Optimal. Leaf size=184 \[ -\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{7 h (g+h x)^{7/2}}-\frac {4 b f^{7/2} p q \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )}{7 h (f g-e h)^{7/2}}+\frac {4 b f^3 p q}{7 h \sqrt {g+h x} (f g-e h)^3}+\frac {4 b f^2 p q}{21 h (g+h x)^{3/2} (f g-e h)^2}+\frac {4 b f p q}{35 h (g+h x)^{5/2} (f g-e h)} \]
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Rubi [A] time = 0.28, antiderivative size = 184, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {2395, 51, 63, 208, 2445} \[ -\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{7 h (g+h x)^{7/2}}+\frac {4 b f^3 p q}{7 h \sqrt {g+h x} (f g-e h)^3}+\frac {4 b f^2 p q}{21 h (g+h x)^{3/2} (f g-e h)^2}-\frac {4 b f^{7/2} p q \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )}{7 h (f g-e h)^{7/2}}+\frac {4 b f p q}{35 h (g+h x)^{5/2} (f g-e h)} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 208
Rule 2395
Rule 2445
Rubi steps
\begin {align*} \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{(g+h x)^{9/2}} \, dx &=\operatorname {Subst}\left (\int \frac {a+b \log \left (c d^q (e+f x)^{p q}\right )}{(g+h x)^{9/2}} \, dx,c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{7 h (g+h x)^{7/2}}+\operatorname {Subst}\left (\frac {(2 b f p q) \int \frac {1}{(e+f x) (g+h x)^{7/2}} \, dx}{7 h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac {4 b f p q}{35 h (f g-e h) (g+h x)^{5/2}}-\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{7 h (g+h x)^{7/2}}+\operatorname {Subst}\left (\frac {\left (2 b f^2 p q\right ) \int \frac {1}{(e+f x) (g+h x)^{5/2}} \, dx}{7 h (f g-e h)},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac {4 b f p q}{35 h (f g-e h) (g+h x)^{5/2}}+\frac {4 b f^2 p q}{21 h (f g-e h)^2 (g+h x)^{3/2}}-\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{7 h (g+h x)^{7/2}}+\operatorname {Subst}\left (\frac {\left (2 b f^3 p q\right ) \int \frac {1}{(e+f x) (g+h x)^{3/2}} \, dx}{7 h (f g-e h)^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac {4 b f p q}{35 h (f g-e h) (g+h x)^{5/2}}+\frac {4 b f^2 p q}{21 h (f g-e h)^2 (g+h x)^{3/2}}+\frac {4 b f^3 p q}{7 h (f g-e h)^3 \sqrt {g+h x}}-\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{7 h (g+h x)^{7/2}}+\operatorname {Subst}\left (\frac {\left (2 b f^4 p q\right ) \int \frac {1}{(e+f x) \sqrt {g+h x}} \, dx}{7 h (f g-e h)^3},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac {4 b f p q}{35 h (f g-e h) (g+h x)^{5/2}}+\frac {4 b f^2 p q}{21 h (f g-e h)^2 (g+h x)^{3/2}}+\frac {4 b f^3 p q}{7 h (f g-e h)^3 \sqrt {g+h x}}-\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{7 h (g+h x)^{7/2}}+\operatorname {Subst}\left (\frac {\left (4 b f^4 p q\right ) \operatorname {Subst}\left (\int \frac {1}{e-\frac {f g}{h}+\frac {f x^2}{h}} \, dx,x,\sqrt {g+h x}\right )}{7 h^2 (f g-e h)^3},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac {4 b f p q}{35 h (f g-e h) (g+h x)^{5/2}}+\frac {4 b f^2 p q}{21 h (f g-e h)^2 (g+h x)^{3/2}}+\frac {4 b f^3 p q}{7 h (f g-e h)^3 \sqrt {g+h x}}-\frac {4 b f^{7/2} p q \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )}{7 h (f g-e h)^{7/2}}-\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{7 h (g+h x)^{7/2}}\\ \end {align*}
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Mathematica [C] time = 0.10, size = 91, normalized size = 0.49 \[ \frac {10 (f g-e h) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )-4 b f p q (g+h x) \, _2F_1\left (-\frac {5}{2},1;-\frac {3}{2};\frac {f (g+h x)}{f g-e h}\right )}{35 h (g+h x)^{7/2} (e h-f g)} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.55, size = 1362, normalized size = 7.40 \[ \text {result too large to display} \]
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 \[ \int \frac {b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a}{{\left (h x + g\right )}^{\frac {9}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.35, size = 0, normalized size = 0.00 \[ \int \frac {b \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )+a}{\left (h x +g \right )^{\frac {9}{2}}}\, dx \]
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 \[ \text {Exception raised: ValueError} \]
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 \[ \int \frac {a+b\,\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )}{{\left (g+h\,x\right )}^{9/2}} \,d x \]
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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