Optimal. Leaf size=145 \[ \frac {4 b e n}{15 g (e f-d g) (f+g x)^{3/2}}+\frac {4 b e^2 n}{5 g (e f-d g)^2 \sqrt {f+g x}}-\frac {4 b e^{5/2} n \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{5 g (e f-d g)^{5/2}}-\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{5 g (f+g x)^{5/2}} \]
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Rubi [A]
time = 0.09, antiderivative size = 145, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2442, 53, 65,
214} \begin {gather*} -\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{5 g (f+g x)^{5/2}}-\frac {4 b e^{5/2} n \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{5 g (e f-d g)^{5/2}}+\frac {4 b e^2 n}{5 g \sqrt {f+g x} (e f-d g)^2}+\frac {4 b e n}{15 g (f+g x)^{3/2} (e f-d g)} \end {gather*}
Antiderivative was successfully verified.
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Rule 53
Rule 65
Rule 214
Rule 2442
Rubi steps
\begin {align*} \int \frac {a+b \log \left (c (d+e x)^n\right )}{(f+g x)^{7/2}} \, dx &=-\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{5 g (f+g x)^{5/2}}+\frac {(2 b e n) \int \frac {1}{(d+e x) (f+g x)^{5/2}} \, dx}{5 g}\\ &=\frac {4 b e n}{15 g (e f-d g) (f+g x)^{3/2}}-\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{5 g (f+g x)^{5/2}}+\frac {\left (2 b e^2 n\right ) \int \frac {1}{(d+e x) (f+g x)^{3/2}} \, dx}{5 g (e f-d g)}\\ &=\frac {4 b e n}{15 g (e f-d g) (f+g x)^{3/2}}+\frac {4 b e^2 n}{5 g (e f-d g)^2 \sqrt {f+g x}}-\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{5 g (f+g x)^{5/2}}+\frac {\left (2 b e^3 n\right ) \int \frac {1}{(d+e x) \sqrt {f+g x}} \, dx}{5 g (e f-d g)^2}\\ &=\frac {4 b e n}{15 g (e f-d g) (f+g x)^{3/2}}+\frac {4 b e^2 n}{5 g (e f-d g)^2 \sqrt {f+g x}}-\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{5 g (f+g x)^{5/2}}+\frac {\left (4 b e^3 n\right ) \text {Subst}\left (\int \frac {1}{d-\frac {e f}{g}+\frac {e x^2}{g}} \, dx,x,\sqrt {f+g x}\right )}{5 g^2 (e f-d g)^2}\\ &=\frac {4 b e n}{15 g (e f-d g) (f+g x)^{3/2}}+\frac {4 b e^2 n}{5 g (e f-d g)^2 \sqrt {f+g x}}-\frac {4 b e^{5/2} n \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{5 g (e f-d g)^{5/2}}-\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{5 g (f+g x)^{5/2}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.03, size = 78, normalized size = 0.54 \begin {gather*} \frac {2 \left (\frac {2 b e n (f+g x) \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};\frac {e (f+g x)}{e f-d g}\right )}{e f-d g}-3 \left (a+b \log \left (c (d+e x)^n\right )\right )\right )}{15 g (f+g x)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.18, size = 0, normalized size = 0.00 \[\int \frac {a +b \ln \left (c \left (e x +d \right )^{n}\right )}{\left (g x +f \right )^{\frac {7}{2}}}\, dx\]
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 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 350 vs.
\(2 (124) = 248\).
time = 0.38, size = 745, normalized size = 5.14 \begin {gather*} \left [\frac {2 \, {\left (3 \, {\left (b g^{3} n x^{3} + 3 \, b f g^{2} n x^{2} + 3 \, b f^{2} g n x + b f^{3} n\right )} \sqrt {-\frac {e}{d g - f e}} e^{2} \log \left (-\frac {d g - 2 \, {\left (d g - f e\right )} \sqrt {g x + f} \sqrt {-\frac {e}{d g - f e}} - {\left (g x + 2 \, f\right )} e}{x e + d}\right ) - {\left (3 \, a d^{2} g^{2} - {\left (6 \, b g^{2} n x^{2} + 14 \, b f g n x + 8 \, b f^{2} n - 3 \, a f^{2}\right )} e^{2} + 2 \, {\left (b d g^{2} n x + b d f g n - 3 \, a d f g\right )} e + 3 \, {\left (b d^{2} g^{2} n - 2 \, b d f g n e + b f^{2} n e^{2}\right )} \log \left (x e + d\right ) + 3 \, {\left (b d^{2} g^{2} - 2 \, b d f g e + b f^{2} e^{2}\right )} \log \left (c\right )\right )} \sqrt {g x + f}\right )}}{15 \, {\left (d^{2} g^{6} x^{3} + 3 \, d^{2} f g^{5} x^{2} + 3 \, d^{2} f^{2} g^{4} x + d^{2} f^{3} g^{3} + {\left (f^{2} g^{4} x^{3} + 3 \, f^{3} g^{3} x^{2} + 3 \, f^{4} g^{2} x + f^{5} g\right )} e^{2} - 2 \, {\left (d f g^{5} x^{3} + 3 \, d f^{2} g^{4} x^{2} + 3 \, d f^{3} g^{3} x + d f^{4} g^{2}\right )} e\right )}}, \frac {2 \, {\left (\frac {6 \, {\left (b g^{3} n x^{3} + 3 \, b f g^{2} n x^{2} + 3 \, b f^{2} g n x + b f^{3} n\right )} \arctan \left (-\frac {\sqrt {d g - f e} e^{\left (-\frac {1}{2}\right )}}{\sqrt {g x + f}}\right ) e^{\frac {5}{2}}}{\sqrt {d g - f e}} - {\left (3 \, a d^{2} g^{2} - {\left (6 \, b g^{2} n x^{2} + 14 \, b f g n x + 8 \, b f^{2} n - 3 \, a f^{2}\right )} e^{2} + 2 \, {\left (b d g^{2} n x + b d f g n - 3 \, a d f g\right )} e + 3 \, {\left (b d^{2} g^{2} n - 2 \, b d f g n e + b f^{2} n e^{2}\right )} \log \left (x e + d\right ) + 3 \, {\left (b d^{2} g^{2} - 2 \, b d f g e + b f^{2} e^{2}\right )} \log \left (c\right )\right )} \sqrt {g x + f}\right )}}{15 \, {\left (d^{2} g^{6} x^{3} + 3 \, d^{2} f g^{5} x^{2} + 3 \, d^{2} f^{2} g^{4} x + d^{2} f^{3} g^{3} + {\left (f^{2} g^{4} x^{3} + 3 \, f^{3} g^{3} x^{2} + 3 \, f^{4} g^{2} x + f^{5} g\right )} e^{2} - 2 \, {\left (d f g^{5} x^{3} + 3 \, d f^{2} g^{4} x^{2} + 3 \, d f^{3} g^{3} x + d f^{4} g^{2}\right )} e\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 123.27, size = 141, normalized size = 0.97 \begin {gather*} \frac {- \frac {2 a}{5 \left (f + g x\right )^{\frac {5}{2}}} + 2 b \left (\frac {2 e n \left (\frac {e g}{\sqrt {f + g x} \left (d g - e f\right )^{2}} + \frac {e g \operatorname {atan}{\left (\frac {\sqrt {f + g x}}{\sqrt {\frac {d g - e f}{e}}} \right )}}{\sqrt {\frac {d g - e f}{e}} \left (d g - e f\right )^{2}} - \frac {g}{3 \left (f + g x\right )^{\frac {3}{2}} \left (d g - e f\right )}\right )}{5 g} - \frac {\log {\left (c \left (d - \frac {e f}{g} + \frac {e \left (f + g x\right )}{g}\right )^{n} \right )}}{5 \left (f + g x\right )^{\frac {5}{2}}}\right )}{g} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )}{{\left (f+g\,x\right )}^{7/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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