Optimal. Leaf size=494 \[ \frac {1}{6} d^2 f x \left (c^2 x^2+1\right )^2 \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )+\frac {5}{16} d^2 f x \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )+\frac {5}{24} d^2 f x \left (c^2 x^2+1\right ) \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )+\frac {5 d^2 f \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{32 b c \sqrt {c^2 x^2+1}}+\frac {d^2 g \left (c^2 x^2+1\right )^3 \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{7 c^2}-\frac {25 b c d^2 f x^2 \sqrt {c^2 d x^2+d}}{96 \sqrt {c^2 x^2+1}}-\frac {b d^2 f \left (c^2 x^2+1\right )^{5/2} \sqrt {c^2 d x^2+d}}{36 c}-\frac {b d^2 g x \sqrt {c^2 d x^2+d}}{7 c \sqrt {c^2 x^2+1}}-\frac {b c d^2 g x^3 \sqrt {c^2 d x^2+d}}{7 \sqrt {c^2 x^2+1}}-\frac {b c^5 d^2 g x^7 \sqrt {c^2 d x^2+d}}{49 \sqrt {c^2 x^2+1}}-\frac {5 b c^3 d^2 f x^4 \sqrt {c^2 d x^2+d}}{96 \sqrt {c^2 x^2+1}}-\frac {3 b c^3 d^2 g x^5 \sqrt {c^2 d x^2+d}}{35 \sqrt {c^2 x^2+1}} \]
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Rubi [A] time = 0.40, antiderivative size = 494, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 10, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {5835, 5821, 5684, 5682, 5675, 30, 14, 261, 5717, 194} \[ \frac {1}{6} d^2 f x \left (c^2 x^2+1\right )^2 \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )+\frac {5}{16} d^2 f x \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )+\frac {5}{24} d^2 f x \left (c^2 x^2+1\right ) \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )+\frac {5 d^2 f \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{32 b c \sqrt {c^2 x^2+1}}+\frac {d^2 g \left (c^2 x^2+1\right )^3 \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{7 c^2}-\frac {5 b c^3 d^2 f x^4 \sqrt {c^2 d x^2+d}}{96 \sqrt {c^2 x^2+1}}-\frac {25 b c d^2 f x^2 \sqrt {c^2 d x^2+d}}{96 \sqrt {c^2 x^2+1}}-\frac {b d^2 f \left (c^2 x^2+1\right )^{5/2} \sqrt {c^2 d x^2+d}}{36 c}-\frac {b c^5 d^2 g x^7 \sqrt {c^2 d x^2+d}}{49 \sqrt {c^2 x^2+1}}-\frac {3 b c^3 d^2 g x^5 \sqrt {c^2 d x^2+d}}{35 \sqrt {c^2 x^2+1}}-\frac {b c d^2 g x^3 \sqrt {c^2 d x^2+d}}{7 \sqrt {c^2 x^2+1}}-\frac {b d^2 g x \sqrt {c^2 d x^2+d}}{7 c \sqrt {c^2 x^2+1}} \]
Antiderivative was successfully verified.
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Rule 14
Rule 30
Rule 194
Rule 261
Rule 5675
Rule 5682
Rule 5684
Rule 5717
Rule 5821
Rule 5835
Rubi steps
\begin {align*} \int (f+g x) \left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx &=\frac {\left (d^2 \sqrt {d+c^2 d x^2}\right ) \int (f+g x) \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{\sqrt {1+c^2 x^2}}\\ &=\frac {\left (d^2 \sqrt {d+c^2 d x^2}\right ) \int \left (f \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )+g x \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )\right ) \, dx}{\sqrt {1+c^2 x^2}}\\ &=\frac {\left (d^2 f \sqrt {d+c^2 d x^2}\right ) \int \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{\sqrt {1+c^2 x^2}}+\frac {\left (d^2 g \sqrt {d+c^2 d x^2}\right ) \int x \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{\sqrt {1+c^2 x^2}}\\ &=\frac {1}{6} d^2 f x \left (1+c^2 x^2\right )^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {d^2 g \left (1+c^2 x^2\right )^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{7 c^2}+\frac {\left (5 d^2 f \sqrt {d+c^2 d x^2}\right ) \int \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{6 \sqrt {1+c^2 x^2}}-\frac {\left (b c d^2 f \sqrt {d+c^2 d x^2}\right ) \int x \left (1+c^2 x^2\right )^2 \, dx}{6 \sqrt {1+c^2 x^2}}-\frac {\left (b d^2 g \sqrt {d+c^2 d x^2}\right ) \int \left (1+c^2 x^2\right )^3 \, dx}{7 c \sqrt {1+c^2 x^2}}\\ &=-\frac {b d^2 f \left (1+c^2 x^2\right )^{5/2} \sqrt {d+c^2 d x^2}}{36 c}+\frac {5}{24} d^2 f x \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{6} d^2 f x \left (1+c^2 x^2\right )^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {d^2 g \left (1+c^2 x^2\right )^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{7 c^2}+\frac {\left (5 d^2 f \sqrt {d+c^2 d x^2}\right ) \int \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{8 \sqrt {1+c^2 x^2}}-\frac {\left (5 b c d^2 f \sqrt {d+c^2 d x^2}\right ) \int x \left (1+c^2 x^2\right ) \, dx}{24 \sqrt {1+c^2 x^2}}-\frac {\left (b d^2 g \sqrt {d+c^2 d x^2}\right ) \int \left (1+3 c^2 x^2+3 c^4 x^4+c^6 x^6\right ) \, dx}{7 c \sqrt {1+c^2 x^2}}\\ &=-\frac {b d^2 g x \sqrt {d+c^2 d x^2}}{7 c \sqrt {1+c^2 x^2}}-\frac {b c d^2 g x^3 \sqrt {d+c^2 d x^2}}{7 \sqrt {1+c^2 x^2}}-\frac {3 b c^3 d^2 g x^5 \sqrt {d+c^2 d x^2}}{35 \sqrt {1+c^2 x^2}}-\frac {b c^5 d^2 g x^7 \sqrt {d+c^2 d x^2}}{49 \sqrt {1+c^2 x^2}}-\frac {b d^2 f \left (1+c^2 x^2\right )^{5/2} \sqrt {d+c^2 d x^2}}{36 c}+\frac {5}{16} d^2 f x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {5}{24} d^2 f x \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{6} d^2 f x \left (1+c^2 x^2\right )^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {d^2 g \left (1+c^2 x^2\right )^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{7 c^2}+\frac {\left (5 d^2 f \sqrt {d+c^2 d x^2}\right ) \int \frac {a+b \sinh ^{-1}(c x)}{\sqrt {1+c^2 x^2}} \, dx}{16 \sqrt {1+c^2 x^2}}-\frac {\left (5 b c d^2 f \sqrt {d+c^2 d x^2}\right ) \int \left (x+c^2 x^3\right ) \, dx}{24 \sqrt {1+c^2 x^2}}-\frac {\left (5 b c d^2 f \sqrt {d+c^2 d x^2}\right ) \int x \, dx}{16 \sqrt {1+c^2 x^2}}\\ &=-\frac {b d^2 g x \sqrt {d+c^2 d x^2}}{7 c \sqrt {1+c^2 x^2}}-\frac {25 b c d^2 f x^2 \sqrt {d+c^2 d x^2}}{96 \sqrt {1+c^2 x^2}}-\frac {b c d^2 g x^3 \sqrt {d+c^2 d x^2}}{7 \sqrt {1+c^2 x^2}}-\frac {5 b c^3 d^2 f x^4 \sqrt {d+c^2 d x^2}}{96 \sqrt {1+c^2 x^2}}-\frac {3 b c^3 d^2 g x^5 \sqrt {d+c^2 d x^2}}{35 \sqrt {1+c^2 x^2}}-\frac {b c^5 d^2 g x^7 \sqrt {d+c^2 d x^2}}{49 \sqrt {1+c^2 x^2}}-\frac {b d^2 f \left (1+c^2 x^2\right )^{5/2} \sqrt {d+c^2 d x^2}}{36 c}+\frac {5}{16} d^2 f x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {5}{24} d^2 f x \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{6} d^2 f x \left (1+c^2 x^2\right )^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {d^2 g \left (1+c^2 x^2\right )^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{7 c^2}+\frac {5 d^2 f \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{32 b c \sqrt {1+c^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 1.34, size = 656, normalized size = 1.33 \[ \frac {d^2 \left (388080 a c^2 f x \sqrt {c^2 x^2+1} \sqrt {c^2 d x^2+d}+176400 a c \sqrt {d} f \sqrt {c^2 x^2+1} \log \left (\sqrt {d} \sqrt {c^2 d x^2+d}+c d x\right )+241920 a c^2 g x^2 \sqrt {c^2 x^2+1} \sqrt {c^2 d x^2+d}+80640 a g \sqrt {c^2 x^2+1} \sqrt {c^2 d x^2+d}+94080 a c^6 f x^5 \sqrt {c^2 x^2+1} \sqrt {c^2 d x^2+d}+80640 a c^6 g x^6 \sqrt {c^2 x^2+1} \sqrt {c^2 d x^2+d}+305760 a c^4 f x^3 \sqrt {c^2 x^2+1} \sqrt {c^2 d x^2+d}+241920 a c^4 g x^4 \sqrt {c^2 x^2+1} \sqrt {c^2 d x^2+d}+88200 b c f \sqrt {c^2 d x^2+d} \sinh ^{-1}(c x)^2-66150 b c f \sqrt {c^2 d x^2+d} \cosh \left (2 \sinh ^{-1}(c x)\right )-6615 b c f \sqrt {c^2 d x^2+d} \cosh \left (4 \sinh ^{-1}(c x)\right )-490 b c f \sqrt {c^2 d x^2+d} \cosh \left (6 \sinh ^{-1}(c x)\right )-80640 b c g x \sqrt {c^2 d x^2+d}-11520 b c^7 g x^7 \sqrt {c^2 d x^2+d}-48384 b c^5 g x^5 \sqrt {c^2 d x^2+d}-80640 b c^3 g x^3 \sqrt {c^2 d x^2+d}+420 b \sqrt {c^2 d x^2+d} \sinh ^{-1}(c x) \left (576 c^2 g x^2 \sqrt {c^2 x^2+1}+192 g \sqrt {c^2 x^2+1}+192 c^6 g x^6 \sqrt {c^2 x^2+1}+576 c^4 g x^4 \sqrt {c^2 x^2+1}+315 c f \sinh \left (2 \sinh ^{-1}(c x)\right )+63 c f \sinh \left (4 \sinh ^{-1}(c x)\right )+7 c f \sinh \left (6 \sinh ^{-1}(c x)\right )\right )\right )}{564480 c^2 \sqrt {c^2 x^2+1}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.56, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a c^{4} d^{2} g x^{5} + a c^{4} d^{2} f x^{4} + 2 \, a c^{2} d^{2} g x^{3} + 2 \, a c^{2} d^{2} f x^{2} + a d^{2} g x + a d^{2} f + {\left (b c^{4} d^{2} g x^{5} + b c^{4} d^{2} f x^{4} + 2 \, b c^{2} d^{2} g x^{3} + 2 \, b c^{2} d^{2} f x^{2} + b d^{2} g x + b d^{2} f\right )} \operatorname {arsinh}\left (c x\right )\right )} \sqrt {c^{2} d x^{2} + d}, x\right ) \]
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 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.57, size = 805, normalized size = 1.63 \[ \frac {a g \left (c^{2} d \,x^{2}+d \right )^{\frac {7}{2}}}{7 c^{2} d}+\frac {a f x \left (c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{6}+\frac {5 a f d x \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{24}+\frac {5 a f \,d^{2} x \sqrt {c^{2} d \,x^{2}+d}}{16}+\frac {5 a f \,d^{3} \ln \left (\frac {x \,c^{2} d}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{16 \sqrt {c^{2} d}}+\frac {5 b \sqrt {d \left (c^{2} x^{2}+1\right )}\, f \arcsinh \left (c x \right )^{2} d^{2}}{32 \sqrt {c^{2} x^{2}+1}\, c}+\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, g \,d^{2} c^{6} \arcsinh \left (c x \right ) x^{8}}{7 c^{2} x^{2}+7}-\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, g \,d^{2} c^{5} x^{7}}{49 \sqrt {c^{2} x^{2}+1}}+\frac {4 b \sqrt {d \left (c^{2} x^{2}+1\right )}\, g \,d^{2} c^{4} \arcsinh \left (c x \right ) x^{6}}{7 \left (c^{2} x^{2}+1\right )}-\frac {3 b \sqrt {d \left (c^{2} x^{2}+1\right )}\, g \,d^{2} c^{3} x^{5}}{35 \sqrt {c^{2} x^{2}+1}}+\frac {6 b \sqrt {d \left (c^{2} x^{2}+1\right )}\, g \,d^{2} c^{2} \arcsinh \left (c x \right ) x^{4}}{7 \left (c^{2} x^{2}+1\right )}-\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, g \,d^{2} c \,x^{3}}{7 \sqrt {c^{2} x^{2}+1}}+\frac {4 b \sqrt {d \left (c^{2} x^{2}+1\right )}\, g \,d^{2} \arcsinh \left (c x \right ) x^{2}}{7 \left (c^{2} x^{2}+1\right )}-\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, g \,d^{2} x}{7 c \sqrt {c^{2} x^{2}+1}}+\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, f \,d^{2} c^{6} \arcsinh \left (c x \right ) x^{7}}{6 c^{2} x^{2}+6}-\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, f \,d^{2} c^{5} x^{6}}{36 \sqrt {c^{2} x^{2}+1}}+\frac {17 b \sqrt {d \left (c^{2} x^{2}+1\right )}\, f \,d^{2} c^{4} \arcsinh \left (c x \right ) x^{5}}{24 \left (c^{2} x^{2}+1\right )}-\frac {13 b \sqrt {d \left (c^{2} x^{2}+1\right )}\, f \,d^{2} c^{3} x^{4}}{96 \sqrt {c^{2} x^{2}+1}}+\frac {59 b \sqrt {d \left (c^{2} x^{2}+1\right )}\, f \,d^{2} c^{2} \arcsinh \left (c x \right ) x^{3}}{48 \left (c^{2} x^{2}+1\right )}-\frac {11 b \sqrt {d \left (c^{2} x^{2}+1\right )}\, f \,d^{2} c \,x^{2}}{32 \sqrt {c^{2} x^{2}+1}}+\frac {11 b \sqrt {d \left (c^{2} x^{2}+1\right )}\, f \,d^{2} \arcsinh \left (c x \right ) x}{16 \left (c^{2} x^{2}+1\right )}+\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, g \,d^{2} \arcsinh \left (c x \right )}{7 c^{2} \left (c^{2} x^{2}+1\right )}-\frac {299 b \sqrt {d \left (c^{2} x^{2}+1\right )}\, f \,d^{2}}{2304 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 \[ \int \left (f+g\,x\right )\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (d\,c^2\,x^2+d\right )}^{5/2} \,d x \]
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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