Optimal. Leaf size=146 \[ -\frac {3 \sqrt {\pi } d^{3/2} e^{\frac {b c}{d}-a} \text {erf}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{8 b^{5/2}}+\frac {3 \sqrt {\pi } d^{3/2} e^{a-\frac {b c}{d}} \text {erfi}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{8 b^{5/2}}-\frac {3 d \sqrt {c+d x} \sinh (a+b x)}{2 b^2}+\frac {(c+d x)^{3/2} \cosh (a+b x)}{b} \]
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Rubi [A] time = 0.25, antiderivative size = 146, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {3296, 3308, 2180, 2204, 2205} \[ -\frac {3 \sqrt {\pi } d^{3/2} e^{\frac {b c}{d}-a} \text {Erf}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{8 b^{5/2}}+\frac {3 \sqrt {\pi } d^{3/2} e^{a-\frac {b c}{d}} \text {Erfi}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{8 b^{5/2}}-\frac {3 d \sqrt {c+d x} \sinh (a+b x)}{2 b^2}+\frac {(c+d x)^{3/2} \cosh (a+b x)}{b} \]
Antiderivative was successfully verified.
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Rule 2180
Rule 2204
Rule 2205
Rule 3296
Rule 3308
Rubi steps
\begin {align*} \int (c+d x)^{3/2} \sinh (a+b x) \, dx &=\frac {(c+d x)^{3/2} \cosh (a+b x)}{b}-\frac {(3 d) \int \sqrt {c+d x} \cosh (a+b x) \, dx}{2 b}\\ &=\frac {(c+d x)^{3/2} \cosh (a+b x)}{b}-\frac {3 d \sqrt {c+d x} \sinh (a+b x)}{2 b^2}+\frac {\left (3 d^2\right ) \int \frac {\sinh (a+b x)}{\sqrt {c+d x}} \, dx}{4 b^2}\\ &=\frac {(c+d x)^{3/2} \cosh (a+b x)}{b}-\frac {3 d \sqrt {c+d x} \sinh (a+b x)}{2 b^2}+\frac {\left (3 d^2\right ) \int \frac {e^{-i (i a+i b x)}}{\sqrt {c+d x}} \, dx}{8 b^2}-\frac {\left (3 d^2\right ) \int \frac {e^{i (i a+i b x)}}{\sqrt {c+d x}} \, dx}{8 b^2}\\ &=\frac {(c+d x)^{3/2} \cosh (a+b x)}{b}-\frac {3 d \sqrt {c+d x} \sinh (a+b x)}{2 b^2}-\frac {(3 d) \operatorname {Subst}\left (\int e^{i \left (i a-\frac {i b c}{d}\right )-\frac {b x^2}{d}} \, dx,x,\sqrt {c+d x}\right )}{4 b^2}+\frac {(3 d) \operatorname {Subst}\left (\int e^{-i \left (i a-\frac {i b c}{d}\right )+\frac {b x^2}{d}} \, dx,x,\sqrt {c+d x}\right )}{4 b^2}\\ &=\frac {(c+d x)^{3/2} \cosh (a+b x)}{b}-\frac {3 d^{3/2} e^{-a+\frac {b c}{d}} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{8 b^{5/2}}+\frac {3 d^{3/2} e^{a-\frac {b c}{d}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{8 b^{5/2}}-\frac {3 d \sqrt {c+d x} \sinh (a+b x)}{2 b^2}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 106, normalized size = 0.73 \[ \frac {d \sqrt {c+d x} e^{-a-\frac {b c}{d}} \left (\frac {e^{\frac {2 b c}{d}} \Gamma \left (\frac {5}{2},\frac {b (c+d x)}{d}\right )}{\sqrt {\frac {b (c+d x)}{d}}}-\frac {e^{2 a} \Gamma \left (\frac {5}{2},-\frac {b (c+d x)}{d}\right )}{\sqrt {-\frac {b (c+d x)}{d}}}\right )}{2 b^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.44, size = 385, normalized size = 2.64 \[ -\frac {3 \, \sqrt {\pi } {\left (d^{2} \cosh \left (b x + a\right ) \cosh \left (-\frac {b c - a d}{d}\right ) - d^{2} \cosh \left (b x + a\right ) \sinh \left (-\frac {b c - a d}{d}\right ) + {\left (d^{2} \cosh \left (-\frac {b c - a d}{d}\right ) - d^{2} \sinh \left (-\frac {b c - a d}{d}\right )\right )} \sinh \left (b x + a\right )\right )} \sqrt {\frac {b}{d}} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {\frac {b}{d}}\right ) + 3 \, \sqrt {\pi } {\left (d^{2} \cosh \left (b x + a\right ) \cosh \left (-\frac {b c - a d}{d}\right ) + d^{2} \cosh \left (b x + a\right ) \sinh \left (-\frac {b c - a d}{d}\right ) + {\left (d^{2} \cosh \left (-\frac {b c - a d}{d}\right ) + d^{2} \sinh \left (-\frac {b c - a d}{d}\right )\right )} \sinh \left (b x + a\right )\right )} \sqrt {-\frac {b}{d}} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {-\frac {b}{d}}\right ) - 2 \, {\left (2 \, b^{2} d x + 2 \, b^{2} c + {\left (2 \, b^{2} d x + 2 \, b^{2} c - 3 \, b d\right )} \cosh \left (b x + a\right )^{2} + 2 \, {\left (2 \, b^{2} d x + 2 \, b^{2} c - 3 \, b d\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + {\left (2 \, b^{2} d x + 2 \, b^{2} c - 3 \, b d\right )} \sinh \left (b x + a\right )^{2} + 3 \, b d\right )} \sqrt {d x + c}}{8 \, {\left (b^{3} \cosh \left (b x + a\right ) + b^{3} \sinh \left (b x + a\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.11, size = 202, normalized size = 1.38 \[ \frac {\frac {3 \, \sqrt {\pi } d^{3} \operatorname {erf}\left (-\frac {\sqrt {b d} \sqrt {d x + c}}{d}\right ) e^{\left (\frac {b c - a d}{d}\right )}}{\sqrt {b d} b^{2}} - \frac {3 \, \sqrt {\pi } d^{3} \operatorname {erf}\left (-\frac {\sqrt {-b d} \sqrt {d x + c}}{d}\right ) e^{\left (-\frac {b c - a d}{d}\right )}}{\sqrt {-b d} b^{2}} + \frac {2 \, {\left (2 \, {\left (d x + c\right )}^{\frac {3}{2}} b d - 3 \, \sqrt {d x + c} d^{2}\right )} e^{\left (\frac {{\left (d x + c\right )} b - b c + a d}{d}\right )}}{b^{2}} + \frac {2 \, {\left (2 \, {\left (d x + c\right )}^{\frac {3}{2}} b d + 3 \, \sqrt {d x + c} d^{2}\right )} e^{\left (-\frac {{\left (d x + c\right )} b - b c + a d}{d}\right )}}{b^{2}}}{8 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.09, size = 0, normalized size = 0.00 \[ \int \left (d x +c \right )^{\frac {3}{2}} \sinh \left (b x +a \right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.35, size = 268, normalized size = 1.84 \[ \frac {16 \, {\left (d x + c\right )}^{\frac {5}{2}} \sinh \left (b x + a\right ) + \frac {{\left (\frac {15 \, \sqrt {\pi } d^{3} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {-\frac {b}{d}}\right ) e^{\left (a - \frac {b c}{d}\right )}}{b^{3} \sqrt {-\frac {b}{d}}} - \frac {15 \, \sqrt {\pi } d^{3} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {\frac {b}{d}}\right ) e^{\left (-a + \frac {b c}{d}\right )}}{b^{3} \sqrt {\frac {b}{d}}} + \frac {2 \, {\left (4 \, {\left (d x + c\right )}^{\frac {5}{2}} b^{2} d e^{\left (\frac {b c}{d}\right )} + 10 \, {\left (d x + c\right )}^{\frac {3}{2}} b d^{2} e^{\left (\frac {b c}{d}\right )} + 15 \, \sqrt {d x + c} d^{3} e^{\left (\frac {b c}{d}\right )}\right )} e^{\left (-a - \frac {{\left (d x + c\right )} b}{d}\right )}}{b^{3}} - \frac {2 \, {\left (4 \, {\left (d x + c\right )}^{\frac {5}{2}} b^{2} d e^{a} - 10 \, {\left (d x + c\right )}^{\frac {3}{2}} b d^{2} e^{a} + 15 \, \sqrt {d x + c} d^{3} e^{a}\right )} e^{\left (\frac {{\left (d x + c\right )} b}{d} - \frac {b c}{d}\right )}}{b^{3}}\right )} b}{d}}{40 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \mathrm {sinh}\left (a+b\,x\right )\,{\left (c+d\,x\right )}^{3/2} \,d x \]
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 \[ \int \left (c + d x\right )^{\frac {3}{2}} \sinh {\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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