Optimal. Leaf size=519 \[ \frac {g \sqrt {1-\frac {h^2 x^2}{g^2}} \sin ^{-1}\left (\frac {h x}{g}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h \sqrt {g-h x} \sqrt {g+h x}}+\frac {i b g p q \sqrt {1-\frac {h^2 x^2}{g^2}} \text {Li}_2\left (-\frac {e^{i \sin ^{-1}\left (\frac {h x}{g}\right )} f g}{i e h-\sqrt {f^2 g^2-e^2 h^2}}\right )}{h \sqrt {g-h x} \sqrt {g+h x}}+\frac {i b g p q \sqrt {1-\frac {h^2 x^2}{g^2}} \text {Li}_2\left (-\frac {e^{i \sin ^{-1}\left (\frac {h x}{g}\right )} f g}{i e h+\sqrt {f^2 g^2-e^2 h^2}}\right )}{h \sqrt {g-h x} \sqrt {g+h x}}-\frac {b g p q \sqrt {1-\frac {h^2 x^2}{g^2}} \sin ^{-1}\left (\frac {h x}{g}\right ) \log \left (1+\frac {f g e^{i \sin ^{-1}\left (\frac {h x}{g}\right )}}{-\sqrt {f^2 g^2-e^2 h^2}+i e h}\right )}{h \sqrt {g-h x} \sqrt {g+h x}}-\frac {b g p q \sqrt {1-\frac {h^2 x^2}{g^2}} \sin ^{-1}\left (\frac {h x}{g}\right ) \log \left (1+\frac {f g e^{i \sin ^{-1}\left (\frac {h x}{g}\right )}}{\sqrt {f^2 g^2-e^2 h^2}+i e h}\right )}{h \sqrt {g-h x} \sqrt {g+h x}}+\frac {i b g p q \sqrt {1-\frac {h^2 x^2}{g^2}} \sin ^{-1}\left (\frac {h x}{g}\right )^2}{2 h \sqrt {g-h x} \sqrt {g+h x}} \]
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Rubi [A] time = 1.41, antiderivative size = 519, normalized size of antiderivative = 1.00, 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 12
Rule 216
Rule 2190
Rule 2279
Rule 2391
Rule 2404
Rule 2407
Rule 2445
Rule 4521
Rule 4741
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*}
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Mathematica [B] time = 4.63, 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 {Li}_2\left (\frac {1}{2}-\frac {1}{2} i \sqrt {\frac {g+h x}{g-h x}}\right )+2 \text {Li}_2\left (\frac {1}{2} i \sqrt {\frac {g+h x}{g-h x}}+\frac {1}{2}\right )+2 \text {Li}_2\left (\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 {Li}_2\left (\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 {Li}_2\left (\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 {Li}_2\left (\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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fricas [F] time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {h x + g} \sqrt {-h x + g} b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + \sqrt {h x + g} \sqrt {-h x + g} a}{h^{2} x^{2} - g^{2}}, x\right ) \]
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 \[ \int \frac {b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a}{\sqrt {h x + g} \sqrt {-h x + g}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.54, size = 0, normalized size = 0.00 \[ \int \frac {b \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )+a}{\sqrt {-h x +g}\, \sqrt {h x +g}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ b \int \frac {q \log \relax (d) + \log \left ({\left ({\left (f x + e\right )}^{p}\right )}^{q}\right ) + \log \relax (c)}{\sqrt {h x + g} \sqrt {-h x + g}}\,{d x} + \frac {a \arcsin \left (\frac {h x}{g}\right )}{h} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {a+b\,\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )}{\sqrt {g+h\,x}\,\sqrt {g-h\,x}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {a + b \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}}{\sqrt {g - h x} \sqrt {g + h x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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