Optimal. Leaf size=169 \[ -\frac {20 b e^2 \sqrt {-1+c+d x} \sqrt {e (c+d x)} \sqrt {1+c+d x}}{147 d}-\frac {4 b \sqrt {-1+c+d x} (e (c+d x))^{5/2} \sqrt {1+c+d x}}{49 d}+\frac {2 (e (c+d x))^{7/2} \left (a+b \cosh ^{-1}(c+d x)\right )}{7 d e}-\frac {20 b e^{5/2} \sqrt {1-c-d x} F\left (\left .\text {ArcSin}\left (\frac {\sqrt {e (c+d x)}}{\sqrt {e}}\right )\right |-1\right )}{147 d \sqrt {-1+c+d x}} \]
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Rubi [A]
time = 0.10, antiderivative size = 169, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {5996, 5883,
104, 12, 118, 117} \begin {gather*} \frac {2 (e (c+d x))^{7/2} \left (a+b \cosh ^{-1}(c+d x)\right )}{7 d e}-\frac {20 b e^{5/2} \sqrt {-c-d x+1} F\left (\left .\text {ArcSin}\left (\frac {\sqrt {e (c+d x)}}{\sqrt {e}}\right )\right |-1\right )}{147 d \sqrt {c+d x-1}}-\frac {20 b e^2 \sqrt {c+d x-1} \sqrt {c+d x+1} \sqrt {e (c+d x)}}{147 d}-\frac {4 b \sqrt {c+d x-1} \sqrt {c+d x+1} (e (c+d x))^{5/2}}{49 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 104
Rule 117
Rule 118
Rule 5883
Rule 5996
Rubi steps
\begin {align*} \int (c e+d e x)^{5/2} \left (a+b \cosh ^{-1}(c+d x)\right ) \, dx &=\frac {\text {Subst}\left (\int (e x)^{5/2} \left (a+b \cosh ^{-1}(x)\right ) \, dx,x,c+d x\right )}{d}\\ &=\frac {2 (e (c+d x))^{7/2} \left (a+b \cosh ^{-1}(c+d x)\right )}{7 d e}-\frac {(2 b) \text {Subst}\left (\int \frac {(e x)^{7/2}}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{7 d e}\\ &=-\frac {4 b \sqrt {-1+c+d x} (e (c+d x))^{5/2} \sqrt {1+c+d x}}{49 d}+\frac {2 (e (c+d x))^{7/2} \left (a+b \cosh ^{-1}(c+d x)\right )}{7 d e}-\frac {(4 b) \text {Subst}\left (\int \frac {5 e^2 (e x)^{3/2}}{2 \sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{49 d e}\\ &=-\frac {4 b \sqrt {-1+c+d x} (e (c+d x))^{5/2} \sqrt {1+c+d x}}{49 d}+\frac {2 (e (c+d x))^{7/2} \left (a+b \cosh ^{-1}(c+d x)\right )}{7 d e}-\frac {(10 b e) \text {Subst}\left (\int \frac {(e x)^{3/2}}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{49 d}\\ &=-\frac {20 b e^2 \sqrt {-1+c+d x} \sqrt {e (c+d x)} \sqrt {1+c+d x}}{147 d}-\frac {4 b \sqrt {-1+c+d x} (e (c+d x))^{5/2} \sqrt {1+c+d x}}{49 d}+\frac {2 (e (c+d x))^{7/2} \left (a+b \cosh ^{-1}(c+d x)\right )}{7 d e}-\frac {(20 b e) \text {Subst}\left (\int \frac {e^2}{2 \sqrt {-1+x} \sqrt {e x} \sqrt {1+x}} \, dx,x,c+d x\right )}{147 d}\\ &=-\frac {20 b e^2 \sqrt {-1+c+d x} \sqrt {e (c+d x)} \sqrt {1+c+d x}}{147 d}-\frac {4 b \sqrt {-1+c+d x} (e (c+d x))^{5/2} \sqrt {1+c+d x}}{49 d}+\frac {2 (e (c+d x))^{7/2} \left (a+b \cosh ^{-1}(c+d x)\right )}{7 d e}-\frac {\left (10 b e^3\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-1+x} \sqrt {e x} \sqrt {1+x}} \, dx,x,c+d x\right )}{147 d}\\ &=-\frac {20 b e^2 \sqrt {-1+c+d x} \sqrt {e (c+d x)} \sqrt {1+c+d x}}{147 d}-\frac {4 b \sqrt {-1+c+d x} (e (c+d x))^{5/2} \sqrt {1+c+d x}}{49 d}+\frac {2 (e (c+d x))^{7/2} \left (a+b \cosh ^{-1}(c+d x)\right )}{7 d e}-\frac {\left (10 b e^3 \sqrt {1-c-d x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x} \sqrt {e x} \sqrt {1+x}} \, dx,x,c+d x\right )}{147 d \sqrt {-1+c+d x}}\\ &=-\frac {20 b e^2 \sqrt {-1+c+d x} \sqrt {e (c+d x)} \sqrt {1+c+d x}}{147 d}-\frac {4 b \sqrt {-1+c+d x} (e (c+d x))^{5/2} \sqrt {1+c+d x}}{49 d}+\frac {2 (e (c+d x))^{7/2} \left (a+b \cosh ^{-1}(c+d x)\right )}{7 d e}-\frac {20 b e^{5/2} \sqrt {1-c-d x} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {e (c+d x)}}{\sqrt {e}}\right )\right |-1\right )}{147 d \sqrt {-1+c+d x}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 0.18, size = 180, normalized size = 1.07 \begin {gather*} \frac {2 (e (c+d x))^{5/2} \left (10 b-4 b (c+d x)^2-6 b (c+d x)^4+21 a \sqrt {-1+c+d x} (c+d x)^3 \sqrt {1+c+d x}+21 b \sqrt {-1+c+d x} (c+d x)^3 \sqrt {1+c+d x} \cosh ^{-1}(c+d x)-10 b \sqrt {1-(c+d x)^2} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};(c+d x)^2\right )\right )}{147 d \sqrt {\frac {-1+c+d x}{c+d x}} (c+d x)^{5/2} \sqrt {1+c+d x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 218, normalized size = 1.29
method | result | size |
derivativedivides | \(\frac {\frac {2 \left (d e x +c e \right )^{\frac {7}{2}} a}{7}+2 b \left (\frac {\left (d e x +c e \right )^{\frac {7}{2}} \mathrm {arccosh}\left (\frac {d e x +c e}{e}\right )}{7}-\frac {2 \left (3 \sqrt {-\frac {1}{e}}\, \left (d e x +c e \right )^{\frac {9}{2}}+2 \sqrt {-\frac {1}{e}}\, e^{2} \left (d e x +c e \right )^{\frac {5}{2}}+5 e^{4} \sqrt {\frac {d e x +c e +e}{e}}\, \sqrt {\frac {-d e x -c e +e}{e}}\, \EllipticF \left (\sqrt {d e x +c e}\, \sqrt {-\frac {1}{e}}, i\right )-5 \sqrt {-\frac {1}{e}}\, e^{4} \sqrt {d e x +c e}\right )}{147 e \sqrt {-\frac {1}{e}}\, \sqrt {\frac {d e x +c e +e}{e}}\, \sqrt {-\frac {-d e x -c e +e}{e}}}\right )}{d e}\) | \(218\) |
default | \(\frac {\frac {2 \left (d e x +c e \right )^{\frac {7}{2}} a}{7}+2 b \left (\frac {\left (d e x +c e \right )^{\frac {7}{2}} \mathrm {arccosh}\left (\frac {d e x +c e}{e}\right )}{7}-\frac {2 \left (3 \sqrt {-\frac {1}{e}}\, \left (d e x +c e \right )^{\frac {9}{2}}+2 \sqrt {-\frac {1}{e}}\, e^{2} \left (d e x +c e \right )^{\frac {5}{2}}+5 e^{4} \sqrt {\frac {d e x +c e +e}{e}}\, \sqrt {\frac {-d e x -c e +e}{e}}\, \EllipticF \left (\sqrt {d e x +c e}\, \sqrt {-\frac {1}{e}}, i\right )-5 \sqrt {-\frac {1}{e}}\, e^{4} \sqrt {d e x +c e}\right )}{147 e \sqrt {-\frac {1}{e}}\, \sqrt {\frac {d e x +c e +e}{e}}\, \sqrt {-\frac {-d e x -c e +e}{e}}}\right )}{d e}\) | \(218\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.11, size = 517, normalized size = 3.06 \begin {gather*} \frac {2 \, {\left (21 \, {\left ({\left (b d^{5} x^{3} + 3 \, b c d^{4} x^{2} + 3 \, b c^{2} d^{3} x + b c^{3} d^{2}\right )} \cosh \left (1\right )^{2} + 2 \, {\left (b d^{5} x^{3} + 3 \, b c d^{4} x^{2} + 3 \, b c^{2} d^{3} x + b c^{3} d^{2}\right )} \cosh \left (1\right ) \sinh \left (1\right ) + {\left (b d^{5} x^{3} + 3 \, b c d^{4} x^{2} + 3 \, b c^{2} d^{3} x + b c^{3} d^{2}\right )} \sinh \left (1\right )^{2}\right )} \sqrt {{\left (d x + c\right )} \cosh \left (1\right ) + {\left (d x + c\right )} \sinh \left (1\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right ) - 10 \, \sqrt {d^{3} \cosh \left (1\right ) + d^{3} \sinh \left (1\right )} {\left (b \cosh \left (1\right )^{2} + 2 \, b \cosh \left (1\right ) \sinh \left (1\right ) + b \sinh \left (1\right )^{2}\right )} {\rm weierstrassPInverse}\left (\frac {4}{d^{2}}, 0, \frac {d x + c}{d}\right ) + {\left (21 \, {\left (a d^{5} x^{3} + 3 \, a c d^{4} x^{2} + 3 \, a c^{2} d^{3} x + a c^{3} d^{2}\right )} \cosh \left (1\right )^{2} + 42 \, {\left (a d^{5} x^{3} + 3 \, a c d^{4} x^{2} + 3 \, a c^{2} d^{3} x + a c^{3} d^{2}\right )} \cosh \left (1\right ) \sinh \left (1\right ) + 21 \, {\left (a d^{5} x^{3} + 3 \, a c d^{4} x^{2} + 3 \, a c^{2} d^{3} x + a c^{3} d^{2}\right )} \sinh \left (1\right )^{2} - 2 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} {\left ({\left (3 \, b d^{4} x^{2} + 6 \, b c d^{3} x + {\left (3 \, b c^{2} + 5 \, b\right )} d^{2}\right )} \cosh \left (1\right )^{2} + 2 \, {\left (3 \, b d^{4} x^{2} + 6 \, b c d^{3} x + {\left (3 \, b c^{2} + 5 \, b\right )} d^{2}\right )} \cosh \left (1\right ) \sinh \left (1\right ) + {\left (3 \, b d^{4} x^{2} + 6 \, b c d^{3} x + {\left (3 \, b c^{2} + 5 \, b\right )} d^{2}\right )} \sinh \left (1\right )^{2}\right )}\right )} \sqrt {{\left (d x + c\right )} \cosh \left (1\right ) + {\left (d x + c\right )} \sinh \left (1\right )}\right )}}{147 \, d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \text {could not integrate} \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 {\left (c\,e+d\,e\,x\right )}^{5/2}\,\left (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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