\(\int (d+e x)^m \log (c (a+b x^3)^p) \, dx\) [206]

Optimal result

Integrand size = 20, antiderivative size = 301 \[ \int (d+e x)^m \log \left (c \left (a+b x^3\right )^p\right ) \, dx=\frac {\sqrt [3]{b} p (d+e x)^{2+m} \operatorname {Hypergeometric2F1}\left (1,2+m,3+m,\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d-\sqrt [3]{a} e}\right )}{e \left (\sqrt [3]{b} d-\sqrt [3]{a} e\right ) (1+m) (2+m)}+\frac {\sqrt [3]{b} p (d+e x)^{2+m} \operatorname {Hypergeometric2F1}\left (1,2+m,3+m,\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d+\sqrt [3]{-1} \sqrt [3]{a} e}\right )}{e \left (\sqrt [3]{b} d+\sqrt [3]{-1} \sqrt [3]{a} e\right ) (1+m) (2+m)}+\frac {\sqrt [3]{b} p (d+e x)^{2+m} \operatorname {Hypergeometric2F1}\left (1,2+m,3+m,\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d-(-1)^{2/3} \sqrt [3]{a} e}\right )}{e \left (\sqrt [3]{b} d-(-1)^{2/3} \sqrt [3]{a} e\right ) (1+m) (2+m)}+\frac {(d+e x)^{1+m} \log \left (c \left (a+b x^3\right )^p\right )}{e (1+m)} \]

[Out]

b^(1/3)*p*(e*x+d)^(2+m)*hypergeom([1, 2+m],[3+m],b^(1/3)*(e*x+d)/(b^(1/3)*d-a^(1/3)*e))/e/(b^(1/3)*d-a^(1/3)*e
)/(1+m)/(2+m)+b^(1/3)*p*(e*x+d)^(2+m)*hypergeom([1, 2+m],[3+m],b^(1/3)*(e*x+d)/(b^(1/3)*d+(-1)^(1/3)*a^(1/3)*e
))/e/(b^(1/3)*d+(-1)^(1/3)*a^(1/3)*e)/(1+m)/(2+m)+b^(1/3)*p*(e*x+d)^(2+m)*hypergeom([1, 2+m],[3+m],b^(1/3)*(e*
x+d)/(b^(1/3)*d-(-1)^(2/3)*a^(1/3)*e))/e/(b^(1/3)*d-(-1)^(2/3)*a^(1/3)*e)/(1+m)/(2+m)+(e*x+d)^(1+m)*ln(c*(b*x^
3+a)^p)/e/(1+m)

Rubi [A] (verified)

Time = 0.48 (sec) , antiderivative size = 301, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {2513, 6857, 70} \[ \int (d+e x)^m \log \left (c \left (a+b x^3\right )^p\right ) \, dx=\frac {(d+e x)^{m+1} \log \left (c \left (a+b x^3\right )^p\right )}{e (m+1)}+\frac {\sqrt [3]{b} p (d+e x)^{m+2} \operatorname {Hypergeometric2F1}\left (1,m+2,m+3,\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d-\sqrt [3]{a} e}\right )}{e (m+1) (m+2) \left (\sqrt [3]{b} d-\sqrt [3]{a} e\right )}+\frac {\sqrt [3]{b} p (d+e x)^{m+2} \operatorname {Hypergeometric2F1}\left (1,m+2,m+3,\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d+\sqrt [3]{-1} \sqrt [3]{a} e}\right )}{e (m+1) (m+2) \left (\sqrt [3]{-1} \sqrt [3]{a} e+\sqrt [3]{b} d\right )}+\frac {\sqrt [3]{b} p (d+e x)^{m+2} \operatorname {Hypergeometric2F1}\left (1,m+2,m+3,\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d-(-1)^{2/3} \sqrt [3]{a} e}\right )}{e (m+1) (m+2) \left (\sqrt [3]{b} d-(-1)^{2/3} \sqrt [3]{a} e\right )} \]

[In]

Int[(d + e*x)^m*Log[c*(a + b*x^3)^p],x]

[Out]

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

Rule 70

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(b*c - a*d)^n*((a + b*x)^(m + 1)/(b^(
n + 1)*(m + 1)))*Hypergeometric2F1[-n, m + 1, m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m
}, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] && IntegerQ[n]

Rule 2513

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

Rule 6857

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {(d+e x)^{1+m} \log \left (c \left (a+b x^3\right )^p\right )}{e (1+m)}-\frac {(3 b p) \int \frac {x^2 (d+e x)^{1+m}}{a+b x^3} \, dx}{e (1+m)} \\ & = \frac {(d+e x)^{1+m} \log \left (c \left (a+b x^3\right )^p\right )}{e (1+m)}-\frac {(3 b p) \int \left (\frac {(d+e x)^{1+m}}{3 b^{2/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac {(d+e x)^{1+m}}{3 b^{2/3} \left (-\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac {(d+e x)^{1+m}}{3 b^{2/3} \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right )}\right ) \, dx}{e (1+m)} \\ & = \frac {(d+e x)^{1+m} \log \left (c \left (a+b x^3\right )^p\right )}{e (1+m)}-\frac {\left (\sqrt [3]{b} p\right ) \int \frac {(d+e x)^{1+m}}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{e (1+m)}-\frac {\left (\sqrt [3]{b} p\right ) \int \frac {(d+e x)^{1+m}}{-\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{e (1+m)}-\frac {\left (\sqrt [3]{b} p\right ) \int \frac {(d+e x)^{1+m}}{(-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{e (1+m)} \\ & = \frac {\sqrt [3]{b} p (d+e x)^{2+m} \, _2F_1\left (1,2+m;3+m;\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d-\sqrt [3]{a} e}\right )}{e \left (\sqrt [3]{b} d-\sqrt [3]{a} e\right ) (1+m) (2+m)}+\frac {\sqrt [3]{b} p (d+e x)^{2+m} \, _2F_1\left (1,2+m;3+m;\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d+\sqrt [3]{-1} \sqrt [3]{a} e}\right )}{e \left (\sqrt [3]{b} d+\sqrt [3]{-1} \sqrt [3]{a} e\right ) (1+m) (2+m)}+\frac {\sqrt [3]{b} p (d+e x)^{2+m} \, _2F_1\left (1,2+m;3+m;\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d-(-1)^{2/3} \sqrt [3]{a} e}\right )}{e \left (\sqrt [3]{b} d-(-1)^{2/3} \sqrt [3]{a} e\right ) (1+m) (2+m)}+\frac {(d+e x)^{1+m} \log \left (c \left (a+b x^3\right )^p\right )}{e (1+m)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.42 (sec) , antiderivative size = 239, normalized size of antiderivative = 0.79 \[ \int (d+e x)^m \log \left (c \left (a+b x^3\right )^p\right ) \, dx=\frac {(d+e x)^{1+m} \left (-\frac {\sqrt [3]{b} p (d+e x) \left (-\frac {\operatorname {Hypergeometric2F1}\left (1,2+m,3+m,\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d-\sqrt [3]{a} e}\right )}{\sqrt [3]{b} d-\sqrt [3]{a} e}-\frac {\operatorname {Hypergeometric2F1}\left (1,2+m,3+m,\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d+\sqrt [3]{-1} \sqrt [3]{a} e}\right )}{\sqrt [3]{b} d+\sqrt [3]{-1} \sqrt [3]{a} e}-\frac {\operatorname {Hypergeometric2F1}\left (1,2+m,3+m,\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d-(-1)^{2/3} \sqrt [3]{a} e}\right )}{\sqrt [3]{b} d-(-1)^{2/3} \sqrt [3]{a} e}\right )}{2+m}+\log \left (c \left (a+b x^3\right )^p\right )\right )}{e (1+m)} \]

[In]

Integrate[(d + e*x)^m*Log[c*(a + b*x^3)^p],x]

[Out]

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

Maple [F]

\[\int \left (e x +d \right )^{m} \ln \left (c \left (b \,x^{3}+a \right )^{p}\right )d x\]

[In]

int((e*x+d)^m*ln(c*(b*x^3+a)^p),x)

[Out]

int((e*x+d)^m*ln(c*(b*x^3+a)^p),x)

Fricas [F]

\[ \int (d+e x)^m \log \left (c \left (a+b x^3\right )^p\right ) \, dx=\int { {\left (e x + d\right )}^{m} \log \left ({\left (b x^{3} + a\right )}^{p} c\right ) \,d x } \]

[In]

integrate((e*x+d)^m*log(c*(b*x^3+a)^p),x, algorithm="fricas")

[Out]

integral((e*x + d)^m*log((b*x^3 + a)^p*c), x)

Sympy [F(-1)]

Timed out. \[ \int (d+e x)^m \log \left (c \left (a+b x^3\right )^p\right ) \, dx=\text {Timed out} \]

[In]

integrate((e*x+d)**m*ln(c*(b*x**3+a)**p),x)

[Out]

Timed out

Maxima [F]

\[ \int (d+e x)^m \log \left (c \left (a+b x^3\right )^p\right ) \, dx=\int { {\left (e x + d\right )}^{m} \log \left ({\left (b x^{3} + a\right )}^{p} c\right ) \,d x } \]

[In]

integrate((e*x+d)^m*log(c*(b*x^3+a)^p),x, algorithm="maxima")

[Out]

(e*x + d)*(e*x + d)^m*log((b*x^3 + a)^p)/(e*(m + 1)) + integrate(-(3*b*d*p*x^2 - (e*(m + 1)*log(c) - 3*e*p)*b*
x^3 - a*e*(m + 1)*log(c))*(e*x + d)^m/(b*e*(m + 1)*x^3 + a*e*(m + 1)), x)

Giac [F]

\[ \int (d+e x)^m \log \left (c \left (a+b x^3\right )^p\right ) \, dx=\int { {\left (e x + d\right )}^{m} \log \left ({\left (b x^{3} + a\right )}^{p} c\right ) \,d x } \]

[In]

integrate((e*x+d)^m*log(c*(b*x^3+a)^p),x, algorithm="giac")

[Out]

integrate((e*x + d)^m*log((b*x^3 + a)^p*c), x)

Mupad [F(-1)]

Timed out. \[ \int (d+e x)^m \log \left (c \left (a+b x^3\right )^p\right ) \, dx=\int \ln \left (c\,{\left (b\,x^3+a\right )}^p\right )\,{\left (d+e\,x\right )}^m \,d x \]

[In]

int(log(c*(a + b*x^3)^p)*(d + e*x)^m,x)

[Out]

int(log(c*(a + b*x^3)^p)*(d + e*x)^m, x)