Optimal. Leaf size=139 \[ \frac {(e+f x) \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{f}-\frac {\sqrt {\pi } \sqrt {b} \sqrt {p} \sqrt {q} (e+f x) e^{-\frac {a}{b p q}} \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {1}{p q}} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt {b} \sqrt {p} \sqrt {q}}\right )}{2 f} \]
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Rubi [A] time = 0.22, antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {2389, 2296, 2300, 2180, 2204, 2445} \[ \frac {(e+f x) \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{f}-\frac {\sqrt {\pi } \sqrt {b} \sqrt {p} \sqrt {q} (e+f x) e^{-\frac {a}{b p q}} \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {1}{p q}} \text {Erfi}\left (\frac {\sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt {b} \sqrt {p} \sqrt {q}}\right )}{2 f} \]
Antiderivative was successfully verified.
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Rule 2180
Rule 2204
Rule 2296
Rule 2300
Rule 2389
Rule 2445
Rubi steps
\begin {align*} \int \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )} \, dx &=\operatorname {Subst}\left (\int \sqrt {a+b \log \left (c d^q (e+f x)^{p q}\right )} \, dx,c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\operatorname {Subst}\left (\frac {\operatorname {Subst}\left (\int \sqrt {a+b \log \left (c d^q x^{p q}\right )} \, dx,x,e+f x\right )}{f},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac {(e+f x) \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{f}-\operatorname {Subst}\left (\frac {(b p q) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b \log \left (c d^q x^{p q}\right )}} \, dx,x,e+f x\right )}{2 f},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac {(e+f x) \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{f}-\operatorname {Subst}\left (\frac {\left (b (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 )}{2 f},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac {(e+f x) \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{f}-\operatorname {Subst}\left (\frac {\left ((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 )}{f},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-\frac {\sqrt {b} e^{-\frac {a}{b p q}} \sqrt {p} \sqrt {\pi } \sqrt {q} (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 )}{2 f}+\frac {(e+f x) \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{f}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 134, normalized size = 0.96 \[ \frac {(e+f x) \left (2 \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}-\sqrt {\pi } \sqrt {b} \sqrt {p} \sqrt {q} e^{-\frac {a}{b p q}} \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {1}{p q}} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt {b} \sqrt {p} \sqrt {q}}\right )\right )}{2 f} \]
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 \[ \int \sqrt {b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.09, size = 0, normalized size = 0.00 \[ \int \sqrt {b \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )+a}\, dx \]
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 \[ \int \sqrt {b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \sqrt {a+b\,\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )} \,d x \]
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 \[ \int \sqrt {a + b \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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