Integrand size = 20, antiderivative size = 83 \[ \int (f x)^m \log \left (c \left (d+e \sqrt {x}\right )^p\right ) \, dx=-\frac {e p x^{3/2} (f x)^m \operatorname {Hypergeometric2F1}\left (1,3+2 m,2 (2+m),-\frac {e \sqrt {x}}{d}\right )}{d \left (3+5 m+2 m^2\right )}+\frac {(f x)^{1+m} \log \left (c \left (d+e \sqrt {x}\right )^p\right )}{f (1+m)} \]
-e*p*x^(3/2)*(f*x)^m*hypergeom([1, 3+2*m],[4+2*m],-e*x^(1/2)/d)/d/(1+m)/(3 +2*m)+(f*x)^(1+m)*ln(c*(d+e*x^(1/2))^p)/f/(1+m)
Time = 0.03 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.92 \[ \int (f x)^m \log \left (c \left (d+e \sqrt {x}\right )^p\right ) \, dx=\frac {x (f x)^m \left (-e p \sqrt {x} \operatorname {Hypergeometric2F1}\left (1,3+2 m,4+2 m,-\frac {e \sqrt {x}}{d}\right )+d (3+2 m) \log \left (c \left (d+e \sqrt {x}\right )^p\right )\right )}{d (1+m) (3+2 m)} \]
(x*(f*x)^m*(-(e*p*Sqrt[x]*Hypergeometric2F1[1, 3 + 2*m, 4 + 2*m, -((e*Sqrt [x])/d)]) + d*(3 + 2*m)*Log[c*(d + e*Sqrt[x])^p]))/(d*(1 + m)*(3 + 2*m))
Time = 0.23 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.12, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2905, 30, 864, 74}
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.
\(\displaystyle \int (f x)^m \log \left (c \left (d+e \sqrt {x}\right )^p\right ) \, dx\) |
\(\Big \downarrow \) 2905 |
\(\displaystyle \frac {(f x)^{m+1} \log \left (c \left (d+e \sqrt {x}\right )^p\right )}{f (m+1)}-\frac {e p \int \frac {(f x)^{m+1}}{\left (d+e \sqrt {x}\right ) \sqrt {x}}dx}{2 f (m+1)}\) |
\(\Big \downarrow \) 30 |
\(\displaystyle \frac {(f x)^{m+1} \log \left (c \left (d+e \sqrt {x}\right )^p\right )}{f (m+1)}-\frac {e p x^{-m} (f x)^m \int \frac {x^{m+\frac {1}{2}}}{d+e \sqrt {x}}dx}{2 (m+1)}\) |
\(\Big \downarrow \) 864 |
\(\displaystyle \frac {(f x)^{m+1} \log \left (c \left (d+e \sqrt {x}\right )^p\right )}{f (m+1)}-\frac {e p x^{-m} (f x)^m \int \frac {x^{m+1}}{d+e \sqrt {x}}d\sqrt {x}}{m+1}\) |
\(\Big \downarrow \) 74 |
\(\displaystyle \frac {(f x)^{m+1} \log \left (c \left (d+e \sqrt {x}\right )^p\right )}{f (m+1)}-\frac {e p x^{\frac {1}{2} (2 m+3)-m} (f x)^m \operatorname {Hypergeometric2F1}\left (1,2 m+3,2 (m+2),-\frac {e \sqrt {x}}{d}\right )}{d (m+1) (2 m+3)}\) |
-((e*p*x^(-m + (3 + 2*m)/2)*(f*x)^m*Hypergeometric2F1[1, 3 + 2*m, 2*(2 + m ), -((e*Sqrt[x])/d)])/(d*(1 + m)*(3 + 2*m))) + ((f*x)^(1 + m)*Log[c*(d + e *Sqrt[x])^p])/(f*(1 + m))
3.1.61.3.1 Defintions of rubi rules used
Int[(u_.)*((a_.)*(x_))^(m_.)*((b_.)*(x_)^(i_.))^(p_), x_Symbol] :> Simp[b^I ntPart[p]*((b*x^i)^FracPart[p]/(a^(i*IntPart[p])*(a*x)^(i*FracPart[p]))) Int[u*(a*x)^(m + i*p), x], x] /; FreeQ[{a, b, i, m, p}, x] && IntegerQ[i] & & !IntegerQ[p]
Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[c^n*((b*x )^(m + 1)/(b*(m + 1)))*Hypergeometric2F1[-n, m + 1, m + 2, (-d)*(x/c)], x] /; FreeQ[{b, c, d, m, n}, x] && !IntegerQ[m] && (IntegerQ[n] || (GtQ[c, 0] && !(EqQ[n, -2^(-1)] && EqQ[c^2 - d^2, 0] && GtQ[-d/(b*c), 0])))
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denomi nator[n]}, Simp[k Subst[Int[x^(k*(m + 1) - 1)*(a + b*x^(k*n))^p, x], x, x ^(1/k)], x]] /; FreeQ[{a, b, m, p}, x] && FractionQ[n]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))*((f_.)*(x_))^ (m_.), x_Symbol] :> Simp[(f*x)^(m + 1)*((a + b*Log[c*(d + e*x^n)^p])/(f*(m + 1))), x] - Simp[b*e*n*(p/(f*(m + 1))) Int[x^(n - 1)*((f*x)^(m + 1)/(d + e*x^n)), x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && NeQ[m, -1]
\[\int \left (f x \right )^{m} \ln \left (c \left (d +e \sqrt {x}\right )^{p}\right )d x\]
\[ \int (f x)^m \log \left (c \left (d+e \sqrt {x}\right )^p\right ) \, dx=\int { \left (f x\right )^{m} \log \left ({\left (e \sqrt {x} + d\right )}^{p} c\right ) \,d x } \]
\[ \int (f x)^m \log \left (c \left (d+e \sqrt {x}\right )^p\right ) \, dx=\int \left (f x\right )^{m} \log {\left (c \left (d + e \sqrt {x}\right )^{p} \right )}\, dx \]
\[ \int (f x)^m \log \left (c \left (d+e \sqrt {x}\right )^p\right ) \, dx=\int { \left (f x\right )^{m} \log \left ({\left (e \sqrt {x} + d\right )}^{p} c\right ) \,d x } \]
e^2*f^m*p*integrate(1/2*x*x^m/(d*e*(m + 1)*sqrt(x) + d^2*(m + 1)), x) + (d *f^m*(2*m + 3)*x*x^m*log((e*sqrt(x) + d)^p) + d*f^m*(2*m + 3)*x*x^m*log(c) - e*f^m*p*x^(3/2)*x^m)/((2*m^2 + 5*m + 3)*d)
\[ \int (f x)^m \log \left (c \left (d+e \sqrt {x}\right )^p\right ) \, dx=\int { \left (f x\right )^{m} \log \left ({\left (e \sqrt {x} + d\right )}^{p} c\right ) \,d x } \]
Timed out. \[ \int (f x)^m \log \left (c \left (d+e \sqrt {x}\right )^p\right ) \, dx=\int \ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^p\right )\,{\left (f\,x\right )}^m \,d x \]