Integrand size = 26, antiderivative size = 422 \[ \int \frac {(f+g x)^3}{\left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}} \, dx=\frac {2 e^{-\frac {a}{b n}} (e f-d g)^3 \sqrt {\pi } (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{b^{3/2} e^4 n^{3/2}}+\frac {4 e^{-\frac {4 a}{b n}} g^3 \sqrt {\pi } (d+e x)^4 \left (c (d+e x)^n\right )^{-4/n} \text {erfi}\left (\frac {2 \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{b^{3/2} e^4 n^{3/2}}+\frac {6 e^{-\frac {2 a}{b n}} g (e f-d g)^2 \sqrt {2 \pi } (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{b^{3/2} e^4 n^{3/2}}+\frac {6 e^{-\frac {3 a}{b n}} g^2 (e f-d g) \sqrt {3 \pi } (d+e x)^3 \left (c (d+e x)^n\right )^{-3/n} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{b^{3/2} e^4 n^{3/2}}-\frac {2 (d+e x) (f+g x)^3}{b e n \sqrt {a+b \log \left (c (d+e x)^n\right )}} \]
2*(-d*g+e*f)^3*(e*x+d)*erfi((a+b*ln(c*(e*x+d)^n))^(1/2)/b^(1/2)/n^(1/2))*P i^(1/2)/b^(3/2)/e^4/exp(a/b/n)/n^(3/2)/((c*(e*x+d)^n)^(1/n))+4*g^3*(e*x+d) ^4*erfi(2*(a+b*ln(c*(e*x+d)^n))^(1/2)/b^(1/2)/n^(1/2))*Pi^(1/2)/b^(3/2)/e^ 4/exp(4*a/b/n)/n^(3/2)/((c*(e*x+d)^n)^(4/n))+6*g*(-d*g+e*f)^2*(e*x+d)^2*er fi(2^(1/2)*(a+b*ln(c*(e*x+d)^n))^(1/2)/b^(1/2)/n^(1/2))*2^(1/2)*Pi^(1/2)/b ^(3/2)/e^4/exp(2*a/b/n)/n^(3/2)/((c*(e*x+d)^n)^(2/n))+6*g^2*(-d*g+e*f)*(e* x+d)^3*erfi(3^(1/2)*(a+b*ln(c*(e*x+d)^n))^(1/2)/b^(1/2)/n^(1/2))*3^(1/2)*P i^(1/2)/b^(3/2)/e^4/exp(3*a/b/n)/n^(3/2)/((c*(e*x+d)^n)^(3/n))-2*(e*x+d)*( g*x+f)^3/b/e/n/(a+b*ln(c*(e*x+d)^n))^(1/2)
Leaf count is larger than twice the leaf count of optimal. \(1281\) vs. \(2(422)=844\).
Time = 1.70 (sec) , antiderivative size = 1281, normalized size of antiderivative = 3.04 \[ \int \frac {(f+g x)^3}{\left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}} \, dx=\frac {2 \left (-\sqrt {b} d e^3 f^3 \sqrt {n}-\sqrt {b} e^4 f^3 \sqrt {n} x-3 \sqrt {b} d e^3 f^2 g \sqrt {n} x-3 \sqrt {b} e^4 f^2 g \sqrt {n} x^2-3 \sqrt {b} d e^3 f g^2 \sqrt {n} x^2-3 \sqrt {b} e^4 f g^2 \sqrt {n} x^3-\sqrt {b} d e^3 g^3 \sqrt {n} x^3-\sqrt {b} e^4 g^3 \sqrt {n} x^4-6 d e^2 e^{-\frac {a}{b n}} f^2 g \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 ) \sqrt {a+b \log \left (c (d+e x)^n\right )}+3 d^2 e e^{-\frac {a}{b n}} f 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 ) \sqrt {a+b \log \left (c (d+e x)^n\right )}-d^3 e^{-\frac {a}{b n}} g^3 \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 ) \sqrt {a+b \log \left (c (d+e x)^n\right )}+2 e^{-\frac {4 a}{b n}} g^3 \sqrt {\pi } (d+e x)^4 \left (c (d+e x)^n\right )^{-4/n} \text {erfi}\left (\frac {2 \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right ) \sqrt {a+b \log \left (c (d+e x)^n\right )}+3 e^2 e^{-\frac {2 a}{b n}} f^2 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 ) \sqrt {a+b \log \left (c (d+e x)^n\right )}-6 d e e^{-\frac {2 a}{b n}} f g^2 \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 ) \sqrt {a+b \log \left (c (d+e x)^n\right )}+3 d^2 e^{-\frac {2 a}{b n}} g^3 \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 ) \sqrt {a+b \log \left (c (d+e x)^n\right )}+3 e e^{-\frac {3 a}{b n}} f 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 ) \sqrt {a+b \log \left (c (d+e x)^n\right )}-3 d e^{-\frac {3 a}{b n}} g^3 \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 ) \sqrt {a+b \log \left (c (d+e x)^n\right )}+\sqrt {b} e^3 e^{-\frac {a}{b n}} f^3 \sqrt {n} (d+e x) \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}}+3 \sqrt {b} d e^2 e^{-\frac {a}{b n}} f^2 g \sqrt {n} (d+e x) \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}}\right )}{b^{3/2} e^4 n^{3/2} \sqrt {a+b \log \left (c (d+e x)^n\right )}} \]
(2*(-(Sqrt[b]*d*e^3*f^3*Sqrt[n]) - Sqrt[b]*e^4*f^3*Sqrt[n]*x - 3*Sqrt[b]*d *e^3*f^2*g*Sqrt[n]*x - 3*Sqrt[b]*e^4*f^2*g*Sqrt[n]*x^2 - 3*Sqrt[b]*d*e^3*f *g^2*Sqrt[n]*x^2 - 3*Sqrt[b]*e^4*f*g^2*Sqrt[n]*x^3 - Sqrt[b]*d*e^3*g^3*Sqr t[n]*x^3 - Sqrt[b]*e^4*g^3*Sqrt[n]*x^4 - (6*d*e^2*f^2*g*Sqrt[Pi]*(d + e*x) *Erfi[Sqrt[a + b*Log[c*(d + e*x)^n]]/(Sqrt[b]*Sqrt[n])]*Sqrt[a + b*Log[c*( d + e*x)^n]])/(E^(a/(b*n))*(c*(d + e*x)^n)^n^(-1)) + (3*d^2*e*f*g^2*Sqrt[P i]*(d + e*x)*Erfi[Sqrt[a + b*Log[c*(d + e*x)^n]]/(Sqrt[b]*Sqrt[n])]*Sqrt[a + b*Log[c*(d + e*x)^n]])/(E^(a/(b*n))*(c*(d + e*x)^n)^n^(-1)) - (d^3*g^3* Sqrt[Pi]*(d + e*x)*Erfi[Sqrt[a + b*Log[c*(d + e*x)^n]]/(Sqrt[b]*Sqrt[n])]* Sqrt[a + b*Log[c*(d + e*x)^n]])/(E^(a/(b*n))*(c*(d + e*x)^n)^n^(-1)) + (2* g^3*Sqrt[Pi]*(d + e*x)^4*Erfi[(2*Sqrt[a + b*Log[c*(d + e*x)^n]])/(Sqrt[b]* Sqrt[n])]*Sqrt[a + b*Log[c*(d + e*x)^n]])/(E^((4*a)/(b*n))*(c*(d + e*x)^n) ^(4/n)) + (3*e^2*f^2*g*Sqrt[2*Pi]*(d + e*x)^2*Erfi[(Sqrt[2]*Sqrt[a + b*Log [c*(d + e*x)^n]])/(Sqrt[b]*Sqrt[n])]*Sqrt[a + b*Log[c*(d + e*x)^n]])/(E^(( 2*a)/(b*n))*(c*(d + e*x)^n)^(2/n)) - (6*d*e*f*g^2*Sqrt[2*Pi]*(d + e*x)^2*E rfi[(Sqrt[2]*Sqrt[a + b*Log[c*(d + e*x)^n]])/(Sqrt[b]*Sqrt[n])]*Sqrt[a + b *Log[c*(d + e*x)^n]])/(E^((2*a)/(b*n))*(c*(d + e*x)^n)^(2/n)) + (3*d^2*g^3 *Sqrt[2*Pi]*(d + e*x)^2*Erfi[(Sqrt[2]*Sqrt[a + b*Log[c*(d + e*x)^n]])/(Sqr t[b]*Sqrt[n])]*Sqrt[a + b*Log[c*(d + e*x)^n]])/(E^((2*a)/(b*n))*(c*(d + e* x)^n)^(2/n)) + (3*e*f*g^2*Sqrt[3*Pi]*(d + e*x)^3*Erfi[(Sqrt[3]*Sqrt[a +...
Time = 1.83 (sec) , antiderivative size = 735, normalized size of antiderivative = 1.74, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {2847, 2848, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {(f+g x)^3}{\left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 2847 |
| \(\displaystyle -\frac {6 (e f-d g) \int \frac {(f+g x)^2}{\sqrt {a+b \log \left (c (d+e x)^n\right )}}dx}{b e n}+\frac {8 \int \frac {(f+g x)^3}{\sqrt {a+b \log \left (c (d+e x)^n\right )}}dx}{b n}-\frac {2 (d+e x) (f+g x)^3}{b e n \sqrt {a+b \log \left (c (d+e x)^n\right )}}\) |
| \(\Big \downarrow \) 2848 |
| \(\displaystyle \frac {8 \int \left (\frac {(e f-d g)^3}{e^3 \sqrt {a+b \log \left (c (d+e x)^n\right )}}+\frac {3 g (d+e x) (e f-d g)^2}{e^3 \sqrt {a+b \log \left (c (d+e x)^n\right )}}+\frac {3 g^2 (d+e x)^2 (e f-d g)}{e^3 \sqrt {a+b \log \left (c (d+e x)^n\right )}}+\frac {g^3 (d+e x)^3}{e^3 \sqrt {a+b \log \left (c (d+e x)^n\right )}}\right )dx}{b n}-\frac {6 (e f-d g) \int \left (\frac {(e f-d g)^2}{e^2 \sqrt {a+b \log \left (c (d+e x)^n\right )}}+\frac {2 g (d+e x) (e f-d g)}{e^2 \sqrt {a+b \log \left (c (d+e x)^n\right )}}+\frac {g^2 (d+e x)^2}{e^2 \sqrt {a+b \log \left (c (d+e x)^n\right )}}\right )dx}{b e n}-\frac {2 (d+e x) (f+g x)^3}{b e n \sqrt {a+b \log \left (c (d+e x)^n\right )}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {8 \left (\frac {\sqrt {3 \pi } g^2 e^{-\frac {3 a}{b n}} (d+e x)^3 (e f-d g) \left (c (d+e x)^n\right )^{-3/n} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{\sqrt {b} e^4 \sqrt {n}}+\frac {3 \sqrt {\frac {\pi }{2}} g e^{-\frac {2 a}{b n}} (d+e x)^2 (e f-d g)^2 \left (c (d+e x)^n\right )^{-2/n} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{\sqrt {b} e^4 \sqrt {n}}+\frac {\sqrt {\pi } e^{-\frac {a}{b n}} (d+e x) (e f-d g)^3 \left (c (d+e x)^n\right )^{-1/n} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{\sqrt {b} e^4 \sqrt {n}}+\frac {\sqrt {\pi } g^3 e^{-\frac {4 a}{b n}} (d+e x)^4 \left (c (d+e x)^n\right )^{-4/n} \text {erfi}\left (\frac {2 \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{2 \sqrt {b} e^4 \sqrt {n}}\right )}{b n}-\frac {6 (e f-d g) \left (\frac {\sqrt {2 \pi } g e^{-\frac {2 a}{b n}} (d+e x)^2 (e f-d g) \left (c (d+e x)^n\right )^{-2/n} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{\sqrt {b} e^3 \sqrt {n}}+\frac {\sqrt {\pi } e^{-\frac {a}{b n}} (d+e x) (e f-d g)^2 \left (c (d+e x)^n\right )^{-1/n} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{\sqrt {b} e^3 \sqrt {n}}+\frac {\sqrt {\frac {\pi }{3}} g^2 e^{-\frac {3 a}{b n}} (d+e x)^3 \left (c (d+e x)^n\right )^{-3/n} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{\sqrt {b} e^3 \sqrt {n}}\right )}{b e n}-\frac {2 (d+e x) (f+g x)^3}{b e n \sqrt {a+b \log \left (c (d+e x)^n\right )}}\) |
(-6*(e*f - d*g)*(((e*f - d*g)^2*Sqrt[Pi]*(d + e*x)*Erfi[Sqrt[a + b*Log[c*( d + e*x)^n]]/(Sqrt[b]*Sqrt[n])])/(Sqrt[b]*e^3*E^(a/(b*n))*Sqrt[n]*(c*(d + e*x)^n)^n^(-1)) + (g*(e*f - d*g)*Sqrt[2*Pi]*(d + e*x)^2*Erfi[(Sqrt[2]*Sqrt [a + b*Log[c*(d + e*x)^n]])/(Sqrt[b]*Sqrt[n])])/(Sqrt[b]*e^3*E^((2*a)/(b*n ))*Sqrt[n]*(c*(d + e*x)^n)^(2/n)) + (g^2*Sqrt[Pi/3]*(d + e*x)^3*Erfi[(Sqrt [3]*Sqrt[a + b*Log[c*(d + e*x)^n]])/(Sqrt[b]*Sqrt[n])])/(Sqrt[b]*e^3*E^((3 *a)/(b*n))*Sqrt[n]*(c*(d + e*x)^n)^(3/n))))/(b*e*n) + (8*(((e*f - d*g)^3*S qrt[Pi]*(d + e*x)*Erfi[Sqrt[a + b*Log[c*(d + e*x)^n]]/(Sqrt[b]*Sqrt[n])])/ (Sqrt[b]*e^4*E^(a/(b*n))*Sqrt[n]*(c*(d + e*x)^n)^n^(-1)) + (g^3*Sqrt[Pi]*( d + e*x)^4*Erfi[(2*Sqrt[a + b*Log[c*(d + e*x)^n]])/(Sqrt[b]*Sqrt[n])])/(2* Sqrt[b]*e^4*E^((4*a)/(b*n))*Sqrt[n]*(c*(d + e*x)^n)^(4/n)) + (3*g*(e*f - d *g)^2*Sqrt[Pi/2]*(d + e*x)^2*Erfi[(Sqrt[2]*Sqrt[a + b*Log[c*(d + e*x)^n]]) /(Sqrt[b]*Sqrt[n])])/(Sqrt[b]*e^4*E^((2*a)/(b*n))*Sqrt[n]*(c*(d + e*x)^n)^ (2/n)) + (g^2*(e*f - d*g)*Sqrt[3*Pi]*(d + e*x)^3*Erfi[(Sqrt[3]*Sqrt[a + b* Log[c*(d + e*x)^n]])/(Sqrt[b]*Sqrt[n])])/(Sqrt[b]*e^4*E^((3*a)/(b*n))*Sqrt [n]*(c*(d + e*x)^n)^(3/n))))/(b*n) - (2*(d + e*x)*(f + g*x)^3)/(b*e*n*Sqrt [a + b*Log[c*(d + e*x)^n]])
3.2.28.3.1 Defintions of rubi rules used
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)*((f_.) + (g_. )*(x_))^(q_.), x_Symbol] :> Simp[(d + e*x)*(f + g*x)^q*((a + b*Log[c*(d + e *x)^n])^(p + 1)/(b*e*n*(p + 1))), x] + (-Simp[(q + 1)/(b*n*(p + 1)) Int[( f + g*x)^q*(a + b*Log[c*(d + e*x)^n])^(p + 1), x], x] + Simp[q*((e*f - d*g) /(b*e*n*(p + 1))) Int[(f + g*x)^(q - 1)*(a + b*Log[c*(d + e*x)^n])^(p + 1 ), x], x]) /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0] && Lt Q[p, -1] && GtQ[q, 0]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)*((f_.) + (g_. )*(x_))^(q_.), x_Symbol] :> Int[ExpandIntegrand[(f + g*x)^q*(a + b*Log[c*(d + e*x)^n])^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && NeQ[e*f - d*g, 0] && IGtQ[q, 0]
\[\int \frac {\left (g x +f \right )^{3}}{{\left (a +b \ln \left (c \left (e x +d \right )^{n}\right )\right )}^{\frac {3}{2}}}d x\]
Exception generated. \[ \int \frac {(f+g x)^3}{\left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int \frac {(f+g x)^3}{\left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}} \, dx=\int \frac {\left (f + g x\right )^{3}}{\left (a + b \log {\left (c \left (d + e x\right )^{n} \right )}\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {(f+g x)^3}{\left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}} \, dx=\int { \frac {{\left (g x + f\right )}^{3}}{{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {(f+g x)^3}{\left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}} \, dx=\int { \frac {{\left (g x + f\right )}^{3}}{{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {(f+g x)^3}{\left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}} \, dx=\int \frac {{\left (f+g\,x\right )}^3}{{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^{3/2}} \,d x \]