Optimal. Leaf size=139 \[ -\frac {4 b (f g-e h) p q \sqrt {g+h x}}{3 f h}-\frac {4 b p q (g+h x)^{3/2}}{9 h}+\frac {4 b (f g-e h)^{3/2} p q \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )}{3 f^{3/2} h}+\frac {2 (g+h x)^{3/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{3 h} \]
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Rubi [A]
time = 0.13, antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {2442, 52, 65,
214, 2495} \begin {gather*} \frac {2 (g+h x)^{3/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{3 h}+\frac {4 b p q (f g-e h)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )}{3 f^{3/2} h}-\frac {4 b p q \sqrt {g+h x} (f g-e h)}{3 f h}-\frac {4 b p q (g+h x)^{3/2}}{9 h} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 65
Rule 214
Rule 2442
Rule 2495
Rubi steps
\begin {align*} \int \sqrt {g+h x} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \, dx &=\text {Subst}\left (\int \sqrt {g+h x} \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right ) \, dx,c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac {2 (g+h x)^{3/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{3 h}-\text {Subst}\left (\frac {(2 b f p q) \int \frac {(g+h x)^{3/2}}{e+f x} \, dx}{3 h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-\frac {4 b p q (g+h x)^{3/2}}{9 h}+\frac {2 (g+h x)^{3/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{3 h}-\text {Subst}\left (\frac {(2 b (f g-e h) p q) \int \frac {\sqrt {g+h x}}{e+f x} \, dx}{3 h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-\frac {4 b (f g-e h) p q \sqrt {g+h x}}{3 f h}-\frac {4 b p q (g+h x)^{3/2}}{9 h}+\frac {2 (g+h x)^{3/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{3 h}-\text {Subst}\left (\frac {\left (2 b (f g-e h)^2 p q\right ) \int \frac {1}{(e+f x) \sqrt {g+h x}} \, dx}{3 f h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-\frac {4 b (f g-e h) p q \sqrt {g+h x}}{3 f h}-\frac {4 b p q (g+h x)^{3/2}}{9 h}+\frac {2 (g+h x)^{3/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{3 h}-\text {Subst}\left (\frac {\left (4 b (f g-e h)^2 p q\right ) \text {Subst}\left (\int \frac {1}{e-\frac {f g}{h}+\frac {f x^2}{h}} \, dx,x,\sqrt {g+h x}\right )}{3 f h^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-\frac {4 b (f g-e h) p q \sqrt {g+h x}}{3 f h}-\frac {4 b p q (g+h x)^{3/2}}{9 h}+\frac {4 b (f g-e h)^{3/2} p q \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )}{3 f^{3/2} h}+\frac {2 (g+h x)^{3/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{3 h}\\ \end {align*}
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Mathematica [A]
time = 0.23, size = 124, normalized size = 0.89 \begin {gather*} \frac {2 \left (6 b (f g-e h)^{3/2} p q \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )+\sqrt {f} \sqrt {g+h x} \left (3 a f (g+h x)-2 b p q (4 f g-3 e h+f h x)+3 b f (g+h x) \log \left (c \left (d (e+f x)^p\right )^q\right )\right )\right )}{9 f^{3/2} h} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.22, size = 0, normalized size = 0.00 \[\int \sqrt {h x +g}\, \left (a +b \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )\right )\, dx\]
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 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.42, size = 366, normalized size = 2.63 \begin {gather*} \left [-\frac {2 \, {\left (3 \, {\left (b f g p q - b h p q e\right )} \sqrt {\frac {f g - h e}{f}} \log \left (\frac {f h x + 2 \, f g - 2 \, \sqrt {h x + g} f \sqrt {\frac {f g - h e}{f}} - h e}{f x + e}\right ) + {\left (8 \, b f g p q - 6 \, b h p q e - 3 \, a f g + {\left (2 \, b f h p q - 3 \, a f h\right )} x - 3 \, {\left (b f h p q x + b f g p q\right )} \log \left (f x + e\right ) - 3 \, {\left (b f h x + b f g\right )} \log \left (c\right ) - 3 \, {\left (b f h q x + b f g q\right )} \log \left (d\right )\right )} \sqrt {h x + g}\right )}}{9 \, f h}, \frac {2 \, {\left (6 \, {\left (b f g p q - b h p q e\right )} \sqrt {-\frac {f g - h e}{f}} \arctan \left (-\frac {\sqrt {h x + g} f \sqrt {-\frac {f g - h e}{f}}}{f g - h e}\right ) - {\left (8 \, b f g p q - 6 \, b h p q e - 3 \, a f g + {\left (2 \, b f h p q - 3 \, a f h\right )} x - 3 \, {\left (b f h p q x + b f g p q\right )} \log \left (f x + e\right ) - 3 \, {\left (b f h x + b f g\right )} \log \left (c\right ) - 3 \, {\left (b f h q x + b f g q\right )} \log \left (d\right )\right )} \sqrt {h x + g}\right )}}{9 \, f h}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 3.05, size = 144, normalized size = 1.04 \begin {gather*} \frac {2 \left (\frac {a \left (g + h x\right )^{\frac {3}{2}}}{3} + b \left (- \frac {2 f p q \left (\frac {h \left (g + h x\right )^{\frac {3}{2}}}{3 f} + \frac {\sqrt {g + h x} \left (- e h^{2} + f g h\right )}{f^{2}} + \frac {h \left (e h - f g\right )^{2} \operatorname {atan}{\left (\frac {\sqrt {g + h x}}{\sqrt {\frac {e h - f g}{f}}} \right )}}{f^{3} \sqrt {\frac {e h - f g}{f}}}\right )}{3 h} + \frac {\left (g + h x\right )^{\frac {3}{2}} \log {\left (c \left (d \left (e - \frac {f g}{h} + \frac {f \left (g + h x\right )}{h}\right )^{p}\right )^{q} \right )}}{3}\right )\right )}{h} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \sqrt {g+h\,x}\,\left (a+b\,\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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