Optimal. Leaf size=176 \[ -\frac{2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{7 g (f+g x)^{7/2}}+\frac{4 b e^3 n}{7 g \sqrt{f+g x} (e f-d g)^3}+\frac{4 b e^2 n}{21 g (f+g x)^{3/2} (e f-d g)^2}-\frac{4 b e^{7/2} n \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right )}{7 g (e f-d g)^{7/2}}+\frac{4 b e n}{35 g (f+g x)^{5/2} (e f-d g)} \]
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Rubi [A] time = 0.144247, antiderivative size = 176, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {2395, 51, 63, 208} \[ -\frac{2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{7 g (f+g x)^{7/2}}+\frac{4 b e^3 n}{7 g \sqrt{f+g x} (e f-d g)^3}+\frac{4 b e^2 n}{21 g (f+g x)^{3/2} (e f-d g)^2}-\frac{4 b e^{7/2} n \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right )}{7 g (e f-d g)^{7/2}}+\frac{4 b e n}{35 g (f+g x)^{5/2} (e f-d g)} \]
Antiderivative was successfully verified.
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Rule 2395
Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{a+b \log \left (c (d+e x)^n\right )}{(f+g x)^{9/2}} \, dx &=-\frac{2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{7 g (f+g x)^{7/2}}+\frac{(2 b e n) \int \frac{1}{(d+e x) (f+g x)^{7/2}} \, dx}{7 g}\\ &=\frac{4 b e n}{35 g (e f-d g) (f+g x)^{5/2}}-\frac{2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{7 g (f+g x)^{7/2}}+\frac{\left (2 b e^2 n\right ) \int \frac{1}{(d+e x) (f+g x)^{5/2}} \, dx}{7 g (e f-d g)}\\ &=\frac{4 b e n}{35 g (e f-d g) (f+g x)^{5/2}}+\frac{4 b e^2 n}{21 g (e f-d g)^2 (f+g x)^{3/2}}-\frac{2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{7 g (f+g x)^{7/2}}+\frac{\left (2 b e^3 n\right ) \int \frac{1}{(d+e x) (f+g x)^{3/2}} \, dx}{7 g (e f-d g)^2}\\ &=\frac{4 b e n}{35 g (e f-d g) (f+g x)^{5/2}}+\frac{4 b e^2 n}{21 g (e f-d g)^2 (f+g x)^{3/2}}+\frac{4 b e^3 n}{7 g (e f-d g)^3 \sqrt{f+g x}}-\frac{2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{7 g (f+g x)^{7/2}}+\frac{\left (2 b e^4 n\right ) \int \frac{1}{(d+e x) \sqrt{f+g x}} \, dx}{7 g (e f-d g)^3}\\ &=\frac{4 b e n}{35 g (e f-d g) (f+g x)^{5/2}}+\frac{4 b e^2 n}{21 g (e f-d g)^2 (f+g x)^{3/2}}+\frac{4 b e^3 n}{7 g (e f-d g)^3 \sqrt{f+g x}}-\frac{2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{7 g (f+g x)^{7/2}}+\frac{\left (4 b e^4 n\right ) \operatorname{Subst}\left (\int \frac{1}{d-\frac{e f}{g}+\frac{e x^2}{g}} \, dx,x,\sqrt{f+g x}\right )}{7 g^2 (e f-d g)^3}\\ &=\frac{4 b e n}{35 g (e f-d g) (f+g x)^{5/2}}+\frac{4 b e^2 n}{21 g (e f-d g)^2 (f+g x)^{3/2}}+\frac{4 b e^3 n}{7 g (e f-d g)^3 \sqrt{f+g x}}-\frac{4 b e^{7/2} n \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right )}{7 g (e f-d g)^{7/2}}-\frac{2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{7 g (f+g x)^{7/2}}\\ \end{align*}
Mathematica [C] time = 0.049949, size = 78, normalized size = 0.44 \[ \frac{2 \left (\frac{2 b e n (f+g x) \, _2F_1\left (-\frac{5}{2},1;-\frac{3}{2};\frac{e (f+g x)}{e f-d g}\right )}{e f-d g}-5 \left (a+b \log \left (c (d+e x)^n\right )\right )\right )}{35 g (f+g x)^{7/2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.9, size = 0, normalized size = 0. \begin{align*} \int{(a+b\ln \left ( c \left ( ex+d \right ) ^{n} \right ) ) \left ( gx+f \right ) ^{-{\frac{9}{2}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.43938, size = 2611, normalized size = 14.84 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b \log \left ({\left (e x + d\right )}^{n} c\right ) + a}{{\left (g x + f\right )}^{\frac{9}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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