Optimal. Leaf size=239 \[ \frac {15 \sqrt {\frac {\pi }{2}} d^{5/2} e^{\frac {2 b c}{d}-2 a} \text {erf}\left (\frac {\sqrt {2} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{256 b^{7/2}}-\frac {15 \sqrt {\frac {\pi }{2}} d^{5/2} e^{2 a-\frac {2 b c}{d}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{256 b^{7/2}}+\frac {15 d^2 \sqrt {c+d x} \sinh (2 a+2 b x)}{64 b^3}-\frac {5 d (c+d x)^{3/2} \sinh ^2(a+b x)}{8 b^2}+\frac {(c+d x)^{5/2} \sinh (a+b x) \cosh (a+b x)}{2 b}-\frac {5 d (c+d x)^{3/2}}{16 b^2}-\frac {(c+d x)^{7/2}}{7 d} \]
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Rubi [A] time = 0.45, antiderivative size = 239, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {3311, 32, 3312, 3296, 3308, 2180, 2204, 2205} \[ \frac {15 \sqrt {\frac {\pi }{2}} d^{5/2} e^{\frac {2 b c}{d}-2 a} \text {Erf}\left (\frac {\sqrt {2} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{256 b^{7/2}}-\frac {15 \sqrt {\frac {\pi }{2}} d^{5/2} e^{2 a-\frac {2 b c}{d}} \text {Erfi}\left (\frac {\sqrt {2} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{256 b^{7/2}}+\frac {15 d^2 \sqrt {c+d x} \sinh (2 a+2 b x)}{64 b^3}-\frac {5 d (c+d x)^{3/2} \sinh ^2(a+b x)}{8 b^2}+\frac {(c+d x)^{5/2} \sinh (a+b x) \cosh (a+b x)}{2 b}-\frac {5 d (c+d x)^{3/2}}{16 b^2}-\frac {(c+d x)^{7/2}}{7 d} \]
Antiderivative was successfully verified.
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Rule 32
Rule 2180
Rule 2204
Rule 2205
Rule 3296
Rule 3308
Rule 3311
Rule 3312
Rubi steps
\begin {align*} \int (c+d x)^{5/2} \sinh ^2(a+b x) \, dx &=\frac {(c+d x)^{5/2} \cosh (a+b x) \sinh (a+b x)}{2 b}-\frac {5 d (c+d x)^{3/2} \sinh ^2(a+b x)}{8 b^2}-\frac {1}{2} \int (c+d x)^{5/2} \, dx+\frac {\left (15 d^2\right ) \int \sqrt {c+d x} \sinh ^2(a+b x) \, dx}{16 b^2}\\ &=-\frac {(c+d x)^{7/2}}{7 d}+\frac {(c+d x)^{5/2} \cosh (a+b x) \sinh (a+b x)}{2 b}-\frac {5 d (c+d x)^{3/2} \sinh ^2(a+b x)}{8 b^2}-\frac {\left (15 d^2\right ) \int \left (\frac {1}{2} \sqrt {c+d x}-\frac {1}{2} \sqrt {c+d x} \cosh (2 a+2 b x)\right ) \, dx}{16 b^2}\\ &=-\frac {5 d (c+d x)^{3/2}}{16 b^2}-\frac {(c+d x)^{7/2}}{7 d}+\frac {(c+d x)^{5/2} \cosh (a+b x) \sinh (a+b x)}{2 b}-\frac {5 d (c+d x)^{3/2} \sinh ^2(a+b x)}{8 b^2}+\frac {\left (15 d^2\right ) \int \sqrt {c+d x} \cosh (2 a+2 b x) \, dx}{32 b^2}\\ &=-\frac {5 d (c+d x)^{3/2}}{16 b^2}-\frac {(c+d x)^{7/2}}{7 d}+\frac {(c+d x)^{5/2} \cosh (a+b x) \sinh (a+b x)}{2 b}-\frac {5 d (c+d x)^{3/2} \sinh ^2(a+b x)}{8 b^2}+\frac {15 d^2 \sqrt {c+d x} \sinh (2 a+2 b x)}{64 b^3}-\frac {\left (15 d^3\right ) \int \frac {\sinh (2 a+2 b x)}{\sqrt {c+d x}} \, dx}{128 b^3}\\ &=-\frac {5 d (c+d x)^{3/2}}{16 b^2}-\frac {(c+d x)^{7/2}}{7 d}+\frac {(c+d x)^{5/2} \cosh (a+b x) \sinh (a+b x)}{2 b}-\frac {5 d (c+d x)^{3/2} \sinh ^2(a+b x)}{8 b^2}+\frac {15 d^2 \sqrt {c+d x} \sinh (2 a+2 b x)}{64 b^3}-\frac {\left (15 d^3\right ) \int \frac {e^{-i (2 i a+2 i b x)}}{\sqrt {c+d x}} \, dx}{256 b^3}+\frac {\left (15 d^3\right ) \int \frac {e^{i (2 i a+2 i b x)}}{\sqrt {c+d x}} \, dx}{256 b^3}\\ &=-\frac {5 d (c+d x)^{3/2}}{16 b^2}-\frac {(c+d x)^{7/2}}{7 d}+\frac {(c+d x)^{5/2} \cosh (a+b x) \sinh (a+b x)}{2 b}-\frac {5 d (c+d x)^{3/2} \sinh ^2(a+b x)}{8 b^2}+\frac {15 d^2 \sqrt {c+d x} \sinh (2 a+2 b x)}{64 b^3}+\frac {\left (15 d^2\right ) \operatorname {Subst}\left (\int e^{i \left (2 i a-\frac {2 i b c}{d}\right )-\frac {2 b x^2}{d}} \, dx,x,\sqrt {c+d x}\right )}{128 b^3}-\frac {\left (15 d^2\right ) \operatorname {Subst}\left (\int e^{-i \left (2 i a-\frac {2 i b c}{d}\right )+\frac {2 b x^2}{d}} \, dx,x,\sqrt {c+d x}\right )}{128 b^3}\\ &=-\frac {5 d (c+d x)^{3/2}}{16 b^2}-\frac {(c+d x)^{7/2}}{7 d}+\frac {15 d^{5/2} e^{-2 a+\frac {2 b c}{d}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {\sqrt {2} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{256 b^{7/2}}-\frac {15 d^{5/2} e^{2 a-\frac {2 b c}{d}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{256 b^{7/2}}+\frac {(c+d x)^{5/2} \cosh (a+b x) \sinh (a+b x)}{2 b}-\frac {5 d (c+d x)^{3/2} \sinh ^2(a+b x)}{8 b^2}+\frac {15 d^2 \sqrt {c+d x} \sinh (2 a+2 b x)}{64 b^3}\\ \end {align*}
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Mathematica [A] time = 6.29, size = 190, normalized size = 0.79 \[ \frac {\sqrt {c+d x} \left (-b (c+d x) \left (7 \sqrt {2} d^3 \Gamma \left (\frac {7}{2},\frac {2 b (c+d x)}{d}\right ) \left (\cosh \left (2 a-\frac {2 b c}{d}\right )-\sinh \left (2 a-\frac {2 b c}{d}\right )\right )+64 b^3 (c+d x)^3 \sqrt {\frac {b (c+d x)}{d}}\right )-7 \sqrt {2} d^4 \sqrt {-\frac {b^2 (c+d x)^2}{d^2}} \Gamma \left (\frac {7}{2},-\frac {2 b (c+d x)}{d}\right ) \left (\sinh \left (2 a-\frac {2 b c}{d}\right )+\cosh \left (2 a-\frac {2 b c}{d}\right )\right )\right )}{448 b^3 d^2 \left (\frac {b (c+d x)}{d}\right )^{3/2}} \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 0.53, size = 1001, normalized size = 4.19 \[ \text {result too large to display} \]
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 \[ \int {\left (d x + c\right )}^{\frac {5}{2}} \sinh \left (b x + a\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.12, size = 0, normalized size = 0.00 \[ \int \left (d x +c \right )^{\frac {5}{2}} \left (\sinh ^{2}\left (b x +a \right )\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.70, size = 281, normalized size = 1.18 \[ -\frac {512 \, {\left (d x + c\right )}^{\frac {7}{2}} + \frac {105 \, \sqrt {2} \sqrt {\pi } d^{3} \operatorname {erf}\left (\sqrt {2} \sqrt {d x + c} \sqrt {-\frac {b}{d}}\right ) e^{\left (2 \, a - \frac {2 \, b c}{d}\right )}}{b^{3} \sqrt {-\frac {b}{d}}} - \frac {105 \, \sqrt {2} \sqrt {\pi } d^{3} \operatorname {erf}\left (\sqrt {2} \sqrt {d x + c} \sqrt {\frac {b}{d}}\right ) e^{\left (-2 \, a + \frac {2 \, b c}{d}\right )}}{b^{3} \sqrt {\frac {b}{d}}} + \frac {28 \, {\left (16 \, {\left (d x + c\right )}^{\frac {5}{2}} b^{2} d e^{\left (\frac {2 \, b c}{d}\right )} + 20 \, {\left (d x + c\right )}^{\frac {3}{2}} b d^{2} e^{\left (\frac {2 \, b c}{d}\right )} + 15 \, \sqrt {d x + c} d^{3} e^{\left (\frac {2 \, b c}{d}\right )}\right )} e^{\left (-2 \, a - \frac {2 \, {\left (d x + c\right )} b}{d}\right )}}{b^{3}} - \frac {28 \, {\left (16 \, {\left (d x + c\right )}^{\frac {5}{2}} b^{2} d e^{\left (2 \, a\right )} - 20 \, {\left (d x + c\right )}^{\frac {3}{2}} b d^{2} e^{\left (2 \, a\right )} + 15 \, \sqrt {d x + c} d^{3} e^{\left (2 \, a\right )}\right )} e^{\left (\frac {2 \, {\left (d x + c\right )} b}{d} - \frac {2 \, b c}{d}\right )}}{b^{3}}}{3584 \, d} \]
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 \[ \int {\mathrm {sinh}\left (a+b\,x\right )}^2\,{\left (c+d\,x\right )}^{5/2} \,d x \]
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 \[ \int \left (c + d x\right )^{\frac {5}{2}} \sinh ^{2}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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