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

Optimal. Leaf size=301 \[ \frac {(d+e x)^{m+1} \log \left (c \left (a+b x^3\right )^p\right )}{e (m+1)}+\frac {\sqrt [3]{b} p (d+e x)^{m+2} \, _2F_1\left (1,m+2;m+3;\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d-\sqrt [3]{a} e}\right )}{e (m+1) (m+2) \left (\sqrt [3]{b} d-\sqrt [3]{a} e\right )}+\frac {\sqrt [3]{b} p (d+e x)^{m+2} \, _2F_1\left (1,m+2;m+3;\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d+\sqrt [3]{-1} \sqrt [3]{a} e}\right )}{e (m+1) (m+2) \left (\sqrt [3]{-1} \sqrt [3]{a} e+\sqrt [3]{b} d\right )}+\frac {\sqrt [3]{b} p (d+e x)^{m+2} \, _2F_1\left (1,m+2;m+3;\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d-(-1)^{2/3} \sqrt [3]{a} e}\right )}{e (m+1) (m+2) \left (\sqrt [3]{b} d-(-1)^{2/3} \sqrt [3]{a} e\right )} \]

[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)

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Rubi [A]  time = 0.76, antiderivative size = 301, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {2463, 6725, 68} \[ \frac {(d+e x)^{m+1} \log \left (c \left (a+b x^3\right )^p\right )}{e (m+1)}+\frac {\sqrt [3]{b} p (d+e x)^{m+2} \, _2F_1\left (1,m+2;m+3;\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d-\sqrt [3]{a} e}\right )}{e (m+1) (m+2) \left (\sqrt [3]{b} d-\sqrt [3]{a} e\right )}+\frac {\sqrt [3]{b} p (d+e x)^{m+2} \, _2F_1\left (1,m+2;m+3;\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d+\sqrt [3]{-1} \sqrt [3]{a} e}\right )}{e (m+1) (m+2) \left (\sqrt [3]{-1} \sqrt [3]{a} e+\sqrt [3]{b} d\right )}+\frac {\sqrt [3]{b} p (d+e x)^{m+2} \, _2F_1\left (1,m+2;m+3;\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d-(-1)^{2/3} \sqrt [3]{a} e}\right )}{e (m+1) (m+2) \left (\sqrt [3]{b} d-(-1)^{2/3} \sqrt [3]{a} e\right )} \]

Antiderivative was successfully verified.

[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 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 2463

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 6725

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

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Mathematica [A]  time = 0.68, size = 239, normalized size = 0.79 \[ \frac {(d+e x)^{m+1} \left (\log \left (c \left (a+b x^3\right )^p\right )-\frac {\sqrt [3]{b} p (d+e x) \left (-\frac {\, _2F_1\left (1,m+2;m+3;\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d-\sqrt [3]{a} e}\right )}{\sqrt [3]{b} d-\sqrt [3]{a} e}-\frac {\, _2F_1\left (1,m+2;m+3;\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d+\sqrt [3]{-1} \sqrt [3]{a} e}\right )}{\sqrt [3]{-1} \sqrt [3]{a} e+\sqrt [3]{b} d}-\frac {\, _2F_1\left (1,m+2;m+3;\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d-(-1)^{2/3} \sqrt [3]{a} e}\right )}{\sqrt [3]{b} d-(-1)^{2/3} \sqrt [3]{a} e}\right )}{m+2}\right )}{e (m+1)} \]

Antiderivative was successfully verified.

[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))

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fricas [F]  time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (e x + d\right )}^{m} \log \left ({\left (b x^{3} + a\right )}^{p} c\right ), x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (e x + d\right )}^{m} \log \left ({\left (b x^{3} + a\right )}^{p} c\right )\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [F]  time = 1.17, size = 0, normalized size = 0.00 \[ \int \left (e x +d \right )^{m} \ln \left (c \left (b \,x^{3}+a \right )^{p}\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {{\left (e x + d\right )} {\left (e x + d\right )}^{m} \log \left ({\left (b x^{3} + a\right )}^{p}\right )}{e {\left (m + 1\right )}} + \int -\frac {{\left (3 \, b d p x^{2} - {\left (e {\left (m + 1\right )} \log \relax (c) - 3 \, e p\right )} b x^{3} - a e {\left (m + 1\right )} \log \relax (c)\right )} {\left (e x + d\right )}^{m}}{b e {\left (m + 1\right )} x^{3} + a e {\left (m + 1\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \ln \left (c\,{\left (b\,x^3+a\right )}^p\right )\,{\left (d+e\,x\right )}^m \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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