Optimal. Leaf size=404 \[ \frac {4 \sqrt {2 \pi } h (e+f x)^2 e^{-\frac {2 a}{b p q}} (f g-e h) \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {2}{p q}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt {b} \sqrt {p} \sqrt {q}}\right )}{b^{3/2} f^3 p^{3/2} q^{3/2}}+\frac {2 \sqrt {\pi } (e+f x) e^{-\frac {a}{b p q}} (f g-e h)^2 \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {1}{p q}} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt {b} \sqrt {p} \sqrt {q}}\right )}{b^{3/2} f^3 p^{3/2} q^{3/2}}+\frac {2 \sqrt {3 \pi } h^2 (e+f x)^3 e^{-\frac {3 a}{b p q}} \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {3}{p q}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt {b} \sqrt {p} \sqrt {q}}\right )}{b^{3/2} f^3 p^{3/2} q^{3/2}}-\frac {2 (e+f x) (g+h x)^2}{b f p q \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}} \]
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Rubi [A] time = 2.25, antiderivative size = 404, normalized size of antiderivative = 1.00, number of steps used = 26, number of rules used = 9, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {2400, 2401, 2389, 2300, 2180, 2204, 2390, 2310, 2445} \[ \frac {4 \sqrt {2 \pi } h (e+f x)^2 e^{-\frac {2 a}{b p q}} (f g-e h) \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {2}{p q}} \text {Erfi}\left (\frac {\sqrt {2} \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt {b} \sqrt {p} \sqrt {q}}\right )}{b^{3/2} f^3 p^{3/2} q^{3/2}}+\frac {2 \sqrt {\pi } (e+f x) e^{-\frac {a}{b p q}} (f g-e h)^2 \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {1}{p q}} \text {Erfi}\left (\frac {\sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt {b} \sqrt {p} \sqrt {q}}\right )}{b^{3/2} f^3 p^{3/2} q^{3/2}}+\frac {2 \sqrt {3 \pi } h^2 (e+f x)^3 e^{-\frac {3 a}{b p q}} \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {3}{p q}} \text {Erfi}\left (\frac {\sqrt {3} \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt {b} \sqrt {p} \sqrt {q}}\right )}{b^{3/2} f^3 p^{3/2} q^{3/2}}-\frac {2 (e+f x) (g+h x)^2}{b f p q \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}} \]
Antiderivative was successfully verified.
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Rule 2180
Rule 2204
Rule 2300
Rule 2310
Rule 2389
Rule 2390
Rule 2400
Rule 2401
Rule 2445
Rubi steps
\begin {align*} \int \frac {(g+h x)^2}{\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}} \, dx &=\operatorname {Subst}\left (\int \frac {(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 )\\ &=-\frac {2 (e+f x) (g+h x)^2}{b f p q \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}+\operatorname {Subst}\left (\frac {6 \int \frac {(g+h x)^2}{\sqrt {a+b \log \left (c d^q (e+f x)^{p q}\right )}} \, dx}{b p q},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\operatorname {Subst}\left (\frac {(4 (f g-e h)) \int \frac {g+h x}{\sqrt {a+b \log \left (c d^q (e+f x)^{p q}\right )}} \, dx}{b f p q},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-\frac {2 (e+f x) (g+h x)^2}{b f p q \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}+\operatorname {Subst}\left (\frac {6 \int \left (\frac {(f g-e h)^2}{f^2 \sqrt {a+b \log \left (c d^q (e+f x)^{p q}\right )}}+\frac {2 h (f g-e h) (e+f x)}{f^2 \sqrt {a+b \log \left (c d^q (e+f x)^{p q}\right )}}+\frac {h^2 (e+f x)^2}{f^2 \sqrt {a+b \log \left (c d^q (e+f x)^{p q}\right )}}\right ) \, dx}{b p q},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\operatorname {Subst}\left (\frac {(4 (f g-e h)) \int \left (\frac {f g-e h}{f \sqrt {a+b \log \left (c d^q (e+f x)^{p q}\right )}}+\frac {h (e+f x)}{f \sqrt {a+b \log \left (c d^q (e+f x)^{p q}\right )}}\right ) \, dx}{b f p q},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-\frac {2 (e+f x) (g+h x)^2}{b f p q \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}+\operatorname {Subst}\left (\frac {\left (6 h^2\right ) \int \frac {(e+f x)^2}{\sqrt {a+b \log \left (c d^q (e+f x)^{p q}\right )}} \, dx}{b f^2 p q},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\operatorname {Subst}\left (\frac {(4 h (f g-e h)) \int \frac {e+f x}{\sqrt {a+b \log \left (c d^q (e+f x)^{p q}\right )}} \, dx}{b f^2 p q},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\operatorname {Subst}\left (\frac {(12 h (f g-e h)) \int \frac {e+f x}{\sqrt {a+b \log \left (c d^q (e+f x)^{p q}\right )}} \, dx}{b f^2 p q},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\operatorname {Subst}\left (\frac {\left (4 (f g-e h)^2\right ) \int \frac {1}{\sqrt {a+b \log \left (c d^q (e+f x)^{p q}\right )}} \, dx}{b f^2 p q},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\operatorname {Subst}\left (\frac {\left (6 (f g-e h)^2\right ) \int \frac {1}{\sqrt {a+b \log \left (c d^q (e+f x)^{p q}\right )}} \, dx}{b f^2 p q},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-\frac {2 (e+f x) (g+h x)^2}{b f p q \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}+\operatorname {Subst}\left (\frac {\left (6 h^2\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {a+b \log \left (c d^q x^{p q}\right )}} \, dx,x,e+f x\right )}{b f^3 p q},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\operatorname {Subst}\left (\frac {(4 h (f g-e h)) \operatorname {Subst}\left (\int \frac {x}{\sqrt {a+b \log \left (c d^q x^{p q}\right )}} \, dx,x,e+f x\right )}{b f^3 p q},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\operatorname {Subst}\left (\frac {(12 h (f g-e h)) \operatorname {Subst}\left (\int \frac {x}{\sqrt {a+b \log \left (c d^q x^{p q}\right )}} \, dx,x,e+f x\right )}{b f^3 p q},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\operatorname {Subst}\left (\frac {\left (4 (f g-e h)^2\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b \log \left (c d^q x^{p q}\right )}} \, dx,x,e+f x\right )}{b f^3 p q},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\operatorname {Subst}\left (\frac {\left (6 (f g-e h)^2\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b \log \left (c d^q x^{p q}\right )}} \, dx,x,e+f x\right )}{b f^3 p q},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-\frac {2 (e+f x) (g+h x)^2}{b f p q \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}+\operatorname {Subst}\left (\frac {\left (6 h^2 (e+f x)^3 \left (c d^q (e+f x)^{p q}\right )^{-\frac {3}{p q}}\right ) \operatorname {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 )}{b f^3 p^2 q^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\operatorname {Subst}\left (\frac {\left (4 h (f g-e h) (e+f x)^2 \left (c d^q (e+f x)^{p q}\right )^{-\frac {2}{p q}}\right ) \operatorname {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 )}{b f^3 p^2 q^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\operatorname {Subst}\left (\frac {\left (12 h (f g-e h) (e+f x)^2 \left (c d^q (e+f x)^{p q}\right )^{-\frac {2}{p q}}\right ) \operatorname {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 )}{b f^3 p^2 q^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\operatorname {Subst}\left (\frac {\left (4 (f g-e h)^2 (e+f x) \left (c d^q (e+f x)^{p q}\right )^{-\frac {1}{p q}}\right ) \operatorname {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 )}{b f^3 p^2 q^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\operatorname {Subst}\left (\frac {\left (6 (f g-e h)^2 (e+f x) \left (c d^q (e+f x)^{p q}\right )^{-\frac {1}{p q}}\right ) \operatorname {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 )}{b f^3 p^2 q^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-\frac {2 (e+f x) (g+h x)^2}{b f p q \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}+\operatorname {Subst}\left (\frac {\left (12 h^2 (e+f x)^3 \left (c d^q (e+f x)^{p q}\right )^{-\frac {3}{p q}}\right ) \operatorname {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 )}{b^2 f^3 p^2 q^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\operatorname {Subst}\left (\frac {\left (8 h (f g-e h) (e+f x)^2 \left (c d^q (e+f x)^{p q}\right )^{-\frac {2}{p q}}\right ) \operatorname {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 )}{b^2 f^3 p^2 q^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\operatorname {Subst}\left (\frac {\left (24 h (f g-e h) (e+f x)^2 \left (c d^q (e+f x)^{p q}\right )^{-\frac {2}{p q}}\right ) \operatorname {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 )}{b^2 f^3 p^2 q^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\operatorname {Subst}\left (\frac {\left (8 (f g-e h)^2 (e+f x) \left (c d^q (e+f x)^{p q}\right )^{-\frac {1}{p q}}\right ) \operatorname {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 )}{b^2 f^3 p^2 q^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\operatorname {Subst}\left (\frac {\left (12 (f g-e h)^2 (e+f x) \left (c d^q (e+f x)^{p q}\right )^{-\frac {1}{p q}}\right ) \operatorname {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 )}{b^2 f^3 p^2 q^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac {2 e^{-\frac {a}{b p q}} (f g-e h)^2 \sqrt {\pi } (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 )}{b^{3/2} f^3 p^{3/2} q^{3/2}}+\frac {4 e^{-\frac {2 a}{b p q}} h (f g-e h) \sqrt {2 \pi } (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 )}{b^{3/2} f^3 p^{3/2} q^{3/2}}+\frac {2 e^{-\frac {3 a}{b p q}} h^2 \sqrt {3 \pi } (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 )}{b^{3/2} f^3 p^{3/2} q^{3/2}}-\frac {2 (e+f x) (g+h x)^2}{b f p q \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}\\ \end {align*}
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Mathematica [B] time = 2.57, size = 1040, normalized size = 2.57 \[ \frac {2 \left (e^{-\frac {3 a}{b p q}} h^2 \sqrt {3 \pi } (e+f x)^3 \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 ) \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )} \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {3}{p q}}-2 e e^{-\frac {2 a}{b p q}} h^2 \sqrt {2 \pi } (e+f x)^2 \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 ) \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )} \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {2}{p q}}+2 e^{-\frac {2 a}{b p q}} f g h \sqrt {2 \pi } (e+f x)^2 \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 ) \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )} \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {2}{p q}}+e^2 e^{-\frac {a}{b p q}} h^2 \sqrt {\pi } (e+f x) \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 ) \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )} \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {1}{p q}}-4 e e^{-\frac {a}{b p q}} f g h \sqrt {\pi } (e+f x) \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 ) \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )} \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {1}{p q}}+\sqrt {b} e^{-\frac {a}{b p q}} f^2 g^2 \sqrt {p} \sqrt {q} (e+f x) \Gamma \left (\frac {1}{2},-\frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{b p q}\right ) \sqrt {-\frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{b p q}} \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {1}{p q}}+2 \sqrt {b} e e^{-\frac {a}{b p q}} f g h \sqrt {p} \sqrt {q} (e+f x) \Gamma \left (\frac {1}{2},-\frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{b p q}\right ) \sqrt {-\frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{b p q}} \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {1}{p q}}-\sqrt {b} f^3 h^2 \sqrt {p} \sqrt {q} x^3-\sqrt {b} e f^2 h^2 \sqrt {p} \sqrt {q} x^2-2 \sqrt {b} f^3 g h \sqrt {p} \sqrt {q} x^2-\sqrt {b} f^3 g^2 \sqrt {p} \sqrt {q} x-2 \sqrt {b} e f^2 g h \sqrt {p} \sqrt {q} x-\sqrt {b} e f^2 g^2 \sqrt {p} \sqrt {q}\right )}{b^{3/2} f^3 p^{3/2} q^{3/2} \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}} \]
Antiderivative was successfully verified.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
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 \[ \int \frac {{\left (h x + g\right )}^{2}}{{\left (b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.33, size = 0, normalized size = 0.00 \[ \int \frac {\left (h x +g \right )^{2}}{\left (b \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )+a \right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (h x + g\right )}^{2}}{{\left (b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {{\left (g+h\,x\right )}^2}{{\left (a+b\,\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )\right )}^{3/2}} \,d x \]
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 \[ \int \frac {\left (g + h x\right )^{2}}{\left (a + b \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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