3.113 \(\int \frac{x^3 (e+f x)^n}{(a+b x) (c+d x)} \, dx\)

Optimal. Leaf size=216 \[ \frac{a^3 (e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{b (e+f x)}{b e-a f}\right )}{b^2 (n+1) (b c-a d) (b e-a f)}-\frac{(a d+b c) (e+f x)^{n+1}}{b^2 d^2 f (n+1)}-\frac{c^3 (e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{d (e+f x)}{d e-c f}\right )}{d^2 (n+1) (b c-a d) (d e-c f)}-\frac{e (e+f x)^{n+1}}{b d f^2 (n+1)}+\frac{(e+f x)^{n+2}}{b d f^2 (n+2)} \]

[Out]

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

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Rubi [A]  time = 0.15416, antiderivative size = 216, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {180, 43, 68} \[ \frac{a^3 (e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{b (e+f x)}{b e-a f}\right )}{b^2 (n+1) (b c-a d) (b e-a f)}-\frac{(a d+b c) (e+f x)^{n+1}}{b^2 d^2 f (n+1)}-\frac{c^3 (e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{d (e+f x)}{d e-c f}\right )}{d^2 (n+1) (b c-a d) (d e-c f)}-\frac{e (e+f x)^{n+1}}{b d f^2 (n+1)}+\frac{(e+f x)^{n+2}}{b d f^2 (n+2)} \]

Antiderivative was successfully verified.

[In]

Int[(x^3*(e + f*x)^n)/((a + b*x)*(c + d*x)),x]

[Out]

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

Rule 180

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_))^(q_), x
_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p*(g + h*x)^q, x], x] /; FreeQ[{a, b, c, d,
e, f, g, h, m, n}, x] && IntegersQ[p, q]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

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]

Rubi steps

\begin{align*} \int \frac{x^3 (e+f x)^n}{(a+b x) (c+d x)} \, dx &=\int \left (\frac{(-b c-a d) (e+f x)^n}{b^2 d^2}+\frac{x (e+f x)^n}{b d}-\frac{a^3 (e+f x)^n}{b^2 (b c-a d) (a+b x)}-\frac{c^3 (e+f x)^n}{d^2 (-b c+a d) (c+d x)}\right ) \, dx\\ &=-\frac{(b c+a d) (e+f x)^{1+n}}{b^2 d^2 f (1+n)}+\frac{\int x (e+f x)^n \, dx}{b d}-\frac{a^3 \int \frac{(e+f x)^n}{a+b x} \, dx}{b^2 (b c-a d)}+\frac{c^3 \int \frac{(e+f x)^n}{c+d x} \, dx}{d^2 (b c-a d)}\\ &=-\frac{(b c+a d) (e+f x)^{1+n}}{b^2 d^2 f (1+n)}+\frac{a^3 (e+f x)^{1+n} \, _2F_1\left (1,1+n;2+n;\frac{b (e+f x)}{b e-a f}\right )}{b^2 (b c-a d) (b e-a f) (1+n)}-\frac{c^3 (e+f x)^{1+n} \, _2F_1\left (1,1+n;2+n;\frac{d (e+f x)}{d e-c f}\right )}{d^2 (b c-a d) (d e-c f) (1+n)}+\frac{\int \left (-\frac{e (e+f x)^n}{f}+\frac{(e+f x)^{1+n}}{f}\right ) \, dx}{b d}\\ &=-\frac{e (e+f x)^{1+n}}{b d f^2 (1+n)}-\frac{(b c+a d) (e+f x)^{1+n}}{b^2 d^2 f (1+n)}+\frac{(e+f x)^{2+n}}{b d f^2 (2+n)}+\frac{a^3 (e+f x)^{1+n} \, _2F_1\left (1,1+n;2+n;\frac{b (e+f x)}{b e-a f}\right )}{b^2 (b c-a d) (b e-a f) (1+n)}-\frac{c^3 (e+f x)^{1+n} \, _2F_1\left (1,1+n;2+n;\frac{d (e+f x)}{d e-c f}\right )}{d^2 (b c-a d) (d e-c f) (1+n)}\\ \end{align*}

Mathematica [A]  time = 0.496155, size = 174, normalized size = 0.81 \[ \frac{(e+f x)^{n+1} \left (\frac{a^3 \, _2F_1\left (1,n+1;n+2;\frac{b (e+f x)}{b e-a f}\right )}{b e-a f}+\frac{(b c-a d) (c f-d e) (a d f (n+2)+b c f (n+2)+b d (e-f (n+1) x))-b^2 c^3 f^2 (n+2) \, _2F_1\left (1,n+1;n+2;\frac{d (e+f x)}{d e-c f}\right )}{d^2 f^2 (n+2) (d e-c f)}\right )}{b^2 (n+1) (b c-a d)} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^3*(e + f*x)^n)/((a + b*x)*(c + d*x)),x]

[Out]

((e + f*x)^(1 + n)*((a^3*Hypergeometric2F1[1, 1 + n, 2 + n, (b*(e + f*x))/(b*e - a*f)])/(b*e - a*f) + ((b*c -
a*d)*(-(d*e) + c*f)*(b*c*f*(2 + n) + a*d*f*(2 + n) + b*d*(e - f*(1 + n)*x)) - b^2*c^3*f^2*(2 + n)*Hypergeometr
ic2F1[1, 1 + n, 2 + n, (d*(e + f*x))/(d*e - c*f)])/(d^2*f^2*(d*e - c*f)*(2 + n))))/(b^2*(b*c - a*d)*(1 + n))

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Maple [F]  time = 0.077, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( fx+e \right ) ^{n}{x}^{3}}{ \left ( bx+a \right ) \left ( dx+c \right ) }}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(f*x+e)^n/(b*x+a)/(d*x+c),x)

[Out]

int(x^3*(f*x+e)^n/(b*x+a)/(d*x+c),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (f x + e\right )}^{n} x^{3}}{{\left (b x + a\right )}{\left (d x + c\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(f*x+e)^n/(b*x+a)/(d*x+c),x, algorithm="maxima")

[Out]

integrate((f*x + e)^n*x^3/((b*x + a)*(d*x + c)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (f x + e\right )}^{n} x^{3}}{b d x^{2} + a c +{\left (b c + a d\right )} x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(f*x+e)^n/(b*x+a)/(d*x+c),x, algorithm="fricas")

[Out]

integral((f*x + e)^n*x^3/(b*d*x^2 + a*c + (b*c + a*d)*x), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3} \left (e + f x\right )^{n}}{\left (a + b x\right ) \left (c + d x\right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(f*x+e)**n/(b*x+a)/(d*x+c),x)

[Out]

Integral(x**3*(e + f*x)**n/((a + b*x)*(c + d*x)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (f x + e\right )}^{n} x^{3}}{{\left (b x + a\right )}{\left (d x + c\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(f*x+e)^n/(b*x+a)/(d*x+c),x, algorithm="giac")

[Out]

integrate((f*x + e)^n*x^3/((b*x + a)*(d*x + c)), x)