Optimal. Leaf size=330 \[ -\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{h \sqrt {g+h x}}-\frac {8 b \sqrt {f} p q \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h \sqrt {f g-e h}}-\frac {8 b^2 \sqrt {f} p^2 q^2 \text {Li}_2\left (1-\frac {2}{1-\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}}\right )}{h \sqrt {f g-e h}}+\frac {8 b^2 \sqrt {f} p^2 q^2 \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )^2}{h \sqrt {f g-e h}}-\frac {16 b^2 \sqrt {f} p^2 q^2 \log \left (\frac {2}{1-\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}}\right ) \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )}{h \sqrt {f g-e h}} \]
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Rubi [A] time = 1.62, antiderivative size = 330, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 13, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.433, Rules used = {2398, 2411, 63, 208, 2348, 12, 1587, 6741, 5984, 5918, 2402, 2315, 2445} \[ -\frac {8 b^2 \sqrt {f} p^2 q^2 \text {PolyLog}\left (2,1-\frac {2}{1-\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}}\right )}{h \sqrt {f g-e h}}-\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{h \sqrt {g+h x}}-\frac {8 b \sqrt {f} p q \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h \sqrt {f g-e h}}+\frac {8 b^2 \sqrt {f} p^2 q^2 \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )^2}{h \sqrt {f g-e h}}-\frac {16 b^2 \sqrt {f} p^2 q^2 \log \left (\frac {2}{1-\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}}\right ) \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )}{h \sqrt {f g-e h}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 63
Rule 208
Rule 1587
Rule 2315
Rule 2348
Rule 2398
Rule 2402
Rule 2411
Rule 2445
Rule 5918
Rule 5984
Rule 6741
Rubi steps
\begin {align*} \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{(g+h x)^{3/2}} \, dx &=\operatorname {Subst}\left (\int \frac {\left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^2}{(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 )^2}{h \sqrt {g+h x}}+\operatorname {Subst}\left (\frac {(4 b f p q) \int \frac {a+b \log \left (c d^q (e+f x)^{p q}\right )}{(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 )^2}{h \sqrt {g+h x}}+\operatorname {Subst}\left (\frac {(4 b p q) \operatorname {Subst}\left (\int \frac {a+b \log \left (c d^q x^{p q}\right )}{x \sqrt {\frac {f g-e h}{f}+\frac {h x}{f}}} \, dx,x,e+f x\right )}{h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-\frac {8 b \sqrt {f} p q \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\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 )^2}{h \sqrt {g+h x}}-\operatorname {Subst}\left (\frac {\left (4 b^2 p^2 q^2\right ) \operatorname {Subst}\left (\int -\frac {2 \sqrt {f} \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g-\frac {e h}{f}+\frac {h x}{f}}}{\sqrt {f g-e h}}\right )}{\sqrt {f g-e h} x} \, dx,x,e+f x\right )}{h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-\frac {8 b \sqrt {f} p q \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\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 )^2}{h \sqrt {g+h x}}+\operatorname {Subst}\left (\frac {\left (8 b^2 \sqrt {f} p^2 q^2\right ) \operatorname {Subst}\left (\int \frac {\tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g-\frac {e h}{f}+\frac {h x}{f}}}{\sqrt {f g-e h}}\right )}{x} \, dx,x,e+f x\right )}{h \sqrt {f g-e h}},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-\frac {8 b \sqrt {f} p q \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\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 )^2}{h \sqrt {g+h x}}+\operatorname {Subst}\left (\frac {\left (16 b^2 f^{3/2} p^2 q^2\right ) \operatorname {Subst}\left (\int \frac {x \tanh ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {f g-e h}}\right )}{e h+f \left (-g+x^2\right )} \, dx,x,\sqrt {g+h x}\right )}{h \sqrt {f g-e h}},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-\frac {8 b \sqrt {f} p q \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\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 )^2}{h \sqrt {g+h x}}+\operatorname {Subst}\left (\frac {\left (16 b^2 f^{3/2} p^2 q^2\right ) \operatorname {Subst}\left (\int \frac {x \tanh ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {f g-e h}}\right )}{-f g+e h+f x^2} \, dx,x,\sqrt {g+h x}\right )}{h \sqrt {f g-e h}},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac {8 b^2 \sqrt {f} p^2 q^2 \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )^2}{h \sqrt {f g-e h}}-\frac {8 b \sqrt {f} p q \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\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 )^2}{h \sqrt {g+h x}}-\operatorname {Subst}\left (\frac {\left (16 b^2 f p^2 q^2\right ) \operatorname {Subst}\left (\int \frac {\tanh ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {f g-e h}}\right )}{1-\frac {\sqrt {f} x}{\sqrt {f g-e h}}} \, dx,x,\sqrt {g+h x}\right )}{h (f g-e h)},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac {8 b^2 \sqrt {f} p^2 q^2 \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )^2}{h \sqrt {f g-e h}}-\frac {8 b \sqrt {f} p q \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\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 )^2}{h \sqrt {g+h x}}-\frac {16 b^2 \sqrt {f} p^2 q^2 \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right ) \log \left (\frac {2}{1-\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}}\right )}{h \sqrt {f g-e h}}+\operatorname {Subst}\left (\frac {\left (16 b^2 f p^2 q^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (\frac {2}{1-\frac {\sqrt {f} x}{\sqrt {f g-e h}}}\right )}{1-\frac {f x^2}{f g-e h}} \, dx,x,\sqrt {g+h x}\right )}{h (f g-e h)},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac {8 b^2 \sqrt {f} p^2 q^2 \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )^2}{h \sqrt {f g-e h}}-\frac {8 b \sqrt {f} p q \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\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 )^2}{h \sqrt {g+h x}}-\frac {16 b^2 \sqrt {f} p^2 q^2 \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right ) \log \left (\frac {2}{1-\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}}\right )}{h \sqrt {f g-e h}}-\operatorname {Subst}\left (\frac {\left (16 b^2 \sqrt {f} p^2 q^2\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}}\right )}{h \sqrt {f g-e h}},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac {8 b^2 \sqrt {f} p^2 q^2 \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )^2}{h \sqrt {f g-e h}}-\frac {8 b \sqrt {f} p q \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\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 )^2}{h \sqrt {g+h x}}-\frac {16 b^2 \sqrt {f} p^2 q^2 \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right ) \log \left (\frac {2}{1-\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}}\right )}{h \sqrt {f g-e h}}-\frac {8 b^2 \sqrt {f} p^2 q^2 \text {Li}_2\left (1-\frac {2}{1-\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}}\right )}{h \sqrt {f g-e h}}\\ \end {align*}
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Mathematica [C] time = 3.77, size = 356, normalized size = 1.08 \[ \frac {2 \left (\frac {b^2 p^2 q^2 \left (h (e+f x) \sqrt {\frac {f (g+h x)}{f g-e h}} \, _4F_3\left (1,1,1,\frac {3}{2};2,2,2;\frac {h (e+f x)}{e h-f g}\right )+(f g-e h) \log (e+f x) \left (\log (e+f x) \left (\sqrt {\frac {f (g+h x)}{f g-e h}}-1\right )-4 \sqrt {\frac {f (g+h x)}{f g-e h}} \log \left (\frac {1}{2} \left (\sqrt {\frac {f (g+h x)}{f g-e h}}+1\right )\right )\right )\right )}{\sqrt {g+h x} (f g-e h)}-\frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )-b p q \log (e+f x)\right )^2}{\sqrt {g+h x}}+\frac {2 b p q \left (\sqrt {g+h x} \sqrt {f g-e h} \log (e+f x)+2 \sqrt {f} (g+h x) \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )\right ) \left (-a-b \log \left (c \left (d (e+f x)^p\right )^q\right )+b p q \log (e+f x)\right )}{(g+h x) \sqrt {f g-e h}}\right )}{h} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {h x + g} b^{2} \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right )^{2} + 2 \, \sqrt {h x + g} a b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + \sqrt {h x + g} a^{2}}{h^{2} x^{2} + 2 \, g h x + g^{2}}, x\right ) \]
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 \[ \int \frac {{\left (b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a\right )}^{2}}{{\left (h x + g\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {\left (b \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )+a \right )^{2}}{\left (h x +g \right )^{\frac {3}{2}}}\, dx \]
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 = -0.00 \[ \int \frac {{\left (a+b\,\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )\right )}^2}{{\left (g+h\,x\right )}^{3/2}} \,d x \]
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 \[ \int \frac {\left (a + b \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}\right )^{2}}{\left (g + h x\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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