Optimal. Leaf size=294 \[ \frac {d \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^3}{8 b c \sqrt {c^2 x^2+1}}+\frac {1}{4} x \left (c^2 d x^2+d\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {3}{8} d x \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {b d \left (c^2 x^2+1\right )^{3/2} \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{8 c}-\frac {3 b c d x^2 \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{8 \sqrt {c^2 x^2+1}}+\frac {15}{64} b^2 d x \sqrt {c^2 d x^2+d}+\frac {1}{32} b^2 d x \left (c^2 x^2+1\right ) \sqrt {c^2 d x^2+d}-\frac {9 b^2 d \sqrt {c^2 d x^2+d} \sinh ^{-1}(c x)}{64 c \sqrt {c^2 x^2+1}} \]
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Rubi [A] time = 0.25, antiderivative size = 294, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {5684, 5682, 5675, 5661, 321, 215, 5717, 195} \[ \frac {d \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^3}{8 b c \sqrt {c^2 x^2+1}}+\frac {1}{4} x \left (c^2 d x^2+d\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {3}{8} d x \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {b d \left (c^2 x^2+1\right )^{3/2} \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{8 c}-\frac {3 b c d x^2 \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{8 \sqrt {c^2 x^2+1}}+\frac {15}{64} b^2 d x \sqrt {c^2 d x^2+d}+\frac {1}{32} b^2 d x \left (c^2 x^2+1\right ) \sqrt {c^2 d x^2+d}-\frac {9 b^2 d \sqrt {c^2 d x^2+d} \sinh ^{-1}(c x)}{64 c \sqrt {c^2 x^2+1}} \]
Antiderivative was successfully verified.
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Rule 195
Rule 215
Rule 321
Rule 5661
Rule 5675
Rule 5682
Rule 5684
Rule 5717
Rubi steps
\begin {align*} \int \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx &=\frac {1}{4} x \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{4} (3 d) \int \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx-\frac {\left (b c d \sqrt {d+c^2 d x^2}\right ) \int x \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{2 \sqrt {1+c^2 x^2}}\\ &=-\frac {b d \left (1+c^2 x^2\right )^{3/2} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{8 c}+\frac {3}{8} d x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{4} x \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {\left (3 d \sqrt {d+c^2 d x^2}\right ) \int \frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt {1+c^2 x^2}} \, dx}{8 \sqrt {1+c^2 x^2}}+\frac {\left (b^2 d \sqrt {d+c^2 d x^2}\right ) \int \left (1+c^2 x^2\right )^{3/2} \, dx}{8 \sqrt {1+c^2 x^2}}-\frac {\left (3 b c d \sqrt {d+c^2 d x^2}\right ) \int x \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{4 \sqrt {1+c^2 x^2}}\\ &=\frac {1}{32} b^2 d x \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2}-\frac {3 b c d x^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{8 \sqrt {1+c^2 x^2}}-\frac {b d \left (1+c^2 x^2\right )^{3/2} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{8 c}+\frac {3}{8} d x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{4} x \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {d \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{8 b c \sqrt {1+c^2 x^2}}+\frac {\left (3 b^2 d \sqrt {d+c^2 d x^2}\right ) \int \sqrt {1+c^2 x^2} \, dx}{32 \sqrt {1+c^2 x^2}}+\frac {\left (3 b^2 c^2 d \sqrt {d+c^2 d x^2}\right ) \int \frac {x^2}{\sqrt {1+c^2 x^2}} \, dx}{8 \sqrt {1+c^2 x^2}}\\ &=\frac {15}{64} b^2 d x \sqrt {d+c^2 d x^2}+\frac {1}{32} b^2 d x \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2}-\frac {3 b c d x^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{8 \sqrt {1+c^2 x^2}}-\frac {b d \left (1+c^2 x^2\right )^{3/2} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{8 c}+\frac {3}{8} d x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{4} x \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {d \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{8 b c \sqrt {1+c^2 x^2}}+\frac {\left (3 b^2 d \sqrt {d+c^2 d x^2}\right ) \int \frac {1}{\sqrt {1+c^2 x^2}} \, dx}{64 \sqrt {1+c^2 x^2}}-\frac {\left (3 b^2 d \sqrt {d+c^2 d x^2}\right ) \int \frac {1}{\sqrt {1+c^2 x^2}} \, dx}{16 \sqrt {1+c^2 x^2}}\\ &=\frac {15}{64} b^2 d x \sqrt {d+c^2 d x^2}+\frac {1}{32} b^2 d x \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2}-\frac {9 b^2 d \sqrt {d+c^2 d x^2} \sinh ^{-1}(c x)}{64 c \sqrt {1+c^2 x^2}}-\frac {3 b c d x^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{8 \sqrt {1+c^2 x^2}}-\frac {b d \left (1+c^2 x^2\right )^{3/2} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{8 c}+\frac {3}{8} d x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{4} x \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {d \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{8 b c \sqrt {1+c^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 1.91, size = 329, normalized size = 1.12 \[ \frac {288 a^2 d^{3/2} \sqrt {c^2 x^2+1} \log \left (\sqrt {d} \sqrt {c^2 d x^2+d}+c d x\right )+96 a^2 c d x \sqrt {c^2 x^2+1} \left (2 c^2 x^2+5\right ) \sqrt {c^2 d x^2+d}-192 a b d \sqrt {c^2 d x^2+d} \left (\cosh \left (2 \sinh ^{-1}(c x)\right )-2 \sinh ^{-1}(c x) \left (\sinh ^{-1}(c x)+\sinh \left (2 \sinh ^{-1}(c x)\right )\right )\right )-12 a b d \sqrt {c^2 d x^2+d} \left (8 \sinh ^{-1}(c x)^2-4 \sinh \left (4 \sinh ^{-1}(c x)\right ) \sinh ^{-1}(c x)+\cosh \left (4 \sinh ^{-1}(c x)\right )\right )+32 b^2 d \sqrt {c^2 d x^2+d} \left (4 \sinh ^{-1}(c x)^3+\left (6 \sinh ^{-1}(c x)^2+3\right ) \sinh \left (2 \sinh ^{-1}(c x)\right )-6 \sinh ^{-1}(c x) \cosh \left (2 \sinh ^{-1}(c x)\right )\right )-b^2 d \sqrt {c^2 d x^2+d} \left (32 \sinh ^{-1}(c x)^3-3 \left (8 \sinh ^{-1}(c x)^2+1\right ) \sinh \left (4 \sinh ^{-1}(c x)\right )+12 \sinh ^{-1}(c x) \cosh \left (4 \sinh ^{-1}(c x)\right )\right )}{768 c \sqrt {c^2 x^2+1}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.79, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a^{2} c^{2} d x^{2} + a^{2} d + {\left (b^{2} c^{2} d x^{2} + b^{2} d\right )} \operatorname {arsinh}\left (c x\right )^{2} + 2 \, {\left (a b c^{2} d x^{2} + a b d\right )} \operatorname {arsinh}\left (c x\right )\right )} \sqrt {c^{2} d x^{2} + d}, x\right ) \]
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 = 0.28, size = 709, normalized size = 2.41 \[ \frac {x \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}} a^{2}}{4}+\frac {3 a^{2} d x \sqrt {c^{2} d \,x^{2}+d}}{8}+\frac {3 a^{2} d^{2} \ln \left (\frac {x \,c^{2} d}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{8 \sqrt {c^{2} d}}+\frac {5 b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, d \arcsinh \left (c x \right )^{2} x}{8 \left (c^{2} x^{2}+1\right )}-\frac {17 b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, d \arcsinh \left (c x \right )}{64 c \sqrt {c^{2} x^{2}+1}}+\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, d \,c^{4} \arcsinh \left (c x \right )^{2} x^{5}}{4 c^{2} x^{2}+4}+\frac {7 b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, d \,c^{2} \arcsinh \left (c x \right )^{2} x^{3}}{8 \left (c^{2} x^{2}+1\right )}+\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, d \,c^{4} x^{5}}{32 c^{2} x^{2}+32}+\frac {19 b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, d \,c^{2} x^{3}}{64 \left (c^{2} x^{2}+1\right )}+\frac {17 b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, d x}{64 \left (c^{2} x^{2}+1\right )}+\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right )^{3} d}{8 \sqrt {c^{2} x^{2}+1}\, c}-\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, d \,c^{3} \arcsinh \left (c x \right ) x^{4}}{8 \sqrt {c^{2} x^{2}+1}}-\frac {5 b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, d c \arcsinh \left (c x \right ) x^{2}}{8 \sqrt {c^{2} x^{2}+1}}+\frac {7 a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, d \,c^{2} \arcsinh \left (c x \right ) x^{3}}{4 \left (c^{2} x^{2}+1\right )}-\frac {5 a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, d c \,x^{2}}{8 \sqrt {c^{2} x^{2}+1}}+\frac {5 a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, d \arcsinh \left (c x \right ) x}{4 \left (c^{2} x^{2}+1\right )}+\frac {a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, d \,c^{4} \arcsinh \left (c x \right ) x^{5}}{2 c^{2} x^{2}+2}-\frac {a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, d \,c^{3} x^{4}}{8 \sqrt {c^{2} x^{2}+1}}+\frac {3 a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right )^{2} d}{8 \sqrt {c^{2} x^{2}+1}\, c}-\frac {17 a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, d}{64 c \sqrt {c^{2} x^{2}+1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
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 \[ \int {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,{\left (d\,c^2\,x^2+d\right )}^{3/2} \,d x \]
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 \[ \int \left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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