Optimal. Leaf size=132 \[ \frac {16 b^2 (e (c+d x))^{5/2} \, _3F_2\left (1,\frac {5}{4},\frac {5}{4};\frac {7}{4},\frac {9}{4};-(c+d x)^2\right )}{15 d e^3}-\frac {8 b (e (c+d x))^{3/2} \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};-(c+d x)^2\right ) \left (a+b \sinh ^{-1}(c+d x)\right )}{3 d e^2}+\frac {2 \sqrt {e (c+d x)} \left (a+b \sinh ^{-1}(c+d x)\right )^2}{d e} \]
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Rubi [A] time = 0.20, antiderivative size = 132, normalized size of antiderivative = 1.00, 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 = {5865, 5661, 5762} \[ \frac {16 b^2 (e (c+d x))^{5/2} \, _3F_2\left (1,\frac {5}{4},\frac {5}{4};\frac {7}{4},\frac {9}{4};-(c+d x)^2\right )}{15 d e^3}-\frac {8 b (e (c+d x))^{3/2} \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};-(c+d x)^2\right ) \left (a+b \sinh ^{-1}(c+d x)\right )}{3 d e^2}+\frac {2 \sqrt {e (c+d x)} \left (a+b \sinh ^{-1}(c+d x)\right )^2}{d e} \]
Antiderivative was successfully verified.
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Rule 5661
Rule 5762
Rule 5865
Rubi steps
\begin {align*} \int \frac {\left (a+b \sinh ^{-1}(c+d x)\right )^2}{\sqrt {c e+d e x}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (a+b \sinh ^{-1}(x)\right )^2}{\sqrt {e x}} \, dx,x,c+d x\right )}{d}\\ &=\frac {2 \sqrt {e (c+d x)} \left (a+b \sinh ^{-1}(c+d x)\right )^2}{d e}-\frac {(4 b) \operatorname {Subst}\left (\int \frac {\sqrt {e x} \left (a+b \sinh ^{-1}(x)\right )}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{d e}\\ &=\frac {2 \sqrt {e (c+d x)} \left (a+b \sinh ^{-1}(c+d x)\right )^2}{d e}-\frac {8 b (e (c+d x))^{3/2} \left (a+b \sinh ^{-1}(c+d x)\right ) \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};-(c+d x)^2\right )}{3 d e^2}+\frac {16 b^2 (e (c+d x))^{5/2} \, _3F_2\left (1,\frac {5}{4},\frac {5}{4};\frac {7}{4},\frac {9}{4};-(c+d x)^2\right )}{15 d e^3}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 110, normalized size = 0.83 \[ \frac {2 \sqrt {e (c+d x)} \left (8 b^2 (c+d x)^2 \, _3F_2\left (1,\frac {5}{4},\frac {5}{4};\frac {7}{4},\frac {9}{4};-(c+d x)^2\right )-20 b (c+d x) \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};-(c+d x)^2\right ) \left (a+b \sinh ^{-1}(c+d x)\right )+15 \left (a+b \sinh ^{-1}(c+d x)\right )^2\right )}{15 d e} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.12, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{2} \operatorname {arsinh}\left (d x + c\right )^{2} + 2 \, a b \operatorname {arsinh}\left (d x + c\right ) + a^{2}}{\sqrt {d e x + c e}}, x\right ) \]
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 \[ \int \frac {{\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{2}}{\sqrt {d e x + c e}}\,{d x} \]
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 \[ \int \frac {\left (a +b \arcsinh \left (d x +c \right )\right )^{2}}{\sqrt {d e x +c e}}\, dx \]
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 \[ \frac {2 \, \sqrt {d x + c} b^{2} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )^{2}}{d \sqrt {e}} + \frac {2 \, \sqrt {d e x + c e} a^{2}}{d e} + \int -\frac {2 \, {\left ({\left (2 \, b^{2} c^{2} \sqrt {e} - {\left (c^{2} \sqrt {e} + \sqrt {e}\right )} a b - {\left (a b d^{2} \sqrt {e} - 2 \, b^{2} d^{2} \sqrt {e}\right )} x^{2} - 2 \, {\left (a b c d \sqrt {e} - 2 \, b^{2} c d \sqrt {e}\right )} x\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1} \sqrt {d x + c} - {\left ({\left (a b d^{3} \sqrt {e} - 2 \, b^{2} d^{3} \sqrt {e}\right )} x^{3} + {\left (c^{3} \sqrt {e} + c \sqrt {e}\right )} a b - 2 \, {\left (c^{3} \sqrt {e} + c \sqrt {e}\right )} b^{2} + 3 \, {\left (a b c d^{2} \sqrt {e} - 2 \, b^{2} c d^{2} \sqrt {e}\right )} x^{2} + {\left ({\left (3 \, c^{2} d \sqrt {e} + d \sqrt {e}\right )} a b - 2 \, {\left (3 \, c^{2} d \sqrt {e} + d \sqrt {e}\right )} b^{2}\right )} x\right )} \sqrt {d x + c}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )}{d^{4} e x^{4} + 4 \, c d^{3} e x^{3} + c^{4} e + c^{2} e + {\left (6 \, c^{2} d^{2} e + d^{2} e\right )} x^{2} + 2 \, {\left (2 \, c^{3} d e + c d e\right )} x + {\left (d^{3} e x^{3} + 3 \, c d^{2} e x^{2} + c^{3} e + c e + {\left (3 \, c^{2} d e + d e\right )} x\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}}\,{d x} \]
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 \[ \int \frac {{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^2}{\sqrt {c\,e+d\,e\,x}} \,d x \]
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 \[ \int \frac {\left (a + b \operatorname {asinh}{\left (c + d x \right )}\right )^{2}}{\sqrt {e \left (c + d x\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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