Optimal. Leaf size=86 \[ -\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h \sqrt {g+h x}}-\frac {4 b \sqrt {f} p q \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )}{h \sqrt {f g-e h}} \]
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Rubi [A] time = 0.13, antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2395, 63, 208, 2445} \[ -\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h \sqrt {g+h x}}-\frac {4 b \sqrt {f} p q \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )}{h \sqrt {f g-e h}} \]
Antiderivative was successfully verified.
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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)^{3/2}} \, dx &=\operatorname {Subst}\left (\int \frac {a+b \log \left (c d^q (e+f x)^{p q}\right )}{(g+h x)^{3/2}} \, 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 )}{h \sqrt {g+h x}}+\operatorname {Subst}\left (\frac {(2 b f p q) \int \frac {1}{(e+f x) \sqrt {g+h x}} \, dx}{h},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 )}{h \sqrt {g+h x}}+\operatorname {Subst}\left (\frac {(4 b f p q) \operatorname {Subst}\left (\int \frac {1}{e-\frac {f g}{h}+\frac {f x^2}{h}} \, dx,x,\sqrt {g+h x}\right )}{h^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-\frac {4 b \sqrt {f} p q \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )}{h \sqrt {f g-e h}}-\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h \sqrt {g+h x}}\\ \end {align*}
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Mathematica [A] time = 0.17, size = 84, normalized size = 0.98 \[ \frac {-\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{\sqrt {g+h x}}-\frac {4 b \sqrt {f} p q \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )}{\sqrt {f g-e h}}}{h} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.55, size = 240, normalized size = 2.79 \[ \left [\frac {2 \, {\left ({\left (b h p q x + b g p q\right )} \sqrt {\frac {f}{f g - e h}} \log \left (\frac {f h x + 2 \, f g - e h - 2 \, {\left (f g - e h\right )} \sqrt {h x + g} \sqrt {\frac {f}{f g - e h}}}{f x + e}\right ) - {\left (b p q \log \left (f x + e\right ) + b q \log \relax (d) + b \log \relax (c) + a\right )} \sqrt {h x + g}\right )}}{h^{2} x + g h}, -\frac {2 \, {\left (2 \, {\left (b h p q x + b g p q\right )} \sqrt {-\frac {f}{f g - e h}} \arctan \left (-\frac {{\left (f g - e h\right )} \sqrt {h x + g} \sqrt {-\frac {f}{f g - e h}}}{f h x + f g}\right ) + {\left (b p q \log \left (f x + e\right ) + b q \log \relax (d) + b \log \relax (c) + a\right )} \sqrt {h x + g}\right )}}{h^{2} x + g h}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 99, normalized size = 1.15 \[ \frac {4 \, b f p q \arctan \left (\frac {\sqrt {h x + g} f}{\sqrt {-f^{2} g + f h e}}\right )}{\sqrt {-f^{2} g + f h e} h} - \frac {2 \, {\left (b p q \log \left ({\left (h x + g\right )} f - f g + h e\right ) - b p q \log \relax (h) + b q \log \relax (d) + b \log \relax (c) + a\right )}}{\sqrt {h x + g} h} \]
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 \[ \int \frac {b \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )+a}{\left (h x +g \right )^{\frac {3}{2}}}\, dx \]
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 \[ \int \frac {a+b\,\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )}{{\left (g+h\,x\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 21.45, size = 90, normalized size = 1.05 \[ \frac {- \frac {2 a}{\sqrt {g + h x}} + 2 b \left (\frac {2 p q \operatorname {atan}{\left (\frac {\sqrt {g + h x}}{\sqrt {\frac {h \left (e - \frac {f g}{h}\right )}{f}}} \right )}}{\sqrt {\frac {h \left (e - \frac {f g}{h}\right )}{f}}} - \frac {\log {\left (c \left (d \left (e - \frac {f g}{h} + \frac {f \left (g + h x\right )}{h}\right )^{p}\right )^{q} \right )}}{\sqrt {g + h x}}\right )}{h} \]
Verification of antiderivative is not currently implemented for this CAS.
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