3.506 \(\int (g+h x)^m (a+b \log (c (d (e+f x)^p)^q)) \, dx\)

Optimal. Leaf size=99 \[ \frac{(g+h x)^{m+1} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h (m+1)}+\frac{b f p q (g+h x)^{m+2} \, _2F_1\left (1,m+2;m+3;\frac{f (g+h x)}{f g-e h}\right )}{h (m+1) (m+2) (f g-e h)} \]

[Out]

(b*f*p*q*(g + h*x)^(2 + m)*Hypergeometric2F1[1, 2 + m, 3 + m, (f*(g + h*x))/(f*g - e*h)])/(h*(f*g - e*h)*(1 +
m)*(2 + m)) + ((g + h*x)^(1 + m)*(a + b*Log[c*(d*(e + f*x)^p)^q]))/(h*(1 + m))

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Rubi [A]  time = 0.105924, antiderivative size = 99, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {2395, 68, 2445} \[ \frac{(g+h x)^{m+1} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h (m+1)}+\frac{b f p q (g+h x)^{m+2} \, _2F_1\left (1,m+2;m+3;\frac{f (g+h x)}{f g-e h}\right )}{h (m+1) (m+2) (f g-e h)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*f*p*q*(g + h*x)^(2 + m)*Hypergeometric2F1[1, 2 + m, 3 + m, (f*(g + h*x))/(f*g - e*h)])/(h*(f*g - e*h)*(1 +
m)*(2 + m)) + ((g + h*x)^(1 + m)*(a + b*Log[c*(d*(e + f*x)^p)^q]))/(h*(1 + m))

Rule 2395

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))*((f_.) + (g_.)*(x_))^(q_.), x_Symbol] :> Simp[((f + g
*x)^(q + 1)*(a + b*Log[c*(d + e*x)^n]))/(g*(q + 1)), x] - Dist[(b*e*n)/(g*(q + 1)), Int[(f + g*x)^(q + 1)/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, q}, x] && NeQ[e*f - d*g, 0] && NeQ[q, -1]

Rule 68

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((b*c - a*d)^n*(a + b*x)^(m + 1)*Hype
rgeometric2F1[-n, m + 1, m + 2, -((d*(a + b*x))/(b*c - a*d))])/(b^(n + 1)*(m + 1)), x] /; FreeQ[{a, b, c, d, m
}, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] && IntegerQ[n]

Rule 2445

Int[((a_.) + Log[(c_.)*((d_.)*((e_.) + (f_.)*(x_))^(m_.))^(n_)]*(b_.))^(p_.)*(u_.), x_Symbol] :> Subst[Int[u*(
a + b*Log[c*d^n*(e + f*x)^(m*n)])^p, x], c*d^n*(e + f*x)^(m*n), c*(d*(e + f*x)^m)^n] /; FreeQ[{a, b, c, d, e,
f, m, n, p}, x] &&  !IntegerQ[n] &&  !(EqQ[d, 1] && EqQ[m, 1]) && IntegralFreeQ[IntHide[u*(a + b*Log[c*d^n*(e
+ f*x)^(m*n)])^p, x]]

Rubi steps

\begin{align*} \int (g+h x)^m \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \, dx &=\operatorname{Subst}\left (\int (g+h x)^m \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right ) \, dx,c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac{(g+h x)^{1+m} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h (1+m)}-\operatorname{Subst}\left (\frac{(b f p q) \int \frac{(g+h x)^{1+m}}{e+f x} \, dx}{h (1+m)},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac{b f p q (g+h x)^{2+m} \, _2F_1\left (1,2+m;3+m;\frac{f (g+h x)}{f g-e h}\right )}{h (f g-e h) (1+m) (2+m)}+\frac{(g+h x)^{1+m} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h (1+m)}\\ \end{align*}

Mathematica [A]  time = 0.119766, size = 86, normalized size = 0.87 \[ \frac{(g+h x)^{m+1} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )+\frac{b f p q (g+h x) \, _2F_1\left (1,m+2;m+3;\frac{f (g+h x)}{f g-e h}\right )}{(m+2) (f g-e h)}\right )}{h (m+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((g + h*x)^(1 + m)*(a + (b*f*p*q*(g + h*x)*Hypergeometric2F1[1, 2 + m, 3 + m, (f*(g + h*x))/(f*g - e*h)])/((f*
g - e*h)*(2 + m)) + b*Log[c*(d*(e + f*x)^p)^q]))/(h*(1 + m))

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Maple [F]  time = 0.713, size = 0, normalized size = 0. \begin{align*} \int \left ( hx+g \right ) ^{m} \left ( a+b\ln \left ( c \left ( d \left ( fx+e \right ) ^{p} \right ) ^{q} \right ) \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Exception raised: ValueError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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