Optimal. Leaf size=120 \[ \frac {4 b f p q}{3 h (f g-e h) \sqrt {g+h x}}-\frac {4 b f^{3/2} p q \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )}{3 h (f g-e h)^{3/2}}-\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{3 h (g+h x)^{3/2}} \]
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Rubi [A]
time = 0.12, antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {2442, 53, 65,
214, 2495} \begin {gather*} -\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{3 h (g+h x)^{3/2}}-\frac {4 b f^{3/2} p q \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )}{3 h (f g-e h)^{3/2}}+\frac {4 b f p q}{3 h \sqrt {g+h x} (f g-e h)} \end {gather*}
Antiderivative was successfully verified.
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Rule 53
Rule 65
Rule 214
Rule 2442
Rule 2495
Rubi steps
\begin {align*} \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{(g+h x)^{5/2}} \, dx &=\text {Subst}\left (\int \frac {a+b \log \left (c d^q (e+f x)^{p q}\right )}{(g+h x)^{5/2}} \, 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 )}{3 h (g+h x)^{3/2}}+\text {Subst}\left (\frac {(2 b f p q) \int \frac {1}{(e+f x) (g+h x)^{3/2}} \, 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 p q}{3 h (f g-e h) \sqrt {g+h x}}-\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{3 h (g+h x)^{3/2}}+\text {Subst}\left (\frac {\left (2 b f^2 p q\right ) \int \frac {1}{(e+f x) \sqrt {g+h x}} \, dx}{3 h (f g-e h)},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac {4 b f p q}{3 h (f g-e h) \sqrt {g+h x}}-\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{3 h (g+h x)^{3/2}}+\text {Subst}\left (\frac {\left (4 b f^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 h^2 (f g-e h)},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac {4 b f p q}{3 h (f g-e h) \sqrt {g+h x}}-\frac {4 b f^{3/2} p q \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )}{3 h (f g-e h)^{3/2}}-\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{3 h (g+h x)^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.32, size = 124, normalized size = 1.03 \begin {gather*} \frac {2 \left (-\frac {2 b f^{3/2} p q \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )}{(f g-e h)^{3/2}}+\frac {a (-f g+e h)+2 b f p q (g+h x)+b (-f g+e h) \log \left (c \left (d (e+f x)^p\right )^q\right )}{(f g-e h) (g+h x)^{3/2}}\right )}{3 h} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.20, size = 0, normalized size = 0.00 \[\int \frac {a +b \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )}{\left (h x +g \right )^{\frac {5}{2}}}\, dx\]
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 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 240 vs.
\(2 (104) = 208\).
time = 0.43, size = 491, normalized size = 4.09 \begin {gather*} \left [-\frac {2 \, {\left ({\left (b f h^{2} p q x^{2} + 2 \, b f g h p q x + b f g^{2} p q\right )} \sqrt {\frac {f}{f g - h e}} \log \left (\frac {f h x + 2 \, f g + 2 \, {\left (f g - h e\right )} \sqrt {h x + g} \sqrt {\frac {f}{f g - h e}} - h e}{f x + e}\right ) - {\left (2 \, b f h p q x + 2 \, b f g p q - a f g + a h e - {\left (b f g p q - b h p q e\right )} \log \left (f x + e\right ) - {\left (b f g - b h e\right )} \log \left (c\right ) - {\left (b f g q - b h q e\right )} \log \left (d\right )\right )} \sqrt {h x + g}\right )}}{3 \, {\left (f g h^{3} x^{2} + 2 \, f g^{2} h^{2} x + f g^{3} h - {\left (h^{4} x^{2} + 2 \, g h^{3} x + g^{2} h^{2}\right )} e\right )}}, -\frac {2 \, {\left (2 \, {\left (b f h^{2} p q x^{2} + 2 \, b f g h p q x + b f g^{2} p q\right )} \sqrt {-\frac {f}{f g - h e}} \arctan \left (-\frac {{\left (f g - h e\right )} \sqrt {h x + g} \sqrt {-\frac {f}{f g - h e}}}{f h x + f g}\right ) - {\left (2 \, b f h p q x + 2 \, b f g p q - a f g + a h e - {\left (b f g p q - b h p q e\right )} \log \left (f x + e\right ) - {\left (b f g - b h e\right )} \log \left (c\right ) - {\left (b f g q - b h q e\right )} \log \left (d\right )\right )} \sqrt {h x + g}\right )}}{3 \, {\left (f g h^{3} x^{2} + 2 \, f g^{2} h^{2} x + f g^{3} h - {\left (h^{4} x^{2} + 2 \, g h^{3} x + g^{2} h^{2}\right )} e\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 95.07, size = 122, normalized size = 1.02 \begin {gather*} \frac {- \frac {2 a}{3 \left (g + h x\right )^{\frac {3}{2}}} + 2 b \left (\frac {2 f p q \left (- \frac {h}{\sqrt {g + h x} \left (e h - f g\right )} - \frac {h \operatorname {atan}{\left (\frac {\sqrt {g + h x}}{\sqrt {\frac {e h - f g}{f}}} \right )}}{\sqrt {\frac {e h - f g}{f}} \left (e h - f g\right )}\right )}{3 h} - \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 )}}{3 \left (g + h x\right )^{\frac {3}{2}}}\right )}{h} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 209 vs.
\(2 (104) = 208\).
time = 2.73, size = 209, normalized size = 1.74 \begin {gather*} \frac {4 \, b f^{2} p q \arctan \left (\frac {\sqrt {h x + g} f}{\sqrt {-f^{2} g + f h e}}\right )}{3 \, \sqrt {-f^{2} g + f h e} {\left (f g h - h^{2} e\right )}} - \frac {2 \, {\left (b f g p q \log \left ({\left (h x + g\right )} f - f g + h e\right ) - b h p q e \log \left ({\left (h x + g\right )} f - f g + h e\right ) - b f g p q \log \left (h\right ) + b h p q e \log \left (h\right ) - 2 \, {\left (h x + g\right )} b f p q + b f g q \log \left (d\right ) - b h q e \log \left (d\right ) + b f g \log \left (c\right ) - b h e \log \left (c\right ) + a f g - a h e\right )}}{3 \, {\left ({\left (h x + g\right )}^{\frac {3}{2}} f g h - {\left (h x + g\right )}^{\frac {3}{2}} h^{2} e\right )}} \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 \frac {a+b\,\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )}{{\left (g+h\,x\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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