3.399 \(\int (f x)^q (a+b \log (c (d+e x^m)^n)) \, dx\)

Optimal. Leaf size=92 \[ \frac{(f x)^{q+1} \left (a+b \log \left (c \left (d+e x^m\right )^n\right )\right )}{f (q+1)}-\frac{b e m n x^{m+1} (f x)^q \, _2F_1\left (1,\frac{m+q+1}{m};\frac{2 m+q+1}{m};-\frac{e x^m}{d}\right )}{d (q+1) (m+q+1)} \]

[Out]

-((b*e*m*n*x^(1 + m)*(f*x)^q*Hypergeometric2F1[1, (1 + m + q)/m, (1 + 2*m + q)/m, -((e*x^m)/d)])/(d*(1 + q)*(1
+ m + q))) + ((f*x)^(1 + q)*(a + b*Log[c*(d + e*x^m)^n]))/(f*(1 + q))

________________________________________________________________________________________

Rubi [A]  time = 0.0499879, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {2455, 20, 364} \[ \frac{(f x)^{q+1} \left (a+b \log \left (c \left (d+e x^m\right )^n\right )\right )}{f (q+1)}-\frac{b e m n x^{m+1} (f x)^q \, _2F_1\left (1,\frac{m+q+1}{m};\frac{2 m+q+1}{m};-\frac{e x^m}{d}\right )}{d (q+1) (m+q+1)} \]

Antiderivative was successfully verified.

[In]

Int[(f*x)^q*(a + b*Log[c*(d + e*x^m)^n]),x]

[Out]

-((b*e*m*n*x^(1 + m)*(f*x)^q*Hypergeometric2F1[1, (1 + m + q)/m, (1 + 2*m + q)/m, -((e*x^m)/d)])/(d*(1 + q)*(1
+ m + q))) + ((f*x)^(1 + q)*(a + b*Log[c*(d + e*x^m)^n]))/(f*(1 + q))

Rule 2455

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] - Dist[(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]

Rule 20

Int[(u_.)*((a_.)*(v_))^(m_)*((b_.)*(v_))^(n_), x_Symbol] :> Dist[(b^IntPart[n]*(b*v)^FracPart[n])/(a^IntPart[n
]*(a*v)^FracPart[n]), Int[u*(a*v)^(m + n), x], x] /; FreeQ[{a, b, m, n}, x] &&  !IntegerQ[m] &&  !IntegerQ[n]
&&  !IntegerQ[m + n]

Rule 364

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-
p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rubi steps

\begin{align*} \int (f x)^q \left (a+b \log \left (c \left (d+e x^m\right )^n\right )\right ) \, dx &=\frac{(f x)^{1+q} \left (a+b \log \left (c \left (d+e x^m\right )^n\right )\right )}{f (1+q)}-\frac{(b e m n) \int \frac{x^{-1+m} (f x)^{1+q}}{d+e x^m} \, dx}{f (1+q)}\\ &=\frac{(f x)^{1+q} \left (a+b \log \left (c \left (d+e x^m\right )^n\right )\right )}{f (1+q)}-\frac{\left (b e m n x^{-q} (f x)^q\right ) \int \frac{x^{m+q}}{d+e x^m} \, dx}{1+q}\\ &=-\frac{b e m n x^{1+m} (f x)^q \, _2F_1\left (1,\frac{1+m+q}{m};\frac{1+2 m+q}{m};-\frac{e x^m}{d}\right )}{d (1+q) (1+m+q)}+\frac{(f x)^{1+q} \left (a+b \log \left (c \left (d+e x^m\right )^n\right )\right )}{f (1+q)}\\ \end{align*}

Mathematica [A]  time = 0.0548306, size = 82, normalized size = 0.89 \[ \frac{x (f x)^q \left (d (m+q+1) \left (a+b \log \left (c \left (d+e x^m\right )^n\right )\right )-b e m n x^m \, _2F_1\left (1,\frac{m+q+1}{m};\frac{2 m+q+1}{m};-\frac{e x^m}{d}\right )\right )}{d (q+1) (m+q+1)} \]

Antiderivative was successfully verified.

[In]

Integrate[(f*x)^q*(a + b*Log[c*(d + e*x^m)^n]),x]

[Out]

(x*(f*x)^q*(-(b*e*m*n*x^m*Hypergeometric2F1[1, (1 + m + q)/m, (1 + 2*m + q)/m, -((e*x^m)/d)]) + d*(1 + m + q)*
(a + b*Log[c*(d + e*x^m)^n])))/(d*(1 + q)*(1 + m + q))

________________________________________________________________________________________

Maple [F]  time = 2.73, size = 0, normalized size = 0. \begin{align*} \int \left ( fx \right ) ^{q} \left ( a+b\ln \left ( c \left ( d+e{x}^{m} \right ) ^{n} \right ) \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x)^q*(a+b*ln(c*(d+e*x^m)^n)),x)

[Out]

int((f*x)^q*(a+b*ln(c*(d+e*x^m)^n)),x)

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x)^q*(a+b*log(c*(d+e*x^m)^n)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\left (f x\right )^{q} b \log \left ({\left (e x^{m} + d\right )}^{n} c\right ) + \left (f x\right )^{q} a, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x)^q*(a+b*log(c*(d+e*x^m)^n)),x, algorithm="fricas")

[Out]

integral((f*x)^q*b*log((e*x^m + d)^n*c) + (f*x)^q*a, x)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (f x\right )^{q} \left (a + b \log{\left (c \left (d + e x^{m}\right )^{n} \right )}\right )\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x)**q*(a+b*ln(c*(d+e*x**m)**n)),x)

[Out]

Integral((f*x)**q*(a + b*log(c*(d + e*x**m)**n)), x)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \log \left ({\left (e x^{m} + d\right )}^{n} c\right ) + a\right )} \left (f x\right )^{q}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x)^q*(a+b*log(c*(d+e*x^m)^n)),x, algorithm="giac")

[Out]

integrate((b*log((e*x^m + d)^n*c) + a)*(f*x)^q, x)