Optimal. Leaf size=380 \[ \frac {4 e^{-\frac {a}{b p q}} (f g-e h) \sqrt {\pi } (e+f x) \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {1}{p q}} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt {b} \sqrt {p} \sqrt {q}}\right )}{3 b^{5/2} f^2 p^{5/2} q^{5/2}}+\frac {8 e^{-\frac {2 a}{b p q}} h \sqrt {2 \pi } (e+f x)^2 \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {2}{p q}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt {b} \sqrt {p} \sqrt {q}}\right )}{3 b^{5/2} f^2 p^{5/2} q^{5/2}}-\frac {2 (e+f x) (g+h x)}{3 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}}+\frac {4 (f g-e h) (e+f x)}{3 b^2 f^2 p^2 q^2 \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}-\frac {8 (e+f x) (g+h x)}{3 b^2 f p^2 q^2 \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}} \]
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Rubi [A]
time = 1.05, antiderivative size = 380, normalized size of antiderivative = 1.00, number of steps
used = 22, number of rules used = 10, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {2447, 2448,
2436, 2337, 2211, 2235, 2437, 2347, 2334, 2495} \begin {gather*} \frac {4 \sqrt {\pi } (e+f x) e^{-\frac {a}{b p q}} (f g-e h) \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 )}{3 b^{5/2} f^2 p^{5/2} q^{5/2}}+\frac {8 \sqrt {2 \pi } h (e+f x)^2 e^{-\frac {2 a}{b p 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 )}{3 b^{5/2} f^2 p^{5/2} q^{5/2}}+\frac {4 (e+f x) (f g-e h)}{3 b^2 f^2 p^2 q^2 \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}-\frac {8 (e+f x) (g+h x)}{3 b^2 f p^2 q^2 \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}-\frac {2 (e+f x) (g+h x)}{3 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2211
Rule 2235
Rule 2334
Rule 2337
Rule 2347
Rule 2436
Rule 2437
Rule 2447
Rule 2448
Rule 2495
Rubi steps
\begin {align*} \int \frac {g+h x}{\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{5/2}} \, dx &=\text {Subst}\left (\int \frac {g+h x}{\left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^{5/2}} \, 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)}{3 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}}+\text {Subst}\left (\frac {4 \int \frac {g+h x}{\left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^{3/2}} \, dx}{3 b p q},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\text {Subst}\left (\frac {(2 (f g-e h)) \int \frac {1}{\left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^{3/2}} \, dx}{3 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)}{3 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}}-\frac {8 (e+f x) (g+h x)}{3 b^2 f p^2 q^2 \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}+\text {Subst}\left (\frac {16 \int \frac {g+h x}{\sqrt {a+b \log \left (c d^q (e+f x)^{p q}\right )}} \, dx}{3 b^2 p^2 q^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\text {Subst}\left (\frac {(8 (f g-e h)) \int \frac {1}{\sqrt {a+b \log \left (c d^q (e+f x)^{p q}\right )}} \, dx}{3 b^2 f p^2 q^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\text {Subst}\left (\frac {(2 (f g-e h)) \text {Subst}\left (\int \frac {1}{\left (a+b \log \left (c d^q x^{p q}\right )\right )^{3/2}} \, dx,x,e+f x\right )}{3 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)}{3 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}}+\frac {4 (f g-e h) (e+f x)}{3 b^2 f^2 p^2 q^2 \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}-\frac {8 (e+f x) (g+h x)}{3 b^2 f p^2 q^2 \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}+\text {Subst}\left (\frac {16 \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}{3 b^2 p^2 q^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\text {Subst}\left (\frac {(4 (f g-e h)) \text {Subst}\left (\int \frac {1}{\sqrt {a+b \log \left (c d^q x^{p q}\right )}} \, dx,x,e+f x\right )}{3 b^2 f^2 p^2 q^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\text {Subst}\left (\frac {(8 (f g-e h)) \text {Subst}\left (\int \frac {1}{\sqrt {a+b \log \left (c d^q x^{p q}\right )}} \, dx,x,e+f x\right )}{3 b^2 f^2 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)}{3 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}}+\frac {4 (f g-e h) (e+f x)}{3 b^2 f^2 p^2 q^2 \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}-\frac {8 (e+f x) (g+h x)}{3 b^2 f p^2 q^2 \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}+\text {Subst}\left (\frac {(16 h) \int \frac {e+f x}{\sqrt {a+b \log \left (c d^q (e+f x)^{p q}\right )}} \, dx}{3 b^2 f p^2 q^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\text {Subst}\left (\frac {(16 (f g-e h)) \int \frac {1}{\sqrt {a+b \log \left (c d^q (e+f x)^{p q}\right )}} \, dx}{3 b^2 f p^2 q^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\text {Subst}\left (\frac {\left (4 (f g-e h) (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 )}{3 b^2 f^2 p^3 q^3},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\text {Subst}\left (\frac {\left (8 (f g-e h) (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 )}{3 b^2 f^2 p^3 q^3},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)}{3 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}}+\frac {4 (f g-e h) (e+f x)}{3 b^2 f^2 p^2 q^2 \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}-\frac {8 (e+f x) (g+h x)}{3 b^2 f p^2 q^2 \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}+\text {Subst}\left (\frac {(16 h) \text {Subst}\left (\int \frac {x}{\sqrt {a+b \log \left (c d^q x^{p q}\right )}} \, dx,x,e+f x\right )}{3 b^2 f^2 p^2 q^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\text {Subst}\left (\frac {(16 (f g-e h)) \text {Subst}\left (\int \frac {1}{\sqrt {a+b \log \left (c d^q x^{p q}\right )}} \, dx,x,e+f x\right )}{3 b^2 f^2 p^2 q^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\text {Subst}\left (\frac {\left (8 (f g-e h) (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 )}{3 b^3 f^2 p^3 q^3},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\text {Subst}\left (\frac {\left (16 (f g-e h) (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 )}{3 b^3 f^2 p^3 q^3},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-\frac {4 e^{-\frac {a}{b p q}} (f g-e h) \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^{5/2} f^2 p^{5/2} q^{5/2}}-\frac {2 (e+f x) (g+h x)}{3 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}}+\frac {4 (f g-e h) (e+f x)}{3 b^2 f^2 p^2 q^2 \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}-\frac {8 (e+f x) (g+h x)}{3 b^2 f p^2 q^2 \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}+\text {Subst}\left (\frac {\left (16 h (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 )}{3 b^2 f^2 p^3 q^3},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\text {Subst}\left (\frac {\left (16 (f g-e h) (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 )}{3 b^2 f^2 p^3 q^3},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-\frac {4 e^{-\frac {a}{b p q}} (f g-e h) \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^{5/2} f^2 p^{5/2} q^{5/2}}-\frac {2 (e+f x) (g+h x)}{3 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}}+\frac {4 (f g-e h) (e+f x)}{3 b^2 f^2 p^2 q^2 \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}-\frac {8 (e+f x) (g+h x)}{3 b^2 f p^2 q^2 \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}+\text {Subst}\left (\frac {\left (32 h (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 )}{3 b^3 f^2 p^3 q^3},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\text {Subst}\left (\frac {\left (32 (f g-e h) (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 )}{3 b^3 f^2 p^3 q^3},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac {4 e^{-\frac {a}{b p q}} (f g-e h) \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 )}{3 b^{5/2} f^2 p^{5/2} q^{5/2}}+\frac {8 e^{-\frac {2 a}{b p q}} 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 )}{3 b^{5/2} f^2 p^{5/2} q^{5/2}}-\frac {2 (e+f x) (g+h x)}{3 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}}+\frac {4 (f g-e h) (e+f x)}{3 b^2 f^2 p^2 q^2 \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}-\frac {8 (e+f x) (g+h x)}{3 b^2 f p^2 q^2 \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}\\ \end {align*}
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Mathematica [A]
time = 1.52, size = 491, normalized size = 1.29 \begin {gather*} -\frac {2 e^{-\frac {2 a}{b p q}} (e+f x) \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {2}{p q}} \left (8 e e^{\frac {a}{b p q}} h \sqrt {\pi } \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 ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}-4 h \sqrt {2 \pi } (e+f x) \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 ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}+\sqrt {b} e^{\frac {a}{b p q}} \sqrt {p} \sqrt {q} \left (c \left (d (e+f x)^p\right )^q\right )^{\frac {1}{p q}} \left (2 b (f g+3 e h) p q \Gamma \left (\frac {1}{2},-\frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{b p q}\right ) \left (-\frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{b p q}\right )^{3/2}+e^{\frac {a}{b p q}} \left (c \left (d (e+f x)^p\right )^q\right )^{\frac {1}{p q}} \left (b f p q (g+h x)+2 a (f g+e h+2 f h x)+2 b (e h+f (g+2 h x)) \log \left (c \left (d (e+f x)^p\right )^q\right )\right )\right )\right )}{3 b^{5/2} f^2 p^{5/2} q^{5/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {h x +g}{\left (a +b \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )\right )^{\frac {5}{2}}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
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 \begin {gather*} \int \frac {g + h x}{\left (a + b \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}\right )^{\frac {5}{2}}}\, dx \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 = -0.00 \begin {gather*} \int \frac {g+h\,x}{{\left (a+b\,\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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