Optimal. Leaf size=152 \[ \frac {4 b f p q}{15 h (f g-e h) (g+h x)^{3/2}}+\frac {4 b f^2 p q}{5 h (f g-e h)^2 \sqrt {g+h x}}-\frac {4 b f^{5/2} p q \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )}{5 h (f g-e h)^{5/2}}-\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{5 h (g+h x)^{5/2}} \]
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Rubi [A]
time = 0.15, antiderivative size = 152, 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, 53, 65,
214, 2495} \begin {gather*} -\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{5 h (g+h x)^{5/2}}-\frac {4 b f^{5/2} p q \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )}{5 h (f g-e h)^{5/2}}+\frac {4 b f^2 p q}{5 h \sqrt {g+h x} (f g-e h)^2}+\frac {4 b f p q}{15 h (g+h x)^{3/2} (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)^{7/2}} \, dx &=\text {Subst}\left (\int \frac {a+b \log \left (c d^q (e+f x)^{p q}\right )}{(g+h x)^{7/2}} \, 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 )}{5 h (g+h x)^{5/2}}+\text {Subst}\left (\frac {(2 b f p q) \int \frac {1}{(e+f x) (g+h x)^{5/2}} \, dx}{5 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}{15 h (f g-e h) (g+h x)^{3/2}}-\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{5 h (g+h x)^{5/2}}+\text {Subst}\left (\frac {\left (2 b f^2 p q\right ) \int \frac {1}{(e+f x) (g+h x)^{3/2}} \, dx}{5 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}{15 h (f g-e h) (g+h x)^{3/2}}+\frac {4 b f^2 p q}{5 h (f g-e h)^2 \sqrt {g+h x}}-\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{5 h (g+h x)^{5/2}}+\text {Subst}\left (\frac {\left (2 b f^3 p q\right ) \int \frac {1}{(e+f x) \sqrt {g+h x}} \, dx}{5 h (f g-e h)^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac {4 b f p q}{15 h (f g-e h) (g+h x)^{3/2}}+\frac {4 b f^2 p q}{5 h (f g-e h)^2 \sqrt {g+h x}}-\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{5 h (g+h x)^{5/2}}+\text {Subst}\left (\frac {\left (4 b f^3 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 )}{5 h^2 (f g-e h)^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac {4 b f p q}{15 h (f g-e h) (g+h x)^{3/2}}+\frac {4 b f^2 p q}{5 h (f g-e h)^2 \sqrt {g+h x}}-\frac {4 b f^{5/2} p q \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )}{5 h (f g-e h)^{5/2}}-\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{5 h (g+h x)^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 0.53, size = 135, normalized size = 0.89 \begin {gather*} \frac {2 \left (-\frac {3 a}{(g+h x)^{5/2}}+\frac {2 b f p q (4 f g-e h+3 f h x)}{(f g-e h)^2 (g+h x)^{3/2}}-\frac {6 b f^{5/2} p q \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )}{(f g-e h)^{5/2}}-\frac {3 b \log \left (c \left (d (e+f x)^p\right )^q\right )}{(g+h x)^{5/2}}\right )}{15 h} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.22, size = 0, normalized size = 0.00 \[\int \frac {a +b \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )}{\left (h x +g \right )^{\frac {7}{2}}}\, dx\]
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. 436 vs.
\(2 (133) = 266\).
time = 0.43, size = 883, normalized size = 5.81 \begin {gather*} \left [\frac {2 \, {\left (3 \, {\left (b f^{2} h^{3} p q x^{3} + 3 \, b f^{2} g h^{2} p q x^{2} + 3 \, b f^{2} g^{2} h p q x + b f^{2} g^{3} 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 (6 \, b f^{2} h^{2} p q x^{2} + 14 \, b f^{2} g h p q x + 8 \, b f^{2} g^{2} p q - 3 \, a f^{2} g^{2} - 3 \, a h^{2} e^{2} - 2 \, {\left (b f h^{2} p q x + b f g h p q - 3 \, a f g h\right )} e - 3 \, {\left (b f^{2} g^{2} p q - 2 \, b f g h p q e + b h^{2} p q e^{2}\right )} \log \left (f x + e\right ) - 3 \, {\left (b f^{2} g^{2} - 2 \, b f g h e + b h^{2} e^{2}\right )} \log \left (c\right ) - 3 \, {\left (b f^{2} g^{2} q - 2 \, b f g h q e + b h^{2} q e^{2}\right )} \log \left (d\right )\right )} \sqrt {h x + g}\right )}}{15 \, {\left (f^{2} g^{2} h^{4} x^{3} + 3 \, f^{2} g^{3} h^{3} x^{2} + 3 \, f^{2} g^{4} h^{2} x + f^{2} g^{5} h + {\left (h^{6} x^{3} + 3 \, g h^{5} x^{2} + 3 \, g^{2} h^{4} x + g^{3} h^{3}\right )} e^{2} - 2 \, {\left (f g h^{5} x^{3} + 3 \, f g^{2} h^{4} x^{2} + 3 \, f g^{3} h^{3} x + f g^{4} h^{2}\right )} e\right )}}, -\frac {2 \, {\left (6 \, {\left (b f^{2} h^{3} p q x^{3} + 3 \, b f^{2} g h^{2} p q x^{2} + 3 \, b f^{2} g^{2} h p q x + b f^{2} g^{3} 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 (6 \, b f^{2} h^{2} p q x^{2} + 14 \, b f^{2} g h p q x + 8 \, b f^{2} g^{2} p q - 3 \, a f^{2} g^{2} - 3 \, a h^{2} e^{2} - 2 \, {\left (b f h^{2} p q x + b f g h p q - 3 \, a f g h\right )} e - 3 \, {\left (b f^{2} g^{2} p q - 2 \, b f g h p q e + b h^{2} p q e^{2}\right )} \log \left (f x + e\right ) - 3 \, {\left (b f^{2} g^{2} - 2 \, b f g h e + b h^{2} e^{2}\right )} \log \left (c\right ) - 3 \, {\left (b f^{2} g^{2} q - 2 \, b f g h q e + b h^{2} q e^{2}\right )} \log \left (d\right )\right )} \sqrt {h x + g}\right )}}{15 \, {\left (f^{2} g^{2} h^{4} x^{3} + 3 \, f^{2} g^{3} h^{3} x^{2} + 3 \, f^{2} g^{4} h^{2} x + f^{2} g^{5} h + {\left (h^{6} x^{3} + 3 \, g h^{5} x^{2} + 3 \, g^{2} h^{4} x + g^{3} h^{3}\right )} e^{2} - 2 \, {\left (f g h^{5} x^{3} + 3 \, f g^{2} h^{4} x^{2} + 3 \, f g^{3} h^{3} x + f g^{4} h^{2}\right )} e\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \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. 378 vs.
\(2 (133) = 266\).
time = 5.71, size = 378, normalized size = 2.49 \begin {gather*} \frac {4 \, b f^{3} h p q \arctan \left (\frac {\sqrt {h x + g} f}{\sqrt {-f^{2} g + f h e}}\right )}{5 \, {\left (f^{2} g^{2} h^{2} - 2 \, f g h^{3} e + h^{4} e^{2}\right )} \sqrt {-f^{2} g + f h e}} - \frac {2 \, {\left (3 \, b f^{2} g^{2} p q \log \left ({\left (h x + g\right )} f - f g + h e\right ) - 6 \, b f g h p q e \log \left ({\left (h x + g\right )} f - f g + h e\right ) - 3 \, b f^{2} g^{2} p q \log \left (h\right ) + 6 \, b f g h p q e \log \left (h\right ) - 6 \, {\left (h x + g\right )}^{2} b f^{2} p q - 2 \, {\left (h x + g\right )} b f^{2} g p q + 2 \, {\left (h x + g\right )} b f h p q e + 3 \, b h^{2} p q e^{2} \log \left ({\left (h x + g\right )} f - f g + h e\right ) + 3 \, b f^{2} g^{2} q \log \left (d\right ) - 6 \, b f g h q e \log \left (d\right ) - 3 \, b h^{2} p q e^{2} \log \left (h\right ) + 3 \, b f^{2} g^{2} \log \left (c\right ) - 6 \, b f g h e \log \left (c\right ) + 3 \, b h^{2} q e^{2} \log \left (d\right ) + 3 \, a f^{2} g^{2} - 6 \, a f g h e + 3 \, b h^{2} e^{2} \log \left (c\right ) + 3 \, a h^{2} e^{2}\right )}}{15 \, {\left ({\left (h x + g\right )}^{\frac {5}{2}} f^{2} g^{2} h - 2 \, {\left (h x + g\right )}^{\frac {5}{2}} f g h^{2} e + {\left (h x + g\right )}^{\frac {5}{2}} h^{3} e^{2}\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 )}^{7/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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