Optimal. Leaf size=183 \[ -\frac {a+b \sinh ^{-1}(c x)}{3 e (d+e x)^3}-\frac {b c \sqrt {c^2 x^2+1}}{6 \left (c^2 d^2+e^2\right ) (d+e x)^2}-\frac {b c^3 d \sqrt {c^2 x^2+1}}{2 \left (c^2 d^2+e^2\right )^2 (d+e x)}-\frac {b c^3 \left (2 c^2 d^2-e^2\right ) \tanh ^{-1}\left (\frac {e-c^2 d x}{\sqrt {c^2 x^2+1} \sqrt {c^2 d^2+e^2}}\right )}{6 e \left (c^2 d^2+e^2\right )^{5/2}} \]
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Rubi [A] time = 0.14, antiderivative size = 183, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {5801, 745, 807, 725, 206} \[ -\frac {a+b \sinh ^{-1}(c x)}{3 e (d+e x)^3}-\frac {b c^3 d \sqrt {c^2 x^2+1}}{2 \left (c^2 d^2+e^2\right )^2 (d+e x)}-\frac {b c \sqrt {c^2 x^2+1}}{6 \left (c^2 d^2+e^2\right ) (d+e x)^2}-\frac {b c^3 \left (2 c^2 d^2-e^2\right ) \tanh ^{-1}\left (\frac {e-c^2 d x}{\sqrt {c^2 x^2+1} \sqrt {c^2 d^2+e^2}}\right )}{6 e \left (c^2 d^2+e^2\right )^{5/2}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 725
Rule 745
Rule 807
Rule 5801
Rubi steps
\begin {align*} \int \frac {a+b \sinh ^{-1}(c x)}{(d+e x)^4} \, dx &=-\frac {a+b \sinh ^{-1}(c x)}{3 e (d+e x)^3}+\frac {(b c) \int \frac {1}{(d+e x)^3 \sqrt {1+c^2 x^2}} \, dx}{3 e}\\ &=-\frac {b c \sqrt {1+c^2 x^2}}{6 \left (c^2 d^2+e^2\right ) (d+e x)^2}-\frac {a+b \sinh ^{-1}(c x)}{3 e (d+e x)^3}-\frac {\left (b c^3\right ) \int \frac {-2 d+e x}{(d+e x)^2 \sqrt {1+c^2 x^2}} \, dx}{6 e \left (c^2 d^2+e^2\right )}\\ &=-\frac {b c \sqrt {1+c^2 x^2}}{6 \left (c^2 d^2+e^2\right ) (d+e x)^2}-\frac {b c^3 d \sqrt {1+c^2 x^2}}{2 \left (c^2 d^2+e^2\right )^2 (d+e x)}-\frac {a+b \sinh ^{-1}(c x)}{3 e (d+e x)^3}+\frac {\left (b c^3 \left (2 c^2 d^2-e^2\right )\right ) \int \frac {1}{(d+e x) \sqrt {1+c^2 x^2}} \, dx}{6 e \left (c^2 d^2+e^2\right )^2}\\ &=-\frac {b c \sqrt {1+c^2 x^2}}{6 \left (c^2 d^2+e^2\right ) (d+e x)^2}-\frac {b c^3 d \sqrt {1+c^2 x^2}}{2 \left (c^2 d^2+e^2\right )^2 (d+e x)}-\frac {a+b \sinh ^{-1}(c x)}{3 e (d+e x)^3}-\frac {\left (b c^3 \left (2 c^2 d^2-e^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{c^2 d^2+e^2-x^2} \, dx,x,\frac {e-c^2 d x}{\sqrt {1+c^2 x^2}}\right )}{6 e \left (c^2 d^2+e^2\right )^2}\\ &=-\frac {b c \sqrt {1+c^2 x^2}}{6 \left (c^2 d^2+e^2\right ) (d+e x)^2}-\frac {b c^3 d \sqrt {1+c^2 x^2}}{2 \left (c^2 d^2+e^2\right )^2 (d+e x)}-\frac {a+b \sinh ^{-1}(c x)}{3 e (d+e x)^3}-\frac {b c^3 \left (2 c^2 d^2-e^2\right ) \tanh ^{-1}\left (\frac {e-c^2 d x}{\sqrt {c^2 d^2+e^2} \sqrt {1+c^2 x^2}}\right )}{6 e \left (c^2 d^2+e^2\right )^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.43, size = 205, normalized size = 1.12 \[ \frac {1}{6} \left (-\frac {2 a}{e (d+e x)^3}-\frac {b c \sqrt {c^2 x^2+1} \left (c^2 d (4 d+3 e x)+e^2\right )}{\left (c^2 d^2+e^2\right )^2 (d+e x)^2}+\frac {b c^3 \left (e^2-2 c^2 d^2\right ) \log \left (\sqrt {c^2 x^2+1} \sqrt {c^2 d^2+e^2}+c^2 (-d) x+e\right )}{e \left (c^2 d^2+e^2\right )^{5/2}}-\frac {b c^3 \left (e^2-2 c^2 d^2\right ) \log (d+e x)}{e \left (c^2 d^2+e^2\right )^{5/2}}-\frac {2 b \sinh ^{-1}(c x)}{e (d+e x)^3}\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 1.67, size = 977, normalized size = 5.34 \[ -\frac {{\left (2 \, a + 3 \, b\right )} c^{6} d^{9} + 3 \, {\left (2 \, a + b\right )} c^{4} d^{7} e^{2} + 6 \, a c^{2} d^{5} e^{4} + 2 \, a d^{3} e^{6} + 3 \, {\left (b c^{6} d^{6} e^{3} + b c^{4} d^{4} e^{5}\right )} x^{3} + 9 \, {\left (b c^{6} d^{7} e^{2} + b c^{4} d^{5} e^{4}\right )} x^{2} + {\left (2 \, b c^{5} d^{8} - b c^{3} d^{6} e^{2} + {\left (2 \, b c^{5} d^{5} e^{3} - b c^{3} d^{3} e^{5}\right )} x^{3} + 3 \, {\left (2 \, b c^{5} d^{6} e^{2} - b c^{3} d^{4} e^{4}\right )} x^{2} + 3 \, {\left (2 \, b c^{5} d^{7} e - b c^{3} d^{5} e^{3}\right )} x\right )} \sqrt {c^{2} d^{2} + e^{2}} \log \left (-\frac {c^{3} d^{2} x - c d e - \sqrt {c^{2} d^{2} + e^{2}} {\left (c^{2} d x - e\right )} + {\left (c^{2} d^{2} - \sqrt {c^{2} d^{2} + e^{2}} c d + e^{2}\right )} \sqrt {c^{2} x^{2} + 1}}{e x + d}\right ) + 9 \, {\left (b c^{6} d^{8} e + b c^{4} d^{6} e^{3}\right )} x - 2 \, {\left ({\left (b c^{6} d^{6} e^{3} + 3 \, b c^{4} d^{4} e^{5} + 3 \, b c^{2} d^{2} e^{7} + b e^{9}\right )} x^{3} + 3 \, {\left (b c^{6} d^{7} e^{2} + 3 \, b c^{4} d^{5} e^{4} + 3 \, b c^{2} d^{3} e^{6} + b d e^{8}\right )} x^{2} + 3 \, {\left (b c^{6} d^{8} e + 3 \, b c^{4} d^{6} e^{3} + 3 \, b c^{2} d^{4} e^{5} + b d^{2} e^{7}\right )} x\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - 2 \, {\left (b c^{6} d^{9} + 3 \, b c^{4} d^{7} e^{2} + 3 \, b c^{2} d^{5} e^{4} + b d^{3} e^{6} + {\left (b c^{6} d^{6} e^{3} + 3 \, b c^{4} d^{4} e^{5} + 3 \, b c^{2} d^{2} e^{7} + b e^{9}\right )} x^{3} + 3 \, {\left (b c^{6} d^{7} e^{2} + 3 \, b c^{4} d^{5} e^{4} + 3 \, b c^{2} d^{3} e^{6} + b d e^{8}\right )} x^{2} + 3 \, {\left (b c^{6} d^{8} e + 3 \, b c^{4} d^{6} e^{3} + 3 \, b c^{2} d^{4} e^{5} + b d^{2} e^{7}\right )} x\right )} \log \left (-c x + \sqrt {c^{2} x^{2} + 1}\right ) + {\left (4 \, b c^{5} d^{8} e + 5 \, b c^{3} d^{6} e^{3} + b c d^{4} e^{5} + 3 \, {\left (b c^{5} d^{6} e^{3} + b c^{3} d^{4} e^{5}\right )} x^{2} + {\left (7 \, b c^{5} d^{7} e^{2} + 8 \, b c^{3} d^{5} e^{4} + b c d^{3} e^{6}\right )} x\right )} \sqrt {c^{2} x^{2} + 1}}{6 \, {\left (c^{6} d^{12} e + 3 \, c^{4} d^{10} e^{3} + 3 \, c^{2} d^{8} e^{5} + d^{6} e^{7} + {\left (c^{6} d^{9} e^{4} + 3 \, c^{4} d^{7} e^{6} + 3 \, c^{2} d^{5} e^{8} + d^{3} e^{10}\right )} x^{3} + 3 \, {\left (c^{6} d^{10} e^{3} + 3 \, c^{4} d^{8} e^{5} + 3 \, c^{2} d^{6} e^{7} + d^{4} e^{9}\right )} x^{2} + 3 \, {\left (c^{6} d^{11} e^{2} + 3 \, c^{4} d^{9} e^{4} + 3 \, c^{2} d^{7} e^{6} + d^{5} e^{8}\right )} 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 \[ \int \frac {b \operatorname {arsinh}\left (c x\right ) + a}{{\left (e x + d\right )}^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 516, normalized size = 2.82 \[ -\frac {c^{3} a}{3 \left (c e x +c d \right )^{3} e}-\frac {c^{3} b \arcsinh \left (c x \right )}{3 \left (c e x +c d \right )^{3} e}-\frac {c^{3} b \sqrt {\left (c x +\frac {c d}{e}\right )^{2}-\frac {2 c d \left (c x +\frac {c d}{e}\right )}{e}+\frac {c^{2} d^{2}+e^{2}}{e^{2}}}}{6 e^{2} \left (c^{2} d^{2}+e^{2}\right ) \left (c x +\frac {c d}{e}\right )^{2}}-\frac {c^{4} b d \sqrt {\left (c x +\frac {c d}{e}\right )^{2}-\frac {2 c d \left (c x +\frac {c d}{e}\right )}{e}+\frac {c^{2} d^{2}+e^{2}}{e^{2}}}}{2 e \left (c^{2} d^{2}+e^{2}\right )^{2} \left (c x +\frac {c d}{e}\right )}-\frac {c^{5} b \,d^{2} \ln \left (\frac {\frac {2 c^{2} d^{2}+2 e^{2}}{e^{2}}-\frac {2 c d \left (c x +\frac {c d}{e}\right )}{e}+2 \sqrt {\frac {c^{2} d^{2}+e^{2}}{e^{2}}}\, \sqrt {\left (c x +\frac {c d}{e}\right )^{2}-\frac {2 c d \left (c x +\frac {c d}{e}\right )}{e}+\frac {c^{2} d^{2}+e^{2}}{e^{2}}}}{c x +\frac {c d}{e}}\right )}{2 e^{2} \left (c^{2} d^{2}+e^{2}\right )^{2} \sqrt {\frac {c^{2} d^{2}+e^{2}}{e^{2}}}}+\frac {c^{3} b \ln \left (\frac {\frac {2 c^{2} d^{2}+2 e^{2}}{e^{2}}-\frac {2 c d \left (c x +\frac {c d}{e}\right )}{e}+2 \sqrt {\frac {c^{2} d^{2}+e^{2}}{e^{2}}}\, \sqrt {\left (c x +\frac {c d}{e}\right )^{2}-\frac {2 c d \left (c x +\frac {c d}{e}\right )}{e}+\frac {c^{2} d^{2}+e^{2}}{e^{2}}}}{c x +\frac {c d}{e}}\right )}{6 e^{2} \left (c^{2} d^{2}+e^{2}\right ) \sqrt {\frac {c^{2} d^{2}+e^{2}}{e^{2}}}} \]
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 {1}{6} \, {\left (6 \, c \int \frac {1}{3 \, {\left (c^{3} e^{4} x^{6} + 3 \, c^{3} d e^{3} x^{5} + 3 \, c d^{2} e^{2} x^{2} + c d^{3} e x + {\left (3 \, c^{3} d^{2} e^{2} + c e^{4}\right )} x^{4} + {\left (c^{3} d^{3} e + 3 \, c d e^{3}\right )} x^{3} + {\left (c^{2} e^{4} x^{5} + 3 \, c^{2} d e^{3} x^{4} + 3 \, d^{2} e^{2} x + d^{3} e + {\left (3 \, c^{2} d^{2} e^{2} + e^{4}\right )} x^{3} + {\left (c^{2} d^{3} e + 3 \, d e^{3}\right )} x^{2}\right )} \sqrt {c^{2} x^{2} + 1}\right )}}\,{d x} - \frac {2 \, {\left (c^{6} d^{3} - 3 \, c^{4} d e^{2}\right )} \log \left (e x + d\right )}{c^{6} d^{6} e + 3 \, c^{4} d^{4} e^{3} + 3 \, c^{2} d^{2} e^{5} + e^{7}} + \frac {3 \, c^{6} d^{6} + 2 \, c^{4} d^{4} e^{2} - c^{2} d^{2} e^{4} + 2 \, {\left (c^{6} d^{4} e^{2} - c^{2} e^{6}\right )} x^{2} + {\left (5 \, c^{6} d^{5} e + 2 \, c^{4} d^{3} e^{3} - 3 \, c^{2} d e^{5}\right )} x + {\left (c^{6} d^{6} - 3 \, c^{4} d^{4} e^{2} + {\left (c^{6} d^{3} e^{3} - 3 \, c^{4} d e^{5}\right )} x^{3} + 3 \, {\left (c^{6} d^{4} e^{2} - 3 \, c^{4} d^{2} e^{4}\right )} x^{2} + 3 \, {\left (c^{6} d^{5} e - 3 \, c^{4} d^{3} e^{3}\right )} x\right )} \log \left (c^{2} x^{2} + 1\right ) - 2 \, {\left (c^{6} d^{6} + 3 \, c^{4} d^{4} e^{2} + 3 \, c^{2} d^{2} e^{4} + e^{6}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )}{c^{6} d^{9} e + 3 \, c^{4} d^{7} e^{3} + 3 \, c^{2} d^{5} e^{5} + d^{3} e^{7} + {\left (c^{6} d^{6} e^{4} + 3 \, c^{4} d^{4} e^{6} + 3 \, c^{2} d^{2} e^{8} + e^{10}\right )} x^{3} + 3 \, {\left (c^{6} d^{7} e^{3} + 3 \, c^{4} d^{5} e^{5} + 3 \, c^{2} d^{3} e^{7} + d e^{9}\right )} x^{2} + 3 \, {\left (c^{6} d^{8} e^{2} + 3 \, c^{4} d^{6} e^{4} + 3 \, c^{2} d^{4} e^{6} + d^{2} e^{8}\right )} x} - \frac {i \, {\left (3 \, c^{6} d^{2} - c^{4} e^{2}\right )} {\left (\log \left (i \, c x + 1\right ) - \log \left (-i \, c x + 1\right )\right )}}{{\left (c^{6} d^{6} + 3 \, c^{4} d^{4} e^{2} + 3 \, c^{2} d^{2} e^{4} + e^{6}\right )} c}\right )} b - \frac {a}{3 \, {\left (e^{4} x^{3} + 3 \, d e^{3} x^{2} + 3 \, d^{2} e^{2} x + d^{3} e\right )}} \]
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 \[ \int \frac {a+b\,\mathrm {asinh}\left (c\,x\right )}{{\left (d+e\,x\right )}^4} \,d x \]
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 \[ \int \frac {a + b \operatorname {asinh}{\left (c x \right )}}{\left (d + e x\right )^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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