Integrand size = 30, antiderivative size = 625 \[ \int (g+h x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2} \, dx=\frac {3 b^{3/2} e^{-\frac {a}{b p q}} (f g-e h)^2 p^{3/2} \sqrt {\pi } q^{3/2} (e+f x) \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {1}{p q}} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt {b} \sqrt {p} \sqrt {q}}\right )}{4 f^3}+\frac {3 b^{3/2} e^{-\frac {2 a}{b p q}} h (f g-e h) p^{3/2} \sqrt {\frac {\pi }{2}} q^{3/2} (e+f x)^2 \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {2}{p q}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt {b} \sqrt {p} \sqrt {q}}\right )}{8 f^3}+\frac {b^{3/2} e^{-\frac {3 a}{b p q}} h^2 p^{3/2} \sqrt {\frac {\pi }{3}} q^{3/2} (e+f x)^3 \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {3}{p q}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt {b} \sqrt {p} \sqrt {q}}\right )}{12 f^3}-\frac {3 b (f g-e h)^2 p q (e+f x) \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{2 f^3}-\frac {3 b h (f g-e h) p q (e+f x)^2 \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{4 f^3}-\frac {b h^2 p q (e+f x)^3 \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{6 f^3}+\frac {(f g-e h)^2 (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}}{f^3}+\frac {h (f g-e h) (e+f x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}}{f^3}+\frac {h^2 (e+f x)^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}}{3 f^3} \]
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Time = 1.34 (sec) , antiderivative size = 625, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2448, 2436, 2333, 2337, 2211, 2235, 2437, 2342, 2347, 2495} \[ \int (g+h x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2} \, dx=\frac {3 \sqrt {\frac {\pi }{2}} b^{3/2} h p^{3/2} q^{3/2} (e+f x)^2 e^{-\frac {2 a}{b p q}} (f g-e h) \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {2}{p q}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt {b} \sqrt {p} \sqrt {q}}\right )}{8 f^3}+\frac {3 \sqrt {\pi } b^{3/2} p^{3/2} q^{3/2} (e+f x) e^{-\frac {a}{b p q}} (f g-e h)^2 \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {1}{p q}} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt {b} \sqrt {p} \sqrt {q}}\right )}{4 f^3}+\frac {\sqrt {\frac {\pi }{3}} b^{3/2} h^2 p^{3/2} q^{3/2} (e+f x)^3 e^{-\frac {3 a}{b p q}} \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {3}{p q}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt {b} \sqrt {p} \sqrt {q}}\right )}{12 f^3}+\frac {h (e+f x)^2 (f g-e h) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}}{f^3}+\frac {(e+f x) (f g-e h)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}}{f^3}-\frac {3 b h p q (e+f x)^2 (f g-e h) \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{4 f^3}-\frac {3 b p q (e+f x) (f g-e h)^2 \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{2 f^3}+\frac {h^2 (e+f x)^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}}{3 f^3}-\frac {b h^2 p q (e+f x)^3 \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{6 f^3} \]
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Rule 2211
Rule 2235
Rule 2333
Rule 2337
Rule 2342
Rule 2347
Rule 2436
Rule 2437
Rule 2448
Rule 2495
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int (g+h x)^2 \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^{3/2} \, dx,c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right ) \\ & = \text {Subst}\left (\int \left (\frac {(f g-e h)^2 \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^{3/2}}{f^2}+\frac {2 h (f g-e h) (e+f x) \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^{3/2}}{f^2}+\frac {h^2 (e+f x)^2 \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^{3/2}}{f^2}\right ) \, dx,c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right ) \\ & = \text {Subst}\left (\frac {h^2 \int (e+f x)^2 \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^{3/2} \, dx}{f^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\text {Subst}\left (\frac {(2 h (f g-e h)) \int (e+f x) \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^{3/2} \, dx}{f^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\text {Subst}\left (\frac {(f g-e h)^2 \int \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^{3/2} \, dx}{f^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right ) \\ & = \text {Subst}\left (\frac {h^2 \text {Subst}\left (\int x^2 \left (a+b \log \left (c d^q x^{p q}\right )\right )^{3/2} \, dx,x,e+f x\right )}{f^3},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\text {Subst}\left (\frac {(2 h (f g-e h)) \text {Subst}\left (\int x \left (a+b \log \left (c d^q x^{p q}\right )\right )^{3/2} \, dx,x,e+f x\right )}{f^3},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\text {Subst}\left (\frac {(f g-e h)^2 \text {Subst}\left (\int \left (a+b \log \left (c d^q x^{p q}\right )\right )^{3/2} \, dx,x,e+f x\right )}{f^3},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right ) \\ & = \frac {(f g-e h)^2 (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}}{f^3}+\frac {h (f g-e h) (e+f x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}}{f^3}+\frac {h^2 (e+f x)^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}}{3 f^3}-\text {Subst}\left (\frac {\left (b h^2 p q\right ) \text {Subst}\left (\int x^2 \sqrt {a+b \log \left (c d^q x^{p q}\right )} \, dx,x,e+f x\right )}{2 f^3},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\text {Subst}\left (\frac {(3 b h (f g-e h) p q) \text {Subst}\left (\int x \sqrt {a+b \log \left (c d^q x^{p q}\right )} \, dx,x,e+f x\right )}{2 f^3},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\text {Subst}\left (\frac {\left (3 b (f g-e h)^2 p q\right ) \text {Subst}\left (\int \sqrt {a+b \log \left (c d^q x^{p q}\right )} \, dx,x,e+f x\right )}{2 f^3},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right ) \\ & = -\frac {3 b (f g-e h)^2 p q (e+f x) \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{2 f^3}-\frac {3 b h (f g-e h) p q (e+f x)^2 \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{4 f^3}-\frac {b h^2 p q (e+f x)^3 \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{6 f^3}+\frac {(f g-e h)^2 (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}}{f^3}+\frac {h (f g-e h) (e+f x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}}{f^3}+\frac {h^2 (e+f x)^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}}{3 f^3}+\text {Subst}\left (\frac {\left (b^2 h^2 p^2 q^2\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {a+b \log \left (c d^q x^{p q}\right )}} \, dx,x,e+f x\right )}{12 f^3},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\text {Subst}\left (\frac {\left (3 b^2 h (f g-e h) p^2 q^2\right ) \text {Subst}\left (\int \frac {x}{\sqrt {a+b \log \left (c d^q x^{p q}\right )}} \, dx,x,e+f x\right )}{8 f^3},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\text {Subst}\left (\frac {\left (3 b^2 (f g-e h)^2 p^2 q^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b \log \left (c d^q x^{p q}\right )}} \, dx,x,e+f x\right )}{4 f^3},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right ) \\ & = -\frac {3 b (f g-e h)^2 p q (e+f x) \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{2 f^3}-\frac {3 b h (f g-e h) p q (e+f x)^2 \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{4 f^3}-\frac {b h^2 p q (e+f x)^3 \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{6 f^3}+\frac {(f g-e h)^2 (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}}{f^3}+\frac {h (f g-e h) (e+f x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}}{f^3}+\frac {h^2 (e+f x)^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}}{3 f^3}+\text {Subst}\left (\frac {\left (b^2 h^2 p q (e+f x)^3 \left (c d^q (e+f x)^{p q}\right )^{-\frac {3}{p q}}\right ) \text {Subst}\left (\int \frac {e^{\frac {3 x}{p q}}}{\sqrt {a+b x}} \, dx,x,\log \left (c d^q (e+f x)^{p q}\right )\right )}{12 f^3},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\text {Subst}\left (\frac {\left (3 b^2 h (f g-e h) p q (e+f x)^2 \left (c d^q (e+f x)^{p q}\right )^{-\frac {2}{p q}}\right ) \text {Subst}\left (\int \frac {e^{\frac {2 x}{p q}}}{\sqrt {a+b x}} \, dx,x,\log \left (c d^q (e+f x)^{p q}\right )\right )}{8 f^3},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\text {Subst}\left (\frac {\left (3 b^2 (f g-e h)^2 p q (e+f x) \left (c d^q (e+f x)^{p q}\right )^{-\frac {1}{p q}}\right ) \text {Subst}\left (\int \frac {e^{\frac {x}{p q}}}{\sqrt {a+b x}} \, dx,x,\log \left (c d^q (e+f x)^{p q}\right )\right )}{4 f^3},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right ) \\ & = -\frac {3 b (f g-e h)^2 p q (e+f x) \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{2 f^3}-\frac {3 b h (f g-e h) p q (e+f x)^2 \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{4 f^3}-\frac {b h^2 p q (e+f x)^3 \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{6 f^3}+\frac {(f g-e h)^2 (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}}{f^3}+\frac {h (f g-e h) (e+f x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}}{f^3}+\frac {h^2 (e+f x)^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}}{3 f^3}+\text {Subst}\left (\frac {\left (b h^2 p q (e+f x)^3 \left (c d^q (e+f x)^{p q}\right )^{-\frac {3}{p q}}\right ) \text {Subst}\left (\int e^{-\frac {3 a}{b p q}+\frac {3 x^2}{b p q}} \, dx,x,\sqrt {a+b \log \left (c d^q (e+f x)^{p q}\right )}\right )}{6 f^3},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\text {Subst}\left (\frac {\left (3 b h (f g-e h) p q (e+f x)^2 \left (c d^q (e+f x)^{p q}\right )^{-\frac {2}{p q}}\right ) \text {Subst}\left (\int e^{-\frac {2 a}{b p q}+\frac {2 x^2}{b p q}} \, dx,x,\sqrt {a+b \log \left (c d^q (e+f x)^{p q}\right )}\right )}{4 f^3},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\text {Subst}\left (\frac {\left (3 b (f g-e h)^2 p q (e+f x) \left (c d^q (e+f x)^{p q}\right )^{-\frac {1}{p q}}\right ) \text {Subst}\left (\int e^{-\frac {a}{b p q}+\frac {x^2}{b p q}} \, dx,x,\sqrt {a+b \log \left (c d^q (e+f x)^{p q}\right )}\right )}{2 f^3},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right ) \\ & = \frac {3 b^{3/2} e^{-\frac {a}{b p q}} (f g-e h)^2 p^{3/2} \sqrt {\pi } q^{3/2} (e+f x) \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {1}{p q}} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt {b} \sqrt {p} \sqrt {q}}\right )}{4 f^3}+\frac {3 b^{3/2} e^{-\frac {2 a}{b p q}} h (f g-e h) p^{3/2} \sqrt {\frac {\pi }{2}} q^{3/2} (e+f x)^2 \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {2}{p q}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt {b} \sqrt {p} \sqrt {q}}\right )}{8 f^3}+\frac {b^{3/2} e^{-\frac {3 a}{b p q}} h^2 p^{3/2} \sqrt {\frac {\pi }{3}} q^{3/2} (e+f x)^3 \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {3}{p q}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt {b} \sqrt {p} \sqrt {q}}\right )}{12 f^3}-\frac {3 b (f g-e h)^2 p q (e+f x) \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{2 f^3}-\frac {3 b h (f g-e h) p q (e+f x)^2 \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{4 f^3}-\frac {b h^2 p q (e+f x)^3 \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{6 f^3}+\frac {(f g-e h)^2 (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}}{f^3}+\frac {h (f g-e h) (e+f x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}}{f^3}+\frac {h^2 (e+f x)^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}}{3 f^3} \\ \end{align*}
Time = 0.94 (sec) , antiderivative size = 545, normalized size of antiderivative = 0.87 \[ \int (g+h x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2} \, dx=\frac {(e+f x) \left (144 (f g-e h)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}+144 h (f g-e h) (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}+48 h^2 (e+f x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}+4 b h^2 p q (e+f x)^2 \left (\sqrt {b} e^{-\frac {3 a}{b p q}} \sqrt {p} \sqrt {3 \pi } \sqrt {q} \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {3}{p q}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt {b} \sqrt {p} \sqrt {q}}\right )-6 \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}\right )+27 b h (f g-e h) p q (e+f x) \left (\sqrt {b} e^{-\frac {2 a}{b p q}} \sqrt {p} \sqrt {2 \pi } \sqrt {q} \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {2}{p q}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt {b} \sqrt {p} \sqrt {q}}\right )-4 \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}\right )+108 b (f g-e h)^2 p q \left (\sqrt {b} e^{-\frac {a}{b p q}} \sqrt {p} \sqrt {\pi } \sqrt {q} \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {1}{p q}} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt {b} \sqrt {p} \sqrt {q}}\right )-2 \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}\right )\right )}{144 f^3} \]
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\[\int \left (h x +g \right )^{2} {\left (a +b \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )\right )}^{\frac {3}{2}}d x\]
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Exception generated. \[ \int (g+h x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2} \, dx=\text {Exception raised: TypeError} \]
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\[ \int (g+h x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2} \, dx=\int \left (a + b \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}\right )^{\frac {3}{2}} \left (g + h x\right )^{2}\, dx \]
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\[ \int (g+h x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2} \, dx=\int { {\left (h x + g\right )}^{2} {\left (b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a\right )}^{\frac {3}{2}} \,d x } \]
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\[ \int (g+h x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2} \, dx=\int { {\left (h x + g\right )}^{2} {\left (b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a\right )}^{\frac {3}{2}} \,d x } \]
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Timed out. \[ \int (g+h x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2} \, dx=\int {\left (g+h\,x\right )}^2\,{\left (a+b\,\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )\right )}^{3/2} \,d x \]
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