Optimal. Leaf size=232 \[ -\frac {i \left (23 a^2-23 a b+8 b^2\right ) \sqrt {a+b \sinh ^2(c+d x)} E\left (i c+i d x\left |\frac {b}{a}\right .\right )}{15 d \sqrt {\frac {b \sinh ^2(c+d x)}{a}+1}}+\frac {b \sinh (c+d x) \cosh (c+d x) \left (a+b \sinh ^2(c+d x)\right )^{3/2}}{5 d}+\frac {4 b (2 a-b) \sinh (c+d x) \cosh (c+d x) \sqrt {a+b \sinh ^2(c+d x)}}{15 d}+\frac {4 i a (a-b) (2 a-b) \sqrt {\frac {b \sinh ^2(c+d x)}{a}+1} F\left (i c+i d x\left |\frac {b}{a}\right .\right )}{15 d \sqrt {a+b \sinh ^2(c+d x)}} \]
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Rubi [A] time = 0.30, antiderivative size = 232, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {3180, 3170, 3172, 3178, 3177, 3183, 3182} \[ -\frac {i \left (23 a^2-23 a b+8 b^2\right ) \sqrt {a+b \sinh ^2(c+d x)} E\left (i c+i d x\left |\frac {b}{a}\right .\right )}{15 d \sqrt {\frac {b \sinh ^2(c+d x)}{a}+1}}+\frac {b \sinh (c+d x) \cosh (c+d x) \left (a+b \sinh ^2(c+d x)\right )^{3/2}}{5 d}+\frac {4 b (2 a-b) \sinh (c+d x) \cosh (c+d x) \sqrt {a+b \sinh ^2(c+d x)}}{15 d}+\frac {4 i a (a-b) (2 a-b) \sqrt {\frac {b \sinh ^2(c+d x)}{a}+1} F\left (i c+i d x\left |\frac {b}{a}\right .\right )}{15 d \sqrt {a+b \sinh ^2(c+d x)}} \]
Antiderivative was successfully verified.
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Rule 3170
Rule 3172
Rule 3177
Rule 3178
Rule 3180
Rule 3182
Rule 3183
Rubi steps
\begin {align*} \int \left (a+b \sinh ^2(c+d x)\right )^{5/2} \, dx &=\frac {b \cosh (c+d x) \sinh (c+d x) \left (a+b \sinh ^2(c+d x)\right )^{3/2}}{5 d}+\frac {1}{5} \int \sqrt {a+b \sinh ^2(c+d x)} \left (a (5 a-b)+4 (2 a-b) b \sinh ^2(c+d x)\right ) \, dx\\ &=\frac {4 (2 a-b) b \cosh (c+d x) \sinh (c+d x) \sqrt {a+b \sinh ^2(c+d x)}}{15 d}+\frac {b \cosh (c+d x) \sinh (c+d x) \left (a+b \sinh ^2(c+d x)\right )^{3/2}}{5 d}+\frac {1}{15} \int \frac {a \left (15 a^2-11 a b+4 b^2\right )+b \left (23 a^2-23 a b+8 b^2\right ) \sinh ^2(c+d x)}{\sqrt {a+b \sinh ^2(c+d x)}} \, dx\\ &=\frac {4 (2 a-b) b \cosh (c+d x) \sinh (c+d x) \sqrt {a+b \sinh ^2(c+d x)}}{15 d}+\frac {b \cosh (c+d x) \sinh (c+d x) \left (a+b \sinh ^2(c+d x)\right )^{3/2}}{5 d}-\frac {1}{15} (4 a (a-b) (2 a-b)) \int \frac {1}{\sqrt {a+b \sinh ^2(c+d x)}} \, dx+\frac {1}{15} \left (23 a^2-23 a b+8 b^2\right ) \int \sqrt {a+b \sinh ^2(c+d x)} \, dx\\ &=\frac {4 (2 a-b) b \cosh (c+d x) \sinh (c+d x) \sqrt {a+b \sinh ^2(c+d x)}}{15 d}+\frac {b \cosh (c+d x) \sinh (c+d x) \left (a+b \sinh ^2(c+d x)\right )^{3/2}}{5 d}+\frac {\left (\left (23 a^2-23 a b+8 b^2\right ) \sqrt {a+b \sinh ^2(c+d x)}\right ) \int \sqrt {1+\frac {b \sinh ^2(c+d x)}{a}} \, dx}{15 \sqrt {1+\frac {b \sinh ^2(c+d x)}{a}}}-\frac {\left (4 a (a-b) (2 a-b) \sqrt {1+\frac {b \sinh ^2(c+d x)}{a}}\right ) \int \frac {1}{\sqrt {1+\frac {b \sinh ^2(c+d x)}{a}}} \, dx}{15 \sqrt {a+b \sinh ^2(c+d x)}}\\ &=\frac {4 (2 a-b) b \cosh (c+d x) \sinh (c+d x) \sqrt {a+b \sinh ^2(c+d x)}}{15 d}+\frac {b \cosh (c+d x) \sinh (c+d x) \left (a+b \sinh ^2(c+d x)\right )^{3/2}}{5 d}-\frac {i \left (23 a^2-23 a b+8 b^2\right ) E\left (i c+i d x\left |\frac {b}{a}\right .\right ) \sqrt {a+b \sinh ^2(c+d x)}}{15 d \sqrt {1+\frac {b \sinh ^2(c+d x)}{a}}}+\frac {4 i a (a-b) (2 a-b) F\left (i c+i d x\left |\frac {b}{a}\right .\right ) \sqrt {1+\frac {b \sinh ^2(c+d x)}{a}}}{15 d \sqrt {a+b \sinh ^2(c+d x)}}\\ \end {align*}
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Mathematica [A] time = 1.41, size = 208, normalized size = 0.90 \[ \frac {\sqrt {2} b \sinh (2 (c+d x)) \left (88 a^2+28 b (2 a-b) \cosh (2 (c+d x))-88 a b+3 b^2 \cosh (4 (c+d x))+25 b^2\right )+64 i a \left (2 a^2-3 a b+b^2\right ) \sqrt {\frac {2 a+b \cosh (2 (c+d x))-b}{a}} F\left (i (c+d x)\left |\frac {b}{a}\right .\right )-16 i a \left (23 a^2-23 a b+8 b^2\right ) \sqrt {\frac {2 a+b \cosh (2 (c+d x))-b}{a}} E\left (i (c+d x)\left |\frac {b}{a}\right .\right )}{240 d \sqrt {2 a+b \cosh (2 (c+d x))-b}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.30, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b^{2} \sinh \left (d x + c\right )^{4} + 2 \, a b \sinh \left (d x + c\right )^{2} + a^{2}\right )} \sqrt {b \sinh \left (d x + c\right )^{2} + a}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.12, size = 609, normalized size = 2.62 \[ \frac {3 \sqrt {-\frac {b}{a}}\, b^{3} \sinh \left (d x +c \right ) \left (\cosh ^{6}\left (d x +c \right )\right )+\left (14 \sqrt {-\frac {b}{a}}\, a \,b^{2}-10 \sqrt {-\frac {b}{a}}\, b^{3}\right ) \left (\cosh ^{4}\left (d x +c \right )\right ) \sinh \left (d x +c \right )+\left (11 \sqrt {-\frac {b}{a}}\, a^{2} b -18 \sqrt {-\frac {b}{a}}\, a \,b^{2}+7 \sqrt {-\frac {b}{a}}\, b^{3}\right ) \left (\cosh ^{2}\left (d x +c \right )\right ) \sinh \left (d x +c \right )+15 a^{3} \sqrt {\frac {b \left (\cosh ^{2}\left (d x +c \right )\right )}{a}+\frac {a -b}{a}}\, \sqrt {\frac {\cosh \left (2 d x +2 c \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sinh \left (d x +c \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right )-34 a^{2} b \sqrt {\frac {b \left (\cosh ^{2}\left (d x +c \right )\right )}{a}+\frac {a -b}{a}}\, \sqrt {\frac {\cosh \left (2 d x +2 c \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sinh \left (d x +c \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right )+27 \sqrt {\frac {b \left (\cosh ^{2}\left (d x +c \right )\right )}{a}+\frac {a -b}{a}}\, \sqrt {\frac {\cosh \left (2 d x +2 c \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sinh \left (d x +c \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) a \,b^{2}-8 \sqrt {\frac {b \left (\cosh ^{2}\left (d x +c \right )\right )}{a}+\frac {a -b}{a}}\, \sqrt {\frac {\cosh \left (2 d x +2 c \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sinh \left (d x +c \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) b^{3}+23 a^{2} b \sqrt {\frac {b \left (\cosh ^{2}\left (d x +c \right )\right )}{a}+\frac {a -b}{a}}\, \sqrt {\frac {\cosh \left (2 d x +2 c \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sinh \left (d x +c \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right )-23 \sqrt {\frac {b \left (\cosh ^{2}\left (d x +c \right )\right )}{a}+\frac {a -b}{a}}\, \sqrt {\frac {\cosh \left (2 d x +2 c \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sinh \left (d x +c \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) a \,b^{2}+8 \sqrt {\frac {b \left (\cosh ^{2}\left (d x +c \right )\right )}{a}+\frac {a -b}{a}}\, \sqrt {\frac {\cosh \left (2 d x +2 c \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sinh \left (d x +c \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) b^{3}}{15 \sqrt {-\frac {b}{a}}\, \cosh \left (d x +c \right ) \sqrt {a +b \left (\sinh ^{2}\left (d x +c \right )\right )}\, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \sinh \left (d x + c\right )^{2} + a\right )}^{\frac {5}{2}}\,{d x} \]
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 \[ \int {\left (b\,{\mathrm {sinh}\left (c+d\,x\right )}^2+a\right )}^{5/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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