Optimal. Leaf size=519 \[ \frac{i b g p q \sqrt{1-\frac{h^2 x^2}{g^2}} \text{PolyLog}\left (2,-\frac{f g e^{i \sin ^{-1}\left (\frac{h x}{g}\right )}}{-\sqrt{f^2 g^2-e^2 h^2}+i e h}\right )}{h \sqrt{g-h x} \sqrt{g+h x}}+\frac{i b g p q \sqrt{1-\frac{h^2 x^2}{g^2}} \text{PolyLog}\left (2,-\frac{f g e^{i \sin ^{-1}\left (\frac{h x}{g}\right )}}{\sqrt{f^2 g^2-e^2 h^2}+i e h}\right )}{h \sqrt{g-h x} \sqrt{g+h x}}+\frac{g \sqrt{1-\frac{h^2 x^2}{g^2}} \sin ^{-1}\left (\frac{h x}{g}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h \sqrt{g-h x} \sqrt{g+h x}}-\frac{b g p q \sqrt{1-\frac{h^2 x^2}{g^2}} \sin ^{-1}\left (\frac{h x}{g}\right ) \log \left (1+\frac{f g e^{i \sin ^{-1}\left (\frac{h x}{g}\right )}}{-\sqrt{f^2 g^2-e^2 h^2}+i e h}\right )}{h \sqrt{g-h x} \sqrt{g+h x}}-\frac{b g p q \sqrt{1-\frac{h^2 x^2}{g^2}} \sin ^{-1}\left (\frac{h x}{g}\right ) \log \left (1+\frac{f g e^{i \sin ^{-1}\left (\frac{h x}{g}\right )}}{\sqrt{f^2 g^2-e^2 h^2}+i e h}\right )}{h \sqrt{g-h x} \sqrt{g+h x}}+\frac{i b g p q \sqrt{1-\frac{h^2 x^2}{g^2}} \sin ^{-1}\left (\frac{h x}{g}\right )^2}{2 h \sqrt{g-h x} \sqrt{g+h x}} \]
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Rubi [A] time = 1.4112, antiderivative size = 519, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 10, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263, Rules used = {2407, 216, 2404, 12, 4741, 4521, 2190, 2279, 2391, 2445} \[ \frac{i b g p q \sqrt{1-\frac{h^2 x^2}{g^2}} \text{PolyLog}\left (2,-\frac{f g e^{i \sin ^{-1}\left (\frac{h x}{g}\right )}}{-\sqrt{f^2 g^2-e^2 h^2}+i e h}\right )}{h \sqrt{g-h x} \sqrt{g+h x}}+\frac{i b g p q \sqrt{1-\frac{h^2 x^2}{g^2}} \text{PolyLog}\left (2,-\frac{f g e^{i \sin ^{-1}\left (\frac{h x}{g}\right )}}{\sqrt{f^2 g^2-e^2 h^2}+i e h}\right )}{h \sqrt{g-h x} \sqrt{g+h x}}+\frac{g \sqrt{1-\frac{h^2 x^2}{g^2}} \sin ^{-1}\left (\frac{h x}{g}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h \sqrt{g-h x} \sqrt{g+h x}}-\frac{b g p q \sqrt{1-\frac{h^2 x^2}{g^2}} \sin ^{-1}\left (\frac{h x}{g}\right ) \log \left (1+\frac{f g e^{i \sin ^{-1}\left (\frac{h x}{g}\right )}}{-\sqrt{f^2 g^2-e^2 h^2}+i e h}\right )}{h \sqrt{g-h x} \sqrt{g+h x}}-\frac{b g p q \sqrt{1-\frac{h^2 x^2}{g^2}} \sin ^{-1}\left (\frac{h x}{g}\right ) \log \left (1+\frac{f g e^{i \sin ^{-1}\left (\frac{h x}{g}\right )}}{\sqrt{f^2 g^2-e^2 h^2}+i e h}\right )}{h \sqrt{g-h x} \sqrt{g+h x}}+\frac{i b g p q \sqrt{1-\frac{h^2 x^2}{g^2}} \sin ^{-1}\left (\frac{h x}{g}\right )^2}{2 h \sqrt{g-h x} \sqrt{g+h x}} \]
Antiderivative was successfully verified.
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Rule 2407
Rule 216
Rule 2404
Rule 12
Rule 4741
Rule 4521
Rule 2190
Rule 2279
Rule 2391
Rule 2445
Rubi steps
\begin{align*} \int \frac{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{\sqrt{g-h x} \sqrt{g+h x}} \, dx &=\operatorname{Subst}\left (\int \frac{a+b \log \left (c d^q (e+f x)^{p q}\right )}{\sqrt{g-h x} \sqrt{g+h x}} \, dx,c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\operatorname{Subst}\left (\frac{\sqrt{1-\frac{h^2 x^2}{g^2}} \int \frac{a+b \log \left (c d^q (e+f x)^{p q}\right )}{\sqrt{1-\frac{h^2 x^2}{g^2}}} \, dx}{\sqrt{g-h x} \sqrt{g+h x}},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac{g \sqrt{1-\frac{h^2 x^2}{g^2}} \sin ^{-1}\left (\frac{h x}{g}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h \sqrt{g-h x} \sqrt{g+h x}}-\operatorname{Subst}\left (\frac{\left (b f p q \sqrt{1-\frac{h^2 x^2}{g^2}}\right ) \int \frac{g \sin ^{-1}\left (\frac{h x}{g}\right )}{e h+f h x} \, dx}{\sqrt{g-h x} \sqrt{g+h x}},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac{g \sqrt{1-\frac{h^2 x^2}{g^2}} \sin ^{-1}\left (\frac{h x}{g}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h \sqrt{g-h x} \sqrt{g+h x}}-\operatorname{Subst}\left (\frac{\left (b f g p q \sqrt{1-\frac{h^2 x^2}{g^2}}\right ) \int \frac{\sin ^{-1}\left (\frac{h x}{g}\right )}{e h+f h x} \, dx}{\sqrt{g-h x} \sqrt{g+h x}},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac{g \sqrt{1-\frac{h^2 x^2}{g^2}} \sin ^{-1}\left (\frac{h x}{g}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h \sqrt{g-h x} \sqrt{g+h x}}-\operatorname{Subst}\left (\frac{\left (b f g p q \sqrt{1-\frac{h^2 x^2}{g^2}}\right ) \operatorname{Subst}\left (\int \frac{x \cos (x)}{\frac{e h^2}{g}+f h \sin (x)} \, dx,x,\sin ^{-1}\left (\frac{h x}{g}\right )\right )}{\sqrt{g-h x} \sqrt{g+h x}},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac{i b g p q \sqrt{1-\frac{h^2 x^2}{g^2}} \sin ^{-1}\left (\frac{h x}{g}\right )^2}{2 h \sqrt{g-h x} \sqrt{g+h x}}+\frac{g \sqrt{1-\frac{h^2 x^2}{g^2}} \sin ^{-1}\left (\frac{h x}{g}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h \sqrt{g-h x} \sqrt{g+h x}}-\operatorname{Subst}\left (\frac{\left (i b f g p q \sqrt{1-\frac{h^2 x^2}{g^2}}\right ) \operatorname{Subst}\left (\int \frac{e^{i x} x}{e^{i x} f h+\frac{i e h^2}{g}-\frac{h \sqrt{f^2 g^2-e^2 h^2}}{g}} \, dx,x,\sin ^{-1}\left (\frac{h x}{g}\right )\right )}{\sqrt{g-h x} \sqrt{g+h x}},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\operatorname{Subst}\left (\frac{\left (i b f g p q \sqrt{1-\frac{h^2 x^2}{g^2}}\right ) \operatorname{Subst}\left (\int \frac{e^{i x} x}{e^{i x} f h+\frac{i e h^2}{g}+\frac{h \sqrt{f^2 g^2-e^2 h^2}}{g}} \, dx,x,\sin ^{-1}\left (\frac{h x}{g}\right )\right )}{\sqrt{g-h x} \sqrt{g+h x}},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac{i b g p q \sqrt{1-\frac{h^2 x^2}{g^2}} \sin ^{-1}\left (\frac{h x}{g}\right )^2}{2 h \sqrt{g-h x} \sqrt{g+h x}}-\frac{b g p q \sqrt{1-\frac{h^2 x^2}{g^2}} \sin ^{-1}\left (\frac{h x}{g}\right ) \log \left (1+\frac{e^{i \sin ^{-1}\left (\frac{h x}{g}\right )} f g}{i e h-\sqrt{f^2 g^2-e^2 h^2}}\right )}{h \sqrt{g-h x} \sqrt{g+h x}}-\frac{b g p q \sqrt{1-\frac{h^2 x^2}{g^2}} \sin ^{-1}\left (\frac{h x}{g}\right ) \log \left (1+\frac{e^{i \sin ^{-1}\left (\frac{h x}{g}\right )} f g}{i e h+\sqrt{f^2 g^2-e^2 h^2}}\right )}{h \sqrt{g-h x} \sqrt{g+h x}}+\frac{g \sqrt{1-\frac{h^2 x^2}{g^2}} \sin ^{-1}\left (\frac{h x}{g}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h \sqrt{g-h x} \sqrt{g+h x}}+\operatorname{Subst}\left (\frac{\left (b g p q \sqrt{1-\frac{h^2 x^2}{g^2}}\right ) \operatorname{Subst}\left (\int \log \left (1+\frac{e^{i x} f h}{\frac{i e h^2}{g}-\frac{h \sqrt{f^2 g^2-e^2 h^2}}{g}}\right ) \, dx,x,\sin ^{-1}\left (\frac{h x}{g}\right )\right )}{h \sqrt{g-h x} \sqrt{g+h x}},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\operatorname{Subst}\left (\frac{\left (b g p q \sqrt{1-\frac{h^2 x^2}{g^2}}\right ) \operatorname{Subst}\left (\int \log \left (1+\frac{e^{i x} f h}{\frac{i e h^2}{g}+\frac{h \sqrt{f^2 g^2-e^2 h^2}}{g}}\right ) \, dx,x,\sin ^{-1}\left (\frac{h x}{g}\right )\right )}{h \sqrt{g-h x} \sqrt{g+h x}},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac{i b g p q \sqrt{1-\frac{h^2 x^2}{g^2}} \sin ^{-1}\left (\frac{h x}{g}\right )^2}{2 h \sqrt{g-h x} \sqrt{g+h x}}-\frac{b g p q \sqrt{1-\frac{h^2 x^2}{g^2}} \sin ^{-1}\left (\frac{h x}{g}\right ) \log \left (1+\frac{e^{i \sin ^{-1}\left (\frac{h x}{g}\right )} f g}{i e h-\sqrt{f^2 g^2-e^2 h^2}}\right )}{h \sqrt{g-h x} \sqrt{g+h x}}-\frac{b g p q \sqrt{1-\frac{h^2 x^2}{g^2}} \sin ^{-1}\left (\frac{h x}{g}\right ) \log \left (1+\frac{e^{i \sin ^{-1}\left (\frac{h x}{g}\right )} f g}{i e h+\sqrt{f^2 g^2-e^2 h^2}}\right )}{h \sqrt{g-h x} \sqrt{g+h x}}+\frac{g \sqrt{1-\frac{h^2 x^2}{g^2}} \sin ^{-1}\left (\frac{h x}{g}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h \sqrt{g-h x} \sqrt{g+h x}}-\operatorname{Subst}\left (\frac{\left (i b g p q \sqrt{1-\frac{h^2 x^2}{g^2}}\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{f h x}{\frac{i e h^2}{g}-\frac{h \sqrt{f^2 g^2-e^2 h^2}}{g}}\right )}{x} \, dx,x,e^{i \sin ^{-1}\left (\frac{h x}{g}\right )}\right )}{h \sqrt{g-h x} \sqrt{g+h x}},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\operatorname{Subst}\left (\frac{\left (i b g p q \sqrt{1-\frac{h^2 x^2}{g^2}}\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{f h x}{\frac{i e h^2}{g}+\frac{h \sqrt{f^2 g^2-e^2 h^2}}{g}}\right )}{x} \, dx,x,e^{i \sin ^{-1}\left (\frac{h x}{g}\right )}\right )}{h \sqrt{g-h x} \sqrt{g+h x}},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac{i b g p q \sqrt{1-\frac{h^2 x^2}{g^2}} \sin ^{-1}\left (\frac{h x}{g}\right )^2}{2 h \sqrt{g-h x} \sqrt{g+h x}}-\frac{b g p q \sqrt{1-\frac{h^2 x^2}{g^2}} \sin ^{-1}\left (\frac{h x}{g}\right ) \log \left (1+\frac{e^{i \sin ^{-1}\left (\frac{h x}{g}\right )} f g}{i e h-\sqrt{f^2 g^2-e^2 h^2}}\right )}{h \sqrt{g-h x} \sqrt{g+h x}}-\frac{b g p q \sqrt{1-\frac{h^2 x^2}{g^2}} \sin ^{-1}\left (\frac{h x}{g}\right ) \log \left (1+\frac{e^{i \sin ^{-1}\left (\frac{h x}{g}\right )} f g}{i e h+\sqrt{f^2 g^2-e^2 h^2}}\right )}{h \sqrt{g-h x} \sqrt{g+h x}}+\frac{g \sqrt{1-\frac{h^2 x^2}{g^2}} \sin ^{-1}\left (\frac{h x}{g}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h \sqrt{g-h x} \sqrt{g+h x}}+\frac{i b g p q \sqrt{1-\frac{h^2 x^2}{g^2}} \text{Li}_2\left (-\frac{e^{i \sin ^{-1}\left (\frac{h x}{g}\right )} f g}{i e h-\sqrt{f^2 g^2-e^2 h^2}}\right )}{h \sqrt{g-h x} \sqrt{g+h x}}+\frac{i b g p q \sqrt{1-\frac{h^2 x^2}{g^2}} \text{Li}_2\left (-\frac{e^{i \sin ^{-1}\left (\frac{h x}{g}\right )} f g}{i e h+\sqrt{f^2 g^2-e^2 h^2}}\right )}{h \sqrt{g-h x} \sqrt{g+h x}}\\ \end{align*}
Mathematica [B] time = 4.46077, size = 1083, normalized size = 2.09 \[ \frac{\tan ^{-1}\left (\frac{h x}{\sqrt{g-h x} \sqrt{g+h x}}\right ) \left (a-b p q \log (e+f x)+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h}-\frac{i b p q \sqrt{g-h x} \sqrt{\frac{g+h x}{g-h x}} \left (\log ^2\left (i-\sqrt{\frac{g+h x}{g-h x}}\right )+2 \log (e+f x) \log \left (i-\sqrt{\frac{g+h x}{g-h x}}\right )+2 \log \left (\frac{1}{2} \left (1-i \sqrt{\frac{g+h x}{g-h x}}\right )\right ) \log \left (i-\sqrt{\frac{g+h x}{g-h x}}\right )-2 \log \left (\frac{\sqrt{f g-e h}-\sqrt{f g+e h} \sqrt{\frac{g+h x}{g-h x}}}{\sqrt{f g-e h}-i \sqrt{f g+e h}}\right ) \log \left (i-\sqrt{\frac{g+h x}{g-h x}}\right )-2 \log \left (\frac{\sqrt{f g-e h}+\sqrt{f g+e h} \sqrt{\frac{g+h x}{g-h x}}}{\sqrt{f g-e h}+i \sqrt{f g+e h}}\right ) \log \left (i-\sqrt{\frac{g+h x}{g-h x}}\right )-\log ^2\left (\sqrt{\frac{g+h x}{g-h x}}+i\right )-2 \log (e+f x) \log \left (\sqrt{\frac{g+h x}{g-h x}}+i\right )-2 \log \left (\frac{1}{2} \left (i \sqrt{\frac{g+h x}{g-h x}}+1\right )\right ) \log \left (\sqrt{\frac{g+h x}{g-h x}}+i\right )+2 \log \left (\sqrt{\frac{g+h x}{g-h x}}+i\right ) \log \left (\frac{\sqrt{f g-e h}-\sqrt{f g+e h} \sqrt{\frac{g+h x}{g-h x}}}{\sqrt{f g-e h}+i \sqrt{f g+e h}}\right )+2 \log \left (\sqrt{\frac{g+h x}{g-h x}}+i\right ) \log \left (\frac{\sqrt{f g-e h}+\sqrt{f g+e h} \sqrt{\frac{g+h x}{g-h x}}}{\sqrt{f g-e h}-i \sqrt{f g+e h}}\right )-2 \text{PolyLog}\left (2,\frac{1}{2}-\frac{1}{2} i \sqrt{\frac{g+h x}{g-h x}}\right )+2 \text{PolyLog}\left (2,\frac{1}{2} i \sqrt{\frac{g+h x}{g-h x}}+\frac{1}{2}\right )+2 \text{PolyLog}\left (2,\frac{\sqrt{f g+e h} \left (1-i \sqrt{\frac{g+h x}{g-h x}}\right )}{i \sqrt{f g-e h}+\sqrt{f g+e h}}\right )-2 \text{PolyLog}\left (2,\frac{\sqrt{f g+e h} \left (i \sqrt{\frac{g+h x}{g-h x}}+1\right )}{\sqrt{f g+e h}-i \sqrt{f g-e h}}\right )-2 \text{PolyLog}\left (2,\frac{\sqrt{f g+e h} \left (i \sqrt{\frac{g+h x}{g-h x}}+1\right )}{i \sqrt{f g-e h}+\sqrt{f g+e h}}\right )+2 \text{PolyLog}\left (2,\frac{\sqrt{f g+e h} \left (\sqrt{\frac{g+h x}{g-h x}}+i\right )}{\sqrt{f g-e h}+i \sqrt{f g+e h}}\right )\right )}{2 h \sqrt{g+h x}} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.06, size = 0, normalized size = 0. \begin{align*} \int{(a+b\ln \left ( c \left ( d \left ( fx+e \right ) ^{p} \right ) ^{q} \right ) ){\frac{1}{\sqrt{-hx+g}}}{\frac{1}{\sqrt{hx+g}}}}\, 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 [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{h x + g} \sqrt{-h x + g} b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + \sqrt{h x + g} \sqrt{-h x + g} a}{h^{2} x^{2} - g^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{a + b \log{\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}}{\sqrt{g - h x} \sqrt{g + h x}}\, dx \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 ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a}{\sqrt{h x + g} \sqrt{-h x + g}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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