\frac {i b}{(c+d x)^{3/2}}\right )}{3 d^2}+\frac {i e^{-i a} f (c+d x)^2 \left (\frac {i b}{(c+d x)^{3/2}}\right )^{4/3} \text {Gamma}\left (-\frac {4}{3},\frac {i b}{(c+d x)^{3/2}}\right )}{3 d^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 2250
Rule 3504
Rule 3514
Rubi steps
\begin {align*} \int (e+f x) \sin \left (a+\frac {b}{(c+d x)^{3/2}}\right ) \, dx &=\frac {2 \text {Subst}\left (\int \left ((d e-c f) x \sin \left (a+\frac {b}{x^3}\right )+f x^3 \sin \left (a+\frac {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+\frac {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+\frac {b}{x^3}\right ) \, dx,x,\sqrt {c+d x}\right )}{d^2}\\ &=\frac {(i f) \text {Subst}\left (\int e^{-i a-\frac {i b}{x^3}} x^3 \, dx,x,\sqrt {c+d x}\right )}{d^2}-\frac {(i f) \text {Subst}\left (\int e^{i a+\frac {i b}{x^3}} x^3 \, dx,x,\sqrt {c+d x}\right )}{d^2}+\frac {(i (d e-c f)) \text {Subst}\left (\int e^{-i a-\frac {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+\frac {i b}{x^3}} x \, dx,x,\sqrt {c+d x}\right )}{d^2}\\ &=-\frac {i e^{i a} f \left (-\frac {i b}{(c+d x)^{3/2}}\right )^{4/3} (c+d x)^2 \Gamma \left (-\frac {4}{3},-\frac {i b}{(c+d x)^{3/2}}\right )}{3 d^2}+\frac {i e^{-i a} f \left (\frac {i b}{(c+d x)^{3/2}}\right )^{4/3} (c+d x)^2 \Gamma \left (-\frac {4}{3},\frac {i b}{(c+d x)^{3/2}}\right )}{3 d^2}-\frac {i e^{i a} (d e-c f) \left (-\frac {i b}{(c+d x)^{3/2}}\right )^{2/3} (c+d x) \Gamma \left (-\frac {2}{3},-\frac {i b}{(c+d x)^{3/2}}\right )}{3 d^2}+\frac {i e^{-i a} (d e-c f) \left (\frac {i b}{(c+d x)^{3/2}}\right )^{2/3} (c+d x) \Gamma \left (-\frac {2}{3},\frac {i b}{(c+d x)^{3/2}}\right )}{3 d^2}\\ \end {align*}
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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice
the leaf count of optimal. \(835\) vs. \(2(251)=502\).
time = 1.74, size = 835, normalized size = 3.33 \begin {gather*} \frac {3 b e \cos (a) \left (\frac {2 \Gamma \left (\frac {1}{3},-\frac {i b}{(c+d x)^{3/2}}\right )}{3 \sqrt [3]{-\frac {i b}{(c+d x)^{3/2}}} \sqrt {c+d x}}+\frac {2 \Gamma \left (\frac {1}{3},\frac {i b}{(c+d x)^{3/2}}\right )}{3 \sqrt [3]{\frac {i b}{(c+d x)^{3/2}}} \sqrt {c+d x}}\right )}{4 d}-\frac {3 b c f \cos (a) \left (\frac {2 \Gamma \left (\frac {1}{3},-\frac {i b}{(c+d x)^{3/2}}\right )}{3 \sqrt [3]{-\frac {i b}{(c+d x)^{3/2}}} \sqrt {c+d x}}+\frac {2 \Gamma \left (\frac {1}{3},\frac {i b}{(c+d x)^{3/2}}\right )}{3 \sqrt [3]{\frac {i b}{(c+d x)^{3/2}}} \sqrt {c+d x}}\right )}{4 d^2}+\frac {9 i b^2 f \cos (a) \left (\frac {2 \Gamma \left (\frac {2}{3},-\frac {i b}{(c+d x)^{3/2}}\right )}{3 \left (-\frac {i b}{(c+d x)^{3/2}}\right )^{2/3} (c+d x)}-\frac {2 \Gamma \left (\frac {2}{3},\frac {i b}{(c+d x)^{3/2}}\right )}{3 \left (\frac {i b}{(c+d x)^{3/2}}\right )^{2/3} (c+d x)}\right )}{8 d^2}+\frac {e (c+d x) \cos \left (\frac {b}{(c+d x)^{3/2}}\right ) \sin (a)}{d}+\frac {3 i b e \left (\frac {2 \Gamma \left (\frac {1}{3},-\frac {i b}{(c+d x)^{3/2}}\right )}{3 \sqrt [3]{-\frac {i b}{(c+d x)^{3/2}}} \sqrt {c+d x}}-\frac {2 \Gamma \left (\frac {1}{3},\frac {i b}{(c+d x)^{3/2}}\right )}{3 \sqrt [3]{\frac {i b}{(c+d x)^{3/2}}} \sqrt {c+d x}}\right ) \sin (a)}{4 d}-\frac {3 i b c f \left (\frac {2 \Gamma \left (\frac {1}{3},-\frac {i b}{(c+d x)^{3/2}}\right )}{3 \sqrt [3]{-\frac {i b}{(c+d x)^{3/2}}} \sqrt {c+d x}}-\frac {2 \Gamma \left (\frac {1}{3},\frac {i b}{(c+d x)^{3/2}}\right )}{3 \sqrt [3]{\frac {i b}{(c+d x)^{3/2}}} \sqrt {c+d x}}\right ) \sin (a)}{4 d^2}-\frac {9 b^2 f \left (\frac {2 \Gamma \left (\frac {2}{3},-\frac {i b}{(c+d x)^{3/2}}\right )}{3 \left (-\frac {i b}{(c+d x)^{3/2}}\right )^{2/3} (c+d x)}+\frac {2 \Gamma \left (\frac {2}{3},\frac {i b}{(c+d x)^{3/2}}\right )}{3 \left (\frac {i b}{(c+d x)^{3/2}}\right )^{2/3} (c+d x)}\right ) \sin (a)}{8 d^2}+\frac {f \sqrt {c+d x} \cos \left (\frac {b}{(c+d x)^{3/2}}\right ) \left (3 b \cos (a)-2 c \sqrt {c+d x} \sin (a)+(c+d x)^{3/2} \sin (a)\right )}{2 d^2}+\frac {e (c+d x) \cos (a) \sin \left (\frac {b}{(c+d x)^{3/2}}\right )}{d}+\frac {f \sqrt {c+d x} \left (-2 c \sqrt {c+d x} \cos (a)+(c+d x)^{3/2} \cos (a)-3 b \sin (a)\right ) \sin \left (\frac {b}{(c+d x)^{3/2}}\right )}{2 d^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \left (f x +e \right ) \sin \left (a +\frac {b}{\left (d x +c \right )^{\frac {3}{2}}}\right )\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than
twice the leaf count of optimal. 505 vs. \(2 (178) = 356\).
time = 0.63, size = 505, normalized size = 2.01 \begin {gather*} -\frac {\frac {2 \, {\left (4 \, {\left (d x + c\right )}^{\frac {3}{2}} \left (\frac {b}{{\left (d x + c\right )}^{\frac {3}{2}}}\right )^{\frac {1}{3}} \sin \left (\frac {{\left (d x + c\right )}^{\frac {3}{2}} a + b}{{\left (d x + c\right )}^{\frac {3}{2}}}\right ) + {\left ({\left ({\left (\sqrt {3} - i\right )} \Gamma \left (\frac {1}{3}, \frac {i \, b}{{\left (d x + c\right )}^{\frac {3}{2}}}\right ) + {\left (\sqrt {3} + i\right )} \Gamma \left (\frac {1}{3}, -\frac {i \, b}{{\left (d x + c\right )}^{\frac {3}{2}}}\right )\right )} \cos \left (a\right ) + {\left ({\left (-i \, \sqrt {3} - 1\right )} \Gamma \left (\frac {1}{3}, \frac {i \, b}{{\left (d x + c\right )}^{\frac {3}{2}}}\right ) + {\left (i \, \sqrt {3} - 1\right )} \Gamma \left (\frac {1}{3}, -\frac {i \, b}{{\left (d x + c\right )}^{\frac {3}{2}}}\right )\right )} \sin \left (a\right )\right )} b\right )} c f}{\sqrt {d x + c} d \left (\frac {b}{{\left (d x + c\right )}^{\frac {3}{2}}}\right )^{\frac {1}{3}}} - \frac {2 \, {\left (4 \, {\left (d x + c\right )}^{\frac {3}{2}} \left (\frac {b}{{\left (d x + c\right )}^{\frac {3}{2}}}\right )^{\frac {1}{3}} \sin \left (\frac {{\left (d x + c\right )}^{\frac {3}{2}} a + b}{{\left (d x + c\right )}^{\frac {3}{2}}}\right ) + {\left ({\left ({\left (\sqrt {3} - i\right )} \Gamma \left (\frac {1}{3}, \frac {i \, b}{{\left (d x + c\right )}^{\frac {3}{2}}}\right ) + {\left (\sqrt {3} + i\right )} \Gamma \left (\frac {1}{3}, -\frac {i \, b}{{\left (d x + c\right )}^{\frac {3}{2}}}\right )\right )} \cos \left (a\right ) + {\left ({\left (-i \, \sqrt {3} - 1\right )} \Gamma \left (\frac {1}{3}, \frac {i \, b}{{\left (d x + c\right )}^{\frac {3}{2}}}\right ) + {\left (i \, \sqrt {3} - 1\right )} \Gamma \left (\frac {1}{3}, -\frac {i \, b}{{\left (d x + c\right )}^{\frac {3}{2}}}\right )\right )} \sin \left (a\right )\right )} b\right )} e}{\sqrt {d x + c} \left (\frac {b}{{\left (d x + c\right )}^{\frac {3}{2}}}\right )^{\frac {1}{3}}} - \frac {{\left (4 \, {\left (d x + c\right )}^{3} \left (\frac {b}{{\left (d x + c\right )}^{\frac {3}{2}}}\right )^{\frac {2}{3}} \sin \left (\frac {{\left (d x + c\right )}^{\frac {3}{2}} a + b}{{\left (d x + c\right )}^{\frac {3}{2}}}\right ) + 12 \, {\left (d x + c\right )}^{\frac {3}{2}} b \left (\frac {b}{{\left (d x + c\right )}^{\frac {3}{2}}}\right )^{\frac {2}{3}} \cos \left (\frac {{\left (d x + c\right )}^{\frac {3}{2}} a + b}{{\left (d x + c\right )}^{\frac {3}{2}}}\right ) - 3 \, {\left ({\left ({\left (\sqrt {3} + i\right )} \Gamma \left (\frac {2}{3}, \frac {i \, b}{{\left (d x + c\right )}^{\frac {3}{2}}}\right ) + {\left (\sqrt {3} - i\right )} \Gamma \left (\frac {2}{3}, -\frac {i \, b}{{\left (d x + c\right )}^{\frac {3}{2}}}\right )\right )} \cos \left (a\right ) + {\left ({\left (-i \, \sqrt {3} + 1\right )} \Gamma \left (\frac {2}{3}, \frac {i \, b}{{\left (d x + c\right )}^{\frac {3}{2}}}\right ) + {\left (i \, \sqrt {3} + 1\right )} \Gamma \left (\frac {2}{3}, -\frac {i \, b}{{\left (d x + c\right )}^{\frac {3}{2}}}\right )\right )} \sin \left (a\right )\right )} b^{2}\right )} f}{{\left (d x + c\right )} d \left (\frac {b}{{\left (d x + c\right )}^{\frac {3}{2}}}\right )^{\frac {2}{3}}}}{8 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.11, size = 333, normalized size = 1.33 \begin {gather*} -\frac {3 \, \left (i \, b\right )^{\frac {1}{3}} b f e^{\left (-i \, a\right )} \Gamma \left (\frac {2}{3}, \frac {i \, \sqrt {d x + c} b}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) + 3 \, \left (-i \, b\right )^{\frac {1}{3}} b f e^{\left (i \, a\right )} \Gamma \left (\frac {2}{3}, -\frac {i \, \sqrt {d x + c} b}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) + 2 \, {\left (-i \, c f + i \, d e\right )} \left (i \, b\right )^{\frac {2}{3}} e^{\left (-i \, a\right )} \Gamma \left (\frac {1}{3}, \frac {i \, \sqrt {d x + c} b}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) + 2 \, {\left (i \, c f - i \, d e\right )} \left (-i \, b\right )^{\frac {2}{3}} e^{\left (i \, a\right )} \Gamma \left (\frac {1}{3}, -\frac {i \, \sqrt {d x + c} b}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) - 6 \, \sqrt {d x + c} b f \cos \left (\frac {a d^{2} x^{2} + 2 \, a c d x + a c^{2} + \sqrt {d x + c} b}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) - 2 \, {\left (d^{2} f x^{2} - c^{2} f + 2 \, {\left (d^{2} x + c d\right )} e\right )} \sin \left (\frac {a d^{2} x^{2} + 2 \, a c d x + a c^{2} + \sqrt {d x + c} b}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}{4 \, d^{2}} \end {gather*}
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 \begin {gather*} \int \left (e + f x\right ) \sin {\left (a + \frac {b}{c \sqrt {c + d x} + d x \sqrt {c + d x}} \right )}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \sin \left (a+\frac {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.
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