3.364 \(\int \frac{x^3 (d+e x)^n}{a+c x^2} \, dx\)

Optimal. Leaf size=209 \[ \frac{a (d+e x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{-a} e}\right )}{2 c^{3/2} (n+1) \left (\sqrt{c} d-\sqrt{-a} e\right )}+\frac{a (d+e x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}\right )}{2 c^{3/2} (n+1) \left (\sqrt{-a} e+\sqrt{c} d\right )}-\frac{d (d+e x)^{n+1}}{c e^2 (n+1)}+\frac{(d+e x)^{n+2}}{c e^2 (n+2)} \]

[Out]

-((d*(d + e*x)^(1 + n))/(c*e^2*(1 + n))) + (d + e*x)^(2 + n)/(c*e^2*(2 + n)) + (a*(d + e*x)^(1 + n)*Hypergeome
tric2F1[1, 1 + n, 2 + n, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d - Sqrt[-a]*e)])/(2*c^(3/2)*(Sqrt[c]*d - Sqrt[-a]*e)*(1
+ n)) + (a*(d + e*x)^(1 + n)*Hypergeometric2F1[1, 1 + n, 2 + n, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)]
)/(2*c^(3/2)*(Sqrt[c]*d + Sqrt[-a]*e)*(1 + n))

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Rubi [A]  time = 0.225294, antiderivative size = 209, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {1629, 831, 68} \[ \frac{a (d+e x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{-a} e}\right )}{2 c^{3/2} (n+1) \left (\sqrt{c} d-\sqrt{-a} e\right )}+\frac{a (d+e x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}\right )}{2 c^{3/2} (n+1) \left (\sqrt{-a} e+\sqrt{c} d\right )}-\frac{d (d+e x)^{n+1}}{c e^2 (n+1)}+\frac{(d+e x)^{n+2}}{c e^2 (n+2)} \]

Antiderivative was successfully verified.

[In]

Int[(x^3*(d + e*x)^n)/(a + c*x^2),x]

[Out]

-((d*(d + e*x)^(1 + n))/(c*e^2*(1 + n))) + (d + e*x)^(2 + n)/(c*e^2*(2 + n)) + (a*(d + e*x)^(1 + n)*Hypergeome
tric2F1[1, 1 + n, 2 + n, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d - Sqrt[-a]*e)])/(2*c^(3/2)*(Sqrt[c]*d - Sqrt[-a]*e)*(1
+ n)) + (a*(d + e*x)^(1 + n)*Hypergeometric2F1[1, 1 + n, 2 + n, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)]
)/(2*c^(3/2)*(Sqrt[c]*d + Sqrt[-a]*e)*(1 + n))

Rule 1629

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*
Pq*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rule 831

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(
d + e*x)^m, (f + g*x)/(a + c*x^2), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] &&  !Ration
alQ[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]

Rubi steps

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

Mathematica [A]  time = 0.230813, size = 168, normalized size = 0.8 \[ \frac{(d+e x)^{n+1} \left (\frac{a \, _2F_1\left (1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{-a} e}\right )}{\sqrt{c} d-\sqrt{-a} e}+\frac{a \, _2F_1\left (1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}\right )}{\sqrt{-a} e+\sqrt{c} d}-\frac{2 \sqrt{c} (d-e (n+1) x)}{e^2 (n+2)}\right )}{2 c^{3/2} (n+1)} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^3*(d + e*x)^n)/(a + c*x^2),x]

[Out]

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

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Maple [F]  time = 0.724, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( ex+d \right ) ^{n}{x}^{3}}{c{x}^{2}+a}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(e*x+d)^n/(c*x^2+a),x)

[Out]

int(x^3*(e*x+d)^n/(c*x^2+a),x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(e*x+d)^n/(c*x^2+a),x, algorithm="maxima")

[Out]

integrate((e*x + d)^n*x^3/(c*x^2 + a), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(e*x+d)^n/(c*x^2+a),x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(e*x+d)**n/(c*x**2+a),x)

[Out]

Integral(x**3*(d + e*x)**n/(a + c*x**2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(e*x+d)^n/(c*x^2+a),x, algorithm="giac")

[Out]

integrate((e*x + d)^n*x^3/(c*x^2 + a), x)