Optimal. Leaf size=325 \[ \frac {2 e^{-\frac {a}{b n}} (e f-d g)^2 \sqrt {\pi } (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{b^{3/2} e^3 n^{3/2}}+\frac {4 e^{-\frac {2 a}{b n}} g (e f-d g) \sqrt {2 \pi } (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{b^{3/2} e^3 n^{3/2}}+\frac {2 e^{-\frac {3 a}{b n}} g^2 \sqrt {3 \pi } (d+e x)^3 \left (c (d+e x)^n\right )^{-3/n} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{b^{3/2} e^3 n^{3/2}}-\frac {2 (d+e x) (f+g x)^2}{b e n \sqrt {a+b \log \left (c (d+e x)^n\right )}} \]
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Rubi [A]
time = 0.64, antiderivative size = 325, normalized size of antiderivative = 1.00, number of steps
used = 25, number of rules used = 8, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {2447, 2448,
2436, 2337, 2211, 2235, 2437, 2347} \begin {gather*} \frac {4 \sqrt {2 \pi } g e^{-\frac {2 a}{b n}} (d+e x)^2 (e f-d g) \left (c (d+e x)^n\right )^{-2/n} \text {Erfi}\left (\frac {\sqrt {2} \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{b^{3/2} e^3 n^{3/2}}+\frac {2 \sqrt {\pi } e^{-\frac {a}{b n}} (d+e x) (e f-d g)^2 \left (c (d+e x)^n\right )^{-1/n} \text {Erfi}\left (\frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{b^{3/2} e^3 n^{3/2}}+\frac {2 \sqrt {3 \pi } g^2 e^{-\frac {3 a}{b n}} (d+e x)^3 \left (c (d+e x)^n\right )^{-3/n} \text {Erfi}\left (\frac {\sqrt {3} \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{b^{3/2} e^3 n^{3/2}}-\frac {2 (d+e x) (f+g x)^2}{b e n \sqrt {a+b \log \left (c (d+e x)^n\right )}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2211
Rule 2235
Rule 2337
Rule 2347
Rule 2436
Rule 2437
Rule 2447
Rule 2448
Rubi steps
\begin {align*} \int \frac {(f+g x)^2}{\left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}} \, dx &=-\frac {2 (d+e x) (f+g x)^2}{b e n \sqrt {a+b \log \left (c (d+e x)^n\right )}}+\frac {6 \int \frac {(f+g x)^2}{\sqrt {a+b \log \left (c (d+e x)^n\right )}} \, dx}{b n}-\frac {(4 (e f-d g)) \int \frac {f+g x}{\sqrt {a+b \log \left (c (d+e x)^n\right )}} \, dx}{b e n}\\ &=-\frac {2 (d+e x) (f+g x)^2}{b e n \sqrt {a+b \log \left (c (d+e x)^n\right )}}+\frac {6 \int \left (\frac {(e f-d g)^2}{e^2 \sqrt {a+b \log \left (c (d+e x)^n\right )}}+\frac {2 g (e f-d g) (d+e x)}{e^2 \sqrt {a+b \log \left (c (d+e x)^n\right )}}+\frac {g^2 (d+e x)^2}{e^2 \sqrt {a+b \log \left (c (d+e x)^n\right )}}\right ) \, dx}{b n}-\frac {(4 (e f-d g)) \int \left (\frac {e f-d g}{e \sqrt {a+b \log \left (c (d+e x)^n\right )}}+\frac {g (d+e x)}{e \sqrt {a+b \log \left (c (d+e x)^n\right )}}\right ) \, dx}{b e n}\\ &=-\frac {2 (d+e x) (f+g x)^2}{b e n \sqrt {a+b \log \left (c (d+e x)^n\right )}}+\frac {\left (6 g^2\right ) \int \frac {(d+e x)^2}{\sqrt {a+b \log \left (c (d+e x)^n\right )}} \, dx}{b e^2 n}-\frac {(4 g (e f-d g)) \int \frac {d+e x}{\sqrt {a+b \log \left (c (d+e x)^n\right )}} \, dx}{b e^2 n}+\frac {(12 g (e f-d g)) \int \frac {d+e x}{\sqrt {a+b \log \left (c (d+e x)^n\right )}} \, dx}{b e^2 n}-\frac {\left (4 (e f-d g)^2\right ) \int \frac {1}{\sqrt {a+b \log \left (c (d+e x)^n\right )}} \, dx}{b e^2 n}+\frac {\left (6 (e f-d g)^2\right ) \int \frac {1}{\sqrt {a+b \log \left (c (d+e x)^n\right )}} \, dx}{b e^2 n}\\ &=-\frac {2 (d+e x) (f+g x)^2}{b e n \sqrt {a+b \log \left (c (d+e x)^n\right )}}+\frac {\left (6 g^2\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {a+b \log \left (c x^n\right )}} \, dx,x,d+e x\right )}{b e^3 n}-\frac {(4 g (e f-d g)) \text {Subst}\left (\int \frac {x}{\sqrt {a+b \log \left (c x^n\right )}} \, dx,x,d+e x\right )}{b e^3 n}+\frac {(12 g (e f-d g)) \text {Subst}\left (\int \frac {x}{\sqrt {a+b \log \left (c x^n\right )}} \, dx,x,d+e x\right )}{b e^3 n}-\frac {\left (4 (e f-d g)^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b \log \left (c x^n\right )}} \, dx,x,d+e x\right )}{b e^3 n}+\frac {\left (6 (e f-d g)^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b \log \left (c x^n\right )}} \, dx,x,d+e x\right )}{b e^3 n}\\ &=-\frac {2 (d+e x) (f+g x)^2}{b e n \sqrt {a+b \log \left (c (d+e x)^n\right )}}+\frac {\left (6 g^2 (d+e x)^3 \left (c (d+e x)^n\right )^{-3/n}\right ) \text {Subst}\left (\int \frac {e^{\frac {3 x}{n}}}{\sqrt {a+b x}} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{b e^3 n^2}-\frac {\left (4 g (e f-d g) (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n}\right ) \text {Subst}\left (\int \frac {e^{\frac {2 x}{n}}}{\sqrt {a+b x}} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{b e^3 n^2}+\frac {\left (12 g (e f-d g) (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n}\right ) \text {Subst}\left (\int \frac {e^{\frac {2 x}{n}}}{\sqrt {a+b x}} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{b e^3 n^2}-\frac {\left (4 (e f-d g)^2 (d+e x) \left (c (d+e x)^n\right )^{-1/n}\right ) \text {Subst}\left (\int \frac {e^{\frac {x}{n}}}{\sqrt {a+b x}} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{b e^3 n^2}+\frac {\left (6 (e f-d g)^2 (d+e x) \left (c (d+e x)^n\right )^{-1/n}\right ) \text {Subst}\left (\int \frac {e^{\frac {x}{n}}}{\sqrt {a+b x}} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{b e^3 n^2}\\ &=-\frac {2 (d+e x) (f+g x)^2}{b e n \sqrt {a+b \log \left (c (d+e x)^n\right )}}+\frac {\left (12 g^2 (d+e x)^3 \left (c (d+e x)^n\right )^{-3/n}\right ) \text {Subst}\left (\int e^{-\frac {3 a}{b n}+\frac {3 x^2}{b n}} \, dx,x,\sqrt {a+b \log \left (c (d+e x)^n\right )}\right )}{b^2 e^3 n^2}-\frac {\left (8 g (e f-d g) (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n}\right ) \text {Subst}\left (\int e^{-\frac {2 a}{b n}+\frac {2 x^2}{b n}} \, dx,x,\sqrt {a+b \log \left (c (d+e x)^n\right )}\right )}{b^2 e^3 n^2}+\frac {\left (24 g (e f-d g) (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n}\right ) \text {Subst}\left (\int e^{-\frac {2 a}{b n}+\frac {2 x^2}{b n}} \, dx,x,\sqrt {a+b \log \left (c (d+e x)^n\right )}\right )}{b^2 e^3 n^2}-\frac {\left (8 (e f-d g)^2 (d+e x) \left (c (d+e x)^n\right )^{-1/n}\right ) \text {Subst}\left (\int e^{-\frac {a}{b n}+\frac {x^2}{b n}} \, dx,x,\sqrt {a+b \log \left (c (d+e x)^n\right )}\right )}{b^2 e^3 n^2}+\frac {\left (12 (e f-d g)^2 (d+e x) \left (c (d+e x)^n\right )^{-1/n}\right ) \text {Subst}\left (\int e^{-\frac {a}{b n}+\frac {x^2}{b n}} \, dx,x,\sqrt {a+b \log \left (c (d+e x)^n\right )}\right )}{b^2 e^3 n^2}\\ &=\frac {2 e^{-\frac {a}{b n}} (e f-d g)^2 \sqrt {\pi } (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{b^{3/2} e^3 n^{3/2}}+\frac {4 e^{-\frac {2 a}{b n}} g (e f-d g) \sqrt {2 \pi } (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{b^{3/2} e^3 n^{3/2}}+\frac {2 e^{-\frac {3 a}{b n}} g^2 \sqrt {3 \pi } (d+e x)^3 \left (c (d+e x)^n\right )^{-3/n} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{b^{3/2} e^3 n^{3/2}}-\frac {2 (d+e x) (f+g x)^2}{b e n \sqrt {a+b \log \left (c (d+e x)^n\right )}}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(901\) vs. \(2(325)=650\).
time = 2.16, size = 901, normalized size = 2.77 \begin {gather*} \frac {2 (d+e x) \left (-\sqrt {b} d e^{-\frac {2 a}{b n}} g^2 \sqrt {n} \sqrt {\pi } \left (c (d+e x)^n\right )^{-2/n} \left (2 d e^{\frac {a}{b n}} \left (c (d+e x)^n\right )^{\frac {1}{n}} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )-\sqrt {2} (d+e x) \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )\right )+2 \sqrt {b} e e^{-\frac {2 a}{b n}} f g \sqrt {n} \sqrt {\pi } \left (c (d+e x)^n\right )^{-2/n} \left (-2 d e^{\frac {a}{b n}} \left (c (d+e x)^n\right )^{\frac {1}{n}} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )+\sqrt {2} (d+e x) \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )\right )-\frac {b e^2 n (f+g x)^2}{\sqrt {a+b \log \left (c (d+e x)^n\right )}}+\frac {e^{-\frac {3 a}{b n}} g^2 \sqrt {\pi } (d+e x)^2 \left (c (d+e x)^n\right )^{-3/n} \left (-\sqrt {3}+\frac {3 \sqrt {2} d e^{\frac {a}{b n}} \left (c (d+e x)^n\right )^{\frac {1}{n}}}{d+e x}-\frac {3 d^2 e^{\frac {2 a}{b n}} \left (c (d+e x)^n\right )^{2/n}}{(d+e x)^2}+\frac {3 d^2 e^{\frac {2 a}{b n}} \left (c (d+e x)^n\right )^{2/n} \text {erf}\left (\sqrt {-\frac {a+b \log \left (c (d+e x)^n\right )}{b n}}\right )}{(d+e x)^2}-\frac {3 \sqrt {2} d e^{\frac {a}{b n}} \left (c (d+e x)^n\right )^{\frac {1}{n}} \text {erf}\left (\sqrt {2} \sqrt {-\frac {a+b \log \left (c (d+e x)^n\right )}{b n}}\right )}{d+e x}+\sqrt {3} \text {erf}\left (\sqrt {3} \sqrt {-\frac {a+b \log \left (c (d+e x)^n\right )}{b n}}\right )\right ) \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {-\frac {a+b \log \left (c (d+e x)^n\right )}{b n}}}-\frac {2 d e e^{-\frac {a}{b n}} f g \left (c (d+e x)^n\right )^{-1/n} \Gamma \left (\frac {1}{2},-\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right ) \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {-\frac {a+b \log \left (c (d+e x)^n\right )}{b n}}}+\frac {b e^2 e^{-\frac {a}{b n}} f^2 n \left (c (d+e x)^n\right )^{-1/n} \Gamma \left (\frac {1}{2},-\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right ) \sqrt {-\frac {a+b \log \left (c (d+e x)^n\right )}{b n}}}{\sqrt {a+b \log \left (c (d+e x)^n\right )}}\right )}{b^2 e^3 n^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.14, size = 0, normalized size = 0.00 \[\int \frac {\left (g x +f \right )^{2}}{\left (a +b \ln \left (c \left (e x +d \right )^{n}\right )\right )^{\frac {3}{2}}}\, dx\]
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 \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 {\left (f + g x\right )^{2}}{\left (a + b \log {\left (c \left (d + e x\right )^{n} \right )}\right )^{\frac {3}{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 {{\left (f+g\,x\right )}^2}{{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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