Optimal. Leaf size=326 \[ \frac {9 \sqrt {\pi } d^{3/2} e^{\frac {b c}{d}-a} \text {erf}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{32 b^{5/2}}+\frac {\sqrt {\frac {\pi }{3}} d^{3/2} e^{\frac {3 b c}{d}-3 a} \text {erf}\left (\frac {\sqrt {3} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{96 b^{5/2}}+\frac {9 \sqrt {\pi } d^{3/2} e^{a-\frac {b c}{d}} \text {erfi}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{32 b^{5/2}}+\frac {\sqrt {\frac {\pi }{3}} d^{3/2} e^{3 a-\frac {3 b c}{d}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{96 b^{5/2}}-\frac {d \sqrt {c+d x} \cosh ^3(a+b x)}{6 b^2}-\frac {d \sqrt {c+d x} \cosh (a+b x)}{b^2}+\frac {2 (c+d x)^{3/2} \sinh (a+b x)}{3 b}+\frac {(c+d x)^{3/2} \sinh (a+b x) \cosh ^2(a+b x)}{3 b} \]
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Rubi [A] time = 0.71, antiderivative size = 326, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 7, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {3311, 3296, 3307, 2180, 2204, 2205, 3312} \[ \frac {9 \sqrt {\pi } d^{3/2} e^{\frac {b c}{d}-a} \text {Erf}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{32 b^{5/2}}+\frac {\sqrt {\frac {\pi }{3}} d^{3/2} e^{\frac {3 b c}{d}-3 a} \text {Erf}\left (\frac {\sqrt {3} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{96 b^{5/2}}+\frac {9 \sqrt {\pi } d^{3/2} e^{a-\frac {b c}{d}} \text {Erfi}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{32 b^{5/2}}+\frac {\sqrt {\frac {\pi }{3}} d^{3/2} e^{3 a-\frac {3 b c}{d}} \text {Erfi}\left (\frac {\sqrt {3} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{96 b^{5/2}}-\frac {d \sqrt {c+d x} \cosh ^3(a+b x)}{6 b^2}-\frac {d \sqrt {c+d x} \cosh (a+b x)}{b^2}+\frac {2 (c+d x)^{3/2} \sinh (a+b x)}{3 b}+\frac {(c+d x)^{3/2} \sinh (a+b x) \cosh ^2(a+b x)}{3 b} \]
Antiderivative was successfully verified.
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Rule 2180
Rule 2204
Rule 2205
Rule 3296
Rule 3307
Rule 3311
Rule 3312
Rubi steps
\begin {align*} \int (c+d x)^{3/2} \cosh ^3(a+b x) \, dx &=-\frac {d \sqrt {c+d x} \cosh ^3(a+b x)}{6 b^2}+\frac {(c+d x)^{3/2} \cosh ^2(a+b x) \sinh (a+b x)}{3 b}+\frac {2}{3} \int (c+d x)^{3/2} \cosh (a+b x) \, dx+\frac {d^2 \int \frac {\cosh ^3(a+b x)}{\sqrt {c+d x}} \, dx}{12 b^2}\\ &=-\frac {d \sqrt {c+d x} \cosh ^3(a+b x)}{6 b^2}+\frac {2 (c+d x)^{3/2} \sinh (a+b x)}{3 b}+\frac {(c+d x)^{3/2} \cosh ^2(a+b x) \sinh (a+b x)}{3 b}-\frac {d \int \sqrt {c+d x} \sinh (a+b x) \, dx}{b}+\frac {d^2 \int \left (\frac {3 \cosh (a+b x)}{4 \sqrt {c+d x}}+\frac {\cosh (3 a+3 b x)}{4 \sqrt {c+d x}}\right ) \, dx}{12 b^2}\\ &=-\frac {d \sqrt {c+d x} \cosh (a+b x)}{b^2}-\frac {d \sqrt {c+d x} \cosh ^3(a+b x)}{6 b^2}+\frac {2 (c+d x)^{3/2} \sinh (a+b x)}{3 b}+\frac {(c+d x)^{3/2} \cosh ^2(a+b x) \sinh (a+b x)}{3 b}+\frac {d^2 \int \frac {\cosh (3 a+3 b x)}{\sqrt {c+d x}} \, dx}{48 b^2}+\frac {d^2 \int \frac {\cosh (a+b x)}{\sqrt {c+d x}} \, dx}{16 b^2}+\frac {d^2 \int \frac {\cosh (a+b x)}{\sqrt {c+d x}} \, dx}{2 b^2}\\ &=-\frac {d \sqrt {c+d x} \cosh (a+b x)}{b^2}-\frac {d \sqrt {c+d x} \cosh ^3(a+b x)}{6 b^2}+\frac {2 (c+d x)^{3/2} \sinh (a+b x)}{3 b}+\frac {(c+d x)^{3/2} \cosh ^2(a+b x) \sinh (a+b x)}{3 b}+\frac {d^2 \int \frac {e^{-i (3 i a+3 i b x)}}{\sqrt {c+d x}} \, dx}{96 b^2}+\frac {d^2 \int \frac {e^{i (3 i a+3 i b x)}}{\sqrt {c+d x}} \, dx}{96 b^2}+\frac {d^2 \int \frac {e^{-i (i a+i b x)}}{\sqrt {c+d x}} \, dx}{32 b^2}+\frac {d^2 \int \frac {e^{i (i a+i b x)}}{\sqrt {c+d x}} \, dx}{32 b^2}+\frac {d^2 \int \frac {e^{-i (i a+i b x)}}{\sqrt {c+d x}} \, dx}{4 b^2}+\frac {d^2 \int \frac {e^{i (i a+i b x)}}{\sqrt {c+d x}} \, dx}{4 b^2}\\ &=-\frac {d \sqrt {c+d x} \cosh (a+b x)}{b^2}-\frac {d \sqrt {c+d x} \cosh ^3(a+b x)}{6 b^2}+\frac {2 (c+d x)^{3/2} \sinh (a+b x)}{3 b}+\frac {(c+d x)^{3/2} \cosh ^2(a+b x) \sinh (a+b x)}{3 b}+\frac {d \operatorname {Subst}\left (\int e^{i \left (3 i a-\frac {3 i b c}{d}\right )-\frac {3 b x^2}{d}} \, dx,x,\sqrt {c+d x}\right )}{48 b^2}+\frac {d \operatorname {Subst}\left (\int e^{-i \left (3 i a-\frac {3 i b c}{d}\right )+\frac {3 b x^2}{d}} \, dx,x,\sqrt {c+d x}\right )}{48 b^2}+\frac {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 )}{16 b^2}+\frac {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 )}{16 b^2}+\frac {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 )}{2 b^2}+\frac {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 )}{2 b^2}\\ &=-\frac {d \sqrt {c+d x} \cosh (a+b x)}{b^2}-\frac {d \sqrt {c+d x} \cosh ^3(a+b x)}{6 b^2}+\frac {9 d^{3/2} e^{-a+\frac {b c}{d}} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{32 b^{5/2}}+\frac {d^{3/2} e^{-3 a+\frac {3 b c}{d}} \sqrt {\frac {\pi }{3}} \text {erf}\left (\frac {\sqrt {3} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{96 b^{5/2}}+\frac {9 d^{3/2} e^{a-\frac {b c}{d}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{32 b^{5/2}}+\frac {d^{3/2} e^{3 a-\frac {3 b c}{d}} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{96 b^{5/2}}+\frac {2 (c+d x)^{3/2} \sinh (a+b x)}{3 b}+\frac {(c+d x)^{3/2} \cosh ^2(a+b x) \sinh (a+b x)}{3 b}\\ \end {align*}
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Mathematica [A] time = 2.01, size = 243, normalized size = 0.75 \[ \frac {d^2 \left (\sqrt {3} \sqrt {-\frac {b (c+d x)}{d}} \Gamma \left (\frac {5}{2},-\frac {3 b (c+d x)}{d}\right ) \left (\sinh \left (3 a-\frac {3 b c}{d}\right )+\cosh \left (3 a-\frac {3 b c}{d}\right )\right )+\left (\cosh \left (a-\frac {b c}{d}\right )-\sinh \left (a-\frac {b c}{d}\right )\right ) \left (81 \sqrt {-\frac {b (c+d x)}{d}} \Gamma \left (\frac {5}{2},-\frac {b (c+d x)}{d}\right ) \left (\sinh \left (2 a-\frac {2 b c}{d}\right )+\cosh \left (2 a-\frac {2 b c}{d}\right )\right )+\sqrt {\frac {b (c+d x)}{d}} \left (\sqrt {3} \Gamma \left (\frac {5}{2},\frac {3 b (c+d x)}{d}\right ) \left (\sinh \left (2 a-\frac {2 b c}{d}\right )-\cosh \left (2 a-\frac {2 b c}{d}\right )\right )-81 \Gamma \left (\frac {5}{2},\frac {b (c+d x)}{d}\right )\right )\right )\right )}{216 b^3 \sqrt {c+d x}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.67, size = 1545, normalized size = 4.74 \[ \text {result too large to display} \]
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 \[ \int {\left (d x + c\right )}^{\frac {3}{2}} \cosh \left (b x + a\right )^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.50, size = 0, normalized size = 0.00 \[ \int \left (d x +c \right )^{\frac {3}{2}} \left (\cosh ^{3}\left (b x +a \right )\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 429, normalized size = 1.32 \[ \frac {\frac {\sqrt {3} \sqrt {\pi } d^{2} \operatorname {erf}\left (\sqrt {3} \sqrt {d x + c} \sqrt {-\frac {b}{d}}\right ) e^{\left (3 \, a - \frac {3 \, b c}{d}\right )}}{b^{2} \sqrt {-\frac {b}{d}}} + \frac {\sqrt {3} \sqrt {\pi } d^{2} \operatorname {erf}\left (\sqrt {3} \sqrt {d x + c} \sqrt {\frac {b}{d}}\right ) e^{\left (-3 \, a + \frac {3 \, b c}{d}\right )}}{b^{2} \sqrt {\frac {b}{d}}} + \frac {81 \, \sqrt {\pi } d^{2} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {-\frac {b}{d}}\right ) e^{\left (a - \frac {b c}{d}\right )}}{b^{2} \sqrt {-\frac {b}{d}}} + \frac {81 \, \sqrt {\pi } d^{2} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {\frac {b}{d}}\right ) e^{\left (-a + \frac {b c}{d}\right )}}{b^{2} \sqrt {\frac {b}{d}}} - \frac {54 \, {\left (2 \, {\left (d x + c\right )}^{\frac {3}{2}} b d e^{\left (\frac {b c}{d}\right )} + 3 \, \sqrt {d x + c} d^{2} e^{\left (\frac {b c}{d}\right )}\right )} e^{\left (-a - \frac {{\left (d x + c\right )} b}{d}\right )}}{b^{2}} - \frac {6 \, {\left (2 \, {\left (d x + c\right )}^{\frac {3}{2}} b d e^{\left (\frac {3 \, b c}{d}\right )} + \sqrt {d x + c} d^{2} e^{\left (\frac {3 \, b c}{d}\right )}\right )} e^{\left (-3 \, a - \frac {3 \, {\left (d x + c\right )} b}{d}\right )}}{b^{2}} + \frac {6 \, {\left (2 \, {\left (d x + c\right )}^{\frac {3}{2}} b d e^{\left (3 \, a\right )} - \sqrt {d x + c} d^{2} e^{\left (3 \, a\right )}\right )} e^{\left (\frac {3 \, {\left (d x + c\right )} b}{d} - \frac {3 \, b c}{d}\right )}}{b^{2}} + \frac {54 \, {\left (2 \, {\left (d x + c\right )}^{\frac {3}{2}} b d e^{a} - 3 \, \sqrt {d x + c} d^{2} e^{a}\right )} e^{\left (\frac {{\left (d x + c\right )} b}{d} - \frac {b c}{d}\right )}}{b^{2}}}{288 \, d} \]
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 \[ \int {\mathrm {cosh}\left (a+b\,x\right )}^3\,{\left (c+d\,x\right )}^{3/2} \,d x \]
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 \[ \int \left (c + d x\right )^{\frac {3}{2}} \cosh ^{3}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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