Optimal. Leaf size=171 \[ \frac {15 d^2 \sqrt {c+d x} \cosh (a+b x)}{4 b^3}+\frac {(c+d x)^{5/2} \cosh (a+b x)}{b}-\frac {15 d^{5/2} e^{-a+\frac {b c}{d}} \sqrt {\pi } \text {Erf}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{16 b^{7/2}}-\frac {15 d^{5/2} e^{a-\frac {b c}{d}} \sqrt {\pi } \text {Erfi}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{16 b^{7/2}}-\frac {5 d (c+d x)^{3/2} \sinh (a+b x)}{2 b^2} \]
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Rubi [A]
time = 0.26, antiderivative size = 171, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 5, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {3377, 3388,
2211, 2235, 2236} \begin {gather*} -\frac {15 \sqrt {\pi } d^{5/2} e^{\frac {b c}{d}-a} \text {Erf}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{16 b^{7/2}}-\frac {15 \sqrt {\pi } d^{5/2} e^{a-\frac {b c}{d}} \text {Erfi}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{16 b^{7/2}}+\frac {15 d^2 \sqrt {c+d x} \cosh (a+b x)}{4 b^3}-\frac {5 d (c+d x)^{3/2} \sinh (a+b x)}{2 b^2}+\frac {(c+d x)^{5/2} \cosh (a+b x)}{b} \end {gather*}
Antiderivative was successfully verified.
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Rule 2211
Rule 2235
Rule 2236
Rule 3377
Rule 3388
Rubi steps
\begin {align*} \int (c+d x)^{5/2} \sinh (a+b x) \, dx &=\frac {(c+d x)^{5/2} \cosh (a+b x)}{b}-\frac {(5 d) \int (c+d x)^{3/2} \cosh (a+b x) \, dx}{2 b}\\ &=\frac {(c+d x)^{5/2} \cosh (a+b x)}{b}-\frac {5 d (c+d x)^{3/2} \sinh (a+b x)}{2 b^2}+\frac {\left (15 d^2\right ) \int \sqrt {c+d x} \sinh (a+b x) \, dx}{4 b^2}\\ &=\frac {15 d^2 \sqrt {c+d x} \cosh (a+b x)}{4 b^3}+\frac {(c+d x)^{5/2} \cosh (a+b x)}{b}-\frac {5 d (c+d x)^{3/2} \sinh (a+b x)}{2 b^2}-\frac {\left (15 d^3\right ) \int \frac {\cosh (a+b x)}{\sqrt {c+d x}} \, dx}{8 b^3}\\ &=\frac {15 d^2 \sqrt {c+d x} \cosh (a+b x)}{4 b^3}+\frac {(c+d x)^{5/2} \cosh (a+b x)}{b}-\frac {5 d (c+d x)^{3/2} \sinh (a+b x)}{2 b^2}-\frac {\left (15 d^3\right ) \int \frac {e^{-i (i a+i b x)}}{\sqrt {c+d x}} \, dx}{16 b^3}-\frac {\left (15 d^3\right ) \int \frac {e^{i (i a+i b x)}}{\sqrt {c+d x}} \, dx}{16 b^3}\\ &=\frac {15 d^2 \sqrt {c+d x} \cosh (a+b x)}{4 b^3}+\frac {(c+d x)^{5/2} \cosh (a+b x)}{b}-\frac {5 d (c+d x)^{3/2} \sinh (a+b x)}{2 b^2}-\frac {\left (15 d^2\right ) \text {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 )}{8 b^3}-\frac {\left (15 d^2\right ) \text {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 )}{8 b^3}\\ &=\frac {15 d^2 \sqrt {c+d x} \cosh (a+b x)}{4 b^3}+\frac {(c+d x)^{5/2} \cosh (a+b x)}{b}-\frac {15 d^{5/2} e^{-a+\frac {b c}{d}} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{16 b^{7/2}}-\frac {15 d^{5/2} e^{a-\frac {b c}{d}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{16 b^{7/2}}-\frac {5 d (c+d x)^{3/2} \sinh (a+b x)}{2 b^2}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 108, normalized size = 0.63 \begin {gather*} \frac {d^3 e^{-a-\frac {b c}{d}} \left (-e^{2 a} \sqrt {-\frac {b (c+d x)}{d}} \Gamma \left (\frac {7}{2},-\frac {b (c+d x)}{d}\right )+e^{\frac {2 b c}{d}} \sqrt {\frac {b (c+d x)}{d}} \Gamma \left (\frac {7}{2},\frac {b (c+d x)}{d}\right )\right )}{2 b^4 \sqrt {c+d x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \left (d x +c \right )^{\frac {5}{2}} \sinh \left (b x +a \right )\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 308 vs.
\(2 (131) = 262\).
time = 0.28, size = 308, normalized size = 1.80 \begin {gather*} \frac {32 \, {\left (d x + c\right )}^{\frac {7}{2}} \sinh \left (b x + a\right ) - \frac {{\left (\frac {105 \, \sqrt {\pi } d^{4} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {-\frac {b}{d}}\right ) e^{\left (a - \frac {b c}{d}\right )}}{b^{4} \sqrt {-\frac {b}{d}}} + \frac {105 \, \sqrt {\pi } d^{4} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {\frac {b}{d}}\right ) e^{\left (-a + \frac {b c}{d}\right )}}{b^{4} \sqrt {\frac {b}{d}}} - \frac {2 \, {\left (8 \, {\left (d x + c\right )}^{\frac {7}{2}} b^{3} d e^{\left (\frac {b c}{d}\right )} + 28 \, {\left (d x + c\right )}^{\frac {5}{2}} b^{2} d^{2} e^{\left (\frac {b c}{d}\right )} + 70 \, {\left (d x + c\right )}^{\frac {3}{2}} b d^{3} e^{\left (\frac {b c}{d}\right )} + 105 \, \sqrt {d x + c} d^{4} e^{\left (\frac {b c}{d}\right )}\right )} e^{\left (-a - \frac {{\left (d x + c\right )} b}{d}\right )}}{b^{4}} + \frac {2 \, {\left (8 \, {\left (d x + c\right )}^{\frac {7}{2}} b^{3} d e^{a} - 28 \, {\left (d x + c\right )}^{\frac {5}{2}} b^{2} d^{2} e^{a} + 70 \, {\left (d x + c\right )}^{\frac {3}{2}} b d^{3} e^{a} - 105 \, \sqrt {d x + c} d^{4} e^{a}\right )} e^{\left (\frac {{\left (d x + c\right )} b}{d} - \frac {b c}{d}\right )}}{b^{4}}\right )} b}{d}}{112 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 521 vs.
\(2 (131) = 262\).
time = 0.35, size = 521, normalized size = 3.05 \begin {gather*} -\frac {15 \, \sqrt {\pi } {\left (d^{3} \cosh \left (b x + a\right ) \cosh \left (-\frac {b c - a d}{d}\right ) - d^{3} \cosh \left (b x + a\right ) \sinh \left (-\frac {b c - a d}{d}\right ) + {\left (d^{3} \cosh \left (-\frac {b c - a d}{d}\right ) - d^{3} \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 ) - 15 \, \sqrt {\pi } {\left (d^{3} \cosh \left (b x + a\right ) \cosh \left (-\frac {b c - a d}{d}\right ) + d^{3} \cosh \left (b x + a\right ) \sinh \left (-\frac {b c - a d}{d}\right ) + {\left (d^{3} \cosh \left (-\frac {b c - a d}{d}\right ) + d^{3} \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 (4 \, b^{3} d^{2} x^{2} + 4 \, b^{3} c^{2} + 10 \, b^{2} c d + 15 \, b d^{2} + {\left (4 \, b^{3} d^{2} x^{2} + 4 \, b^{3} c^{2} - 10 \, b^{2} c d + 15 \, b d^{2} + 2 \, {\left (4 \, b^{3} c d - 5 \, b^{2} d^{2}\right )} x\right )} \cosh \left (b x + a\right )^{2} + 2 \, {\left (4 \, b^{3} d^{2} x^{2} + 4 \, b^{3} c^{2} - 10 \, b^{2} c d + 15 \, b d^{2} + 2 \, {\left (4 \, b^{3} c d - 5 \, b^{2} d^{2}\right )} x\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + {\left (4 \, b^{3} d^{2} x^{2} + 4 \, b^{3} c^{2} - 10 \, b^{2} c d + 15 \, b d^{2} + 2 \, {\left (4 \, b^{3} c d - 5 \, b^{2} d^{2}\right )} x\right )} \sinh \left (b x + a\right )^{2} + 2 \, {\left (4 \, b^{3} c d + 5 \, b^{2} d^{2}\right )} x\right )} \sqrt {d x + c}}{16 \, {\left (b^{4} \cosh \left (b x + a\right ) + b^{4} \sinh \left (b x + a\right )\right )}} \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 (c + d x\right )^{\frac {5}{2}} \sinh {\left (a + b x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.47, size = 232, normalized size = 1.36 \begin {gather*} \frac {\frac {15 \, \sqrt {\pi } d^{4} \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^{3}} + \frac {15 \, \sqrt {\pi } d^{4} \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^{3}} + \frac {2 \, {\left (4 \, {\left (d x + c\right )}^{\frac {5}{2}} b^{2} d - 10 \, {\left (d x + c\right )}^{\frac {3}{2}} b d^{2} + 15 \, \sqrt {d x + c} d^{3}\right )} e^{\left (\frac {{\left (d x + c\right )} b - b c + a d}{d}\right )}}{b^{3}} + \frac {2 \, {\left (4 \, {\left (d x + c\right )}^{\frac {5}{2}} b^{2} d + 10 \, {\left (d x + c\right )}^{\frac {3}{2}} b d^{2} + 15 \, \sqrt {d x + c} d^{3}\right )} e^{\left (-\frac {{\left (d x + c\right )} b - b c + a d}{d}\right )}}{b^{3}}}{16 \, d} \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 = -0.01 \begin {gather*} \int \mathrm {sinh}\left (a+b\,x\right )\,{\left (c+d\,x\right )}^{5/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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