Optimal. Leaf size=153 \[ -\frac {16 b^2 (e (c+d x))^{13/2} \, _3F_2\left (1,\frac {13}{4},\frac {13}{4};\frac {15}{4},\frac {17}{4};(c+d x)^2\right )}{1287 d e^3}-\frac {8 b \sqrt {-c-d x+1} (e (c+d x))^{11/2} \, _2F_1\left (\frac {1}{2},\frac {11}{4};\frac {15}{4};(c+d x)^2\right ) \left (a+b \cosh ^{-1}(c+d x)\right )}{99 d e^2 \sqrt {c+d x-1}}+\frac {2 (e (c+d x))^{9/2} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{9 d e} \]
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Rubi [A] time = 0.32, antiderivative size = 165, normalized size of antiderivative = 1.08, number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {5866, 5662, 5763} \[ -\frac {16 b^2 (e (c+d x))^{13/2} \, _3F_2\left (1,\frac {13}{4},\frac {13}{4};\frac {15}{4},\frac {17}{4};(c+d x)^2\right )}{1287 d e^3}-\frac {8 b \sqrt {1-(c+d x)^2} (e (c+d x))^{11/2} \, _2F_1\left (\frac {1}{2},\frac {11}{4};\frac {15}{4};(c+d x)^2\right ) \left (a+b \cosh ^{-1}(c+d x)\right )}{99 d e^2 \sqrt {c+d x-1} \sqrt {c+d x+1}}+\frac {2 (e (c+d x))^{9/2} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{9 d e} \]
Antiderivative was successfully verified.
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Rule 5662
Rule 5763
Rule 5866
Rubi steps
\begin {align*} \int (c e+d e x)^{7/2} \left (a+b \cosh ^{-1}(c+d x)\right )^2 \, dx &=\frac {\operatorname {Subst}\left (\int (e x)^{7/2} \left (a+b \cosh ^{-1}(x)\right )^2 \, dx,x,c+d x\right )}{d}\\ &=\frac {2 (e (c+d x))^{9/2} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{9 d e}-\frac {(4 b) \operatorname {Subst}\left (\int \frac {(e x)^{9/2} \left (a+b \cosh ^{-1}(x)\right )}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{9 d e}\\ &=\frac {2 (e (c+d x))^{9/2} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{9 d e}-\frac {8 b (e (c+d x))^{11/2} \sqrt {1-(c+d x)^2} \left (a+b \cosh ^{-1}(c+d x)\right ) \, _2F_1\left (\frac {1}{2},\frac {11}{4};\frac {15}{4};(c+d x)^2\right )}{99 d e^2 \sqrt {-1+c+d x} \sqrt {1+c+d x}}-\frac {16 b^2 (e (c+d x))^{13/2} \, _3F_2\left (1,\frac {13}{4},\frac {13}{4};\frac {15}{4},\frac {17}{4};(c+d x)^2\right )}{1287 d e^3}\\ \end {align*}
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Mathematica [A] time = 0.54, size = 140, normalized size = 0.92 \[ \frac {2 (e (c+d x))^{9/2} \left (143 \left (a+b \cosh ^{-1}(c+d x)\right )^2-4 b (c+d x) \left (2 b (c+d x) \, _3F_2\left (1,\frac {13}{4},\frac {13}{4};\frac {15}{4},\frac {17}{4};(c+d x)^2\right )+\frac {13 \sqrt {1-(c+d x)^2} \, _2F_1\left (\frac {1}{2},\frac {11}{4};\frac {15}{4};(c+d x)^2\right ) \left (a+b \cosh ^{-1}(c+d x)\right )}{\sqrt {c+d x-1} \sqrt {c+d x+1}}\right )\right )}{1287 d e} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.78, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a^{2} d^{3} e^{3} x^{3} + 3 \, a^{2} c d^{2} e^{3} x^{2} + 3 \, a^{2} c^{2} d e^{3} x + a^{2} c^{3} e^{3} + {\left (b^{2} d^{3} e^{3} x^{3} + 3 \, b^{2} c d^{2} e^{3} x^{2} + 3 \, b^{2} c^{2} d e^{3} x + b^{2} c^{3} e^{3}\right )} \operatorname {arcosh}\left (d x + c\right )^{2} + 2 \, {\left (a b d^{3} e^{3} x^{3} + 3 \, a b c d^{2} e^{3} x^{2} + 3 \, a b c^{2} d e^{3} x + a b c^{3} e^{3}\right )} \operatorname {arcosh}\left (d x + c\right )\right )} \sqrt {d e x + c e}, x\right ) \]
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 \[ \int {\left (d e x + c e\right )}^{\frac {7}{2}} {\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F(-2)] time = 180.00, size = 0, normalized size = 0.00 \[ \int \left (d e x +c e \right )^{\frac {7}{2}} \left (a +b \,\mathrm {arccosh}\left (d x +c \right )\right )^{2}\, dx \]
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 \[ \frac {2 \, {\left (d e x + c e\right )}^{\frac {9}{2}} a^{2}}{9 \, d e} + \frac {2 \, {\left (b^{2} d^{4} e^{\frac {7}{2}} x^{4} + 4 \, b^{2} c d^{3} e^{\frac {7}{2}} x^{3} + 6 \, b^{2} c^{2} d^{2} e^{\frac {7}{2}} x^{2} + 4 \, b^{2} c^{3} d e^{\frac {7}{2}} x + b^{2} c^{4} e^{\frac {7}{2}}\right )} \sqrt {d x + c} \log \left (d x + \sqrt {d x + c + 1} \sqrt {d x + c - 1} + c\right )^{2}}{9 \, d} + \int -\frac {2 \, {\left ({\left (2 \, b^{2} c^{5} e^{\frac {7}{2}} - {\left (9 \, a b d^{5} e^{\frac {7}{2}} - 2 \, b^{2} d^{5} e^{\frac {7}{2}}\right )} x^{5} - 5 \, {\left (9 \, a b c d^{4} e^{\frac {7}{2}} - 2 \, b^{2} c d^{4} e^{\frac {7}{2}}\right )} x^{4} + {\left (20 \, b^{2} c^{2} d^{3} e^{\frac {7}{2}} - 9 \, {\left (10 \, c^{2} d^{3} e^{\frac {7}{2}} - d^{3} e^{\frac {7}{2}}\right )} a b\right )} x^{3} - 9 \, {\left (c^{5} e^{\frac {7}{2}} - c^{3} e^{\frac {7}{2}}\right )} a b + {\left (20 \, b^{2} c^{3} d^{2} e^{\frac {7}{2}} - 9 \, {\left (10 \, c^{3} d^{2} e^{\frac {7}{2}} - 3 \, c d^{2} e^{\frac {7}{2}}\right )} a b\right )} x^{2} + {\left (10 \, b^{2} c^{4} d e^{\frac {7}{2}} - 9 \, {\left (5 \, c^{4} d e^{\frac {7}{2}} - 3 \, c^{2} d e^{\frac {7}{2}}\right )} a b\right )} x\right )} \sqrt {d x + c + 1} \sqrt {d x + c} \sqrt {d x + c - 1} - {\left ({\left (9 \, a b d^{6} e^{\frac {7}{2}} - 2 \, b^{2} d^{6} e^{\frac {7}{2}}\right )} x^{6} + 6 \, {\left (9 \, a b c d^{5} e^{\frac {7}{2}} - 2 \, b^{2} c d^{5} e^{\frac {7}{2}}\right )} x^{5} + {\left (9 \, {\left (15 \, c^{2} d^{4} e^{\frac {7}{2}} - d^{4} e^{\frac {7}{2}}\right )} a b - 2 \, {\left (15 \, c^{2} d^{4} e^{\frac {7}{2}} - d^{4} e^{\frac {7}{2}}\right )} b^{2}\right )} x^{4} + 4 \, {\left (9 \, {\left (5 \, c^{3} d^{3} e^{\frac {7}{2}} - c d^{3} e^{\frac {7}{2}}\right )} a b - 2 \, {\left (5 \, c^{3} d^{3} e^{\frac {7}{2}} - c d^{3} e^{\frac {7}{2}}\right )} b^{2}\right )} x^{3} + 9 \, {\left (c^{6} e^{\frac {7}{2}} - c^{4} e^{\frac {7}{2}}\right )} a b - 2 \, {\left (c^{6} e^{\frac {7}{2}} - c^{4} e^{\frac {7}{2}}\right )} b^{2} + 3 \, {\left (9 \, {\left (5 \, c^{4} d^{2} e^{\frac {7}{2}} - 2 \, c^{2} d^{2} e^{\frac {7}{2}}\right )} a b - 2 \, {\left (5 \, c^{4} d^{2} e^{\frac {7}{2}} - 2 \, c^{2} d^{2} e^{\frac {7}{2}}\right )} b^{2}\right )} x^{2} + 2 \, {\left (9 \, {\left (3 \, c^{5} d e^{\frac {7}{2}} - 2 \, c^{3} d e^{\frac {7}{2}}\right )} a b - 2 \, {\left (3 \, c^{5} d e^{\frac {7}{2}} - 2 \, c^{3} d e^{\frac {7}{2}}\right )} b^{2}\right )} x\right )} \sqrt {d x + c}\right )} \log \left (d x + \sqrt {d x + c + 1} \sqrt {d x + c - 1} + c\right )}{9 \, {\left (d^{3} x^{3} + 3 \, c d^{2} x^{2} + c^{3} + {\left (d^{2} x^{2} + 2 \, c d x + c^{2} - 1\right )} \sqrt {d x + c + 1} \sqrt {d x + c - 1} + {\left (3 \, c^{2} d - d\right )} x - c\right )}}\,{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 = -0.01 \[ \int {\left (c\,e+d\,e\,x\right )}^{7/2}\,{\left (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right )}^2 \,d x \]
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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