3.363 \(\int \frac{x^4 (d+e x)^n}{a+c x^2} \, dx\)

Optimal. Leaf size=250 \[ \frac{\left (c d^2-a e^2\right ) (d+e x)^{n+1}}{c^2 e^3 (n+1)}+\frac{(-a)^{3/2} (d+e x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{-a} e}\right )}{2 c^2 (n+1) \left (\sqrt{c} d-\sqrt{-a} e\right )}-\frac{(-a)^{3/2} (d+e x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}\right )}{2 c^2 (n+1) \left (\sqrt{-a} e+\sqrt{c} d\right )}-\frac{2 d (d+e x)^{n+2}}{c e^3 (n+2)}+\frac{(d+e x)^{n+3}}{c e^3 (n+3)} \]

[Out]

((c*d^2 - a*e^2)*(d + e*x)^(1 + n))/(c^2*e^3*(1 + n)) - (2*d*(d + e*x)^(2 + n))/(c*e^3*(2 + n)) + (d + e*x)^(3
+ n)/(c*e^3*(3 + n)) + ((-a)^(3/2)*(d + e*x)^(1 + n)*Hypergeometric2F1[1, 1 + n, 2 + n, (Sqrt[c]*(d + e*x))/(
Sqrt[c]*d - Sqrt[-a]*e)])/(2*c^2*(Sqrt[c]*d - Sqrt[-a]*e)*(1 + n)) - ((-a)^(3/2)*(d + e*x)^(1 + n)*Hypergeomet
ric2F1[1, 1 + n, 2 + n, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)])/(2*c^2*(Sqrt[c]*d + Sqrt[-a]*e)*(1 + n)
)

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Rubi [A]  time = 0.403131, antiderivative size = 250, 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, 712, 68} \[ \frac{\left (c d^2-a e^2\right ) (d+e x)^{n+1}}{c^2 e^3 (n+1)}+\frac{(-a)^{3/2} (d+e x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{-a} e}\right )}{2 c^2 (n+1) \left (\sqrt{c} d-\sqrt{-a} e\right )}-\frac{(-a)^{3/2} (d+e x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}\right )}{2 c^2 (n+1) \left (\sqrt{-a} e+\sqrt{c} d\right )}-\frac{2 d (d+e x)^{n+2}}{c e^3 (n+2)}+\frac{(d+e x)^{n+3}}{c e^3 (n+3)} \]

Antiderivative was successfully verified.

[In]

Int[(x^4*(d + e*x)^n)/(a + c*x^2),x]

[Out]

((c*d^2 - a*e^2)*(d + e*x)^(1 + n))/(c^2*e^3*(1 + n)) - (2*d*(d + e*x)^(2 + n))/(c*e^3*(2 + n)) + (d + e*x)^(3
+ n)/(c*e^3*(3 + n)) + ((-a)^(3/2)*(d + e*x)^(1 + n)*Hypergeometric2F1[1, 1 + n, 2 + n, (Sqrt[c]*(d + e*x))/(
Sqrt[c]*d - Sqrt[-a]*e)])/(2*c^2*(Sqrt[c]*d - Sqrt[-a]*e)*(1 + n)) - ((-a)^(3/2)*(d + e*x)^(1 + n)*Hypergeomet
ric2F1[1, 1 + n, 2 + n, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)])/(2*c^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 712

Int[((d_) + (e_.)*(x_))^(m_)/((a_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m, 1/(a + c*x^2
), x], x] /; FreeQ[{a, c, d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[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^4 (d+e x)^n}{a+c x^2} \, dx &=\int \left (\frac{\left (c d^2-a e^2\right ) (d+e x)^n}{c^2 e^2}-\frac{2 d (d+e x)^{1+n}}{c e^2}+\frac{(d+e x)^{2+n}}{c e^2}+\frac{a^2 (d+e x)^n}{c^2 \left (a+c x^2\right )}\right ) \, dx\\ &=\frac{\left (c d^2-a e^2\right ) (d+e x)^{1+n}}{c^2 e^3 (1+n)}-\frac{2 d (d+e x)^{2+n}}{c e^3 (2+n)}+\frac{(d+e x)^{3+n}}{c e^3 (3+n)}+\frac{a^2 \int \frac{(d+e x)^n}{a+c x^2} \, dx}{c^2}\\ &=\frac{\left (c d^2-a e^2\right ) (d+e x)^{1+n}}{c^2 e^3 (1+n)}-\frac{2 d (d+e x)^{2+n}}{c e^3 (2+n)}+\frac{(d+e x)^{3+n}}{c e^3 (3+n)}+\frac{a^2 \int \left (\frac{\sqrt{-a} (d+e x)^n}{2 a \left (\sqrt{-a}-\sqrt{c} x\right )}+\frac{\sqrt{-a} (d+e x)^n}{2 a \left (\sqrt{-a}+\sqrt{c} x\right )}\right ) \, dx}{c^2}\\ &=\frac{\left (c d^2-a e^2\right ) (d+e x)^{1+n}}{c^2 e^3 (1+n)}-\frac{2 d (d+e x)^{2+n}}{c e^3 (2+n)}+\frac{(d+e x)^{3+n}}{c e^3 (3+n)}-\frac{(-a)^{3/2} \int \frac{(d+e x)^n}{\sqrt{-a}-\sqrt{c} x} \, dx}{2 c^2}-\frac{(-a)^{3/2} \int \frac{(d+e x)^n}{\sqrt{-a}+\sqrt{c} x} \, dx}{2 c^2}\\ &=\frac{\left (c d^2-a e^2\right ) (d+e x)^{1+n}}{c^2 e^3 (1+n)}-\frac{2 d (d+e x)^{2+n}}{c e^3 (2+n)}+\frac{(d+e x)^{3+n}}{c e^3 (3+n)}+\frac{(-a)^{3/2} (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^2 \left (\sqrt{c} d-\sqrt{-a} e\right ) (1+n)}-\frac{(-a)^{3/2} (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^2 \left (\sqrt{c} d+\sqrt{-a} e\right ) (1+n)}\\ \end{align*}

Mathematica [A]  time = 0.519903, size = 217, normalized size = 0.87 \[ \frac{(d+e x)^{n+1} \left (\frac{2 \left (c d^2-a e^2\right )}{e^3 (n+1)}+\frac{(-a)^{3/2} \, _2F_1\left (1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{-a} e}\right )}{(n+1) \left (\sqrt{c} d-\sqrt{-a} e\right )}+\frac{\sqrt{-a} a \, _2F_1\left (1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}\right )}{(n+1) \left (\sqrt{-a} e+\sqrt{c} d\right )}+\frac{2 c (d+e x)^2}{e^3 (n+3)}-\frac{4 c d (d+e x)}{e^3 (n+2)}\right )}{2 c^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^4*(d + e*x)^n)/(a + c*x^2),x]

[Out]

((d + e*x)^(1 + n)*((2*(c*d^2 - a*e^2))/(e^3*(1 + n)) - (4*c*d*(d + e*x))/(e^3*(2 + n)) + (2*c*(d + e*x)^2)/(e
^3*(3 + n)) + ((-a)^(3/2)*Hypergeometric2F1[1, 1 + n, 2 + n, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d - Sqrt[-a]*e)])/((
Sqrt[c]*d - Sqrt[-a]*e)*(1 + n)) + (Sqrt[-a]*a*Hypergeometric2F1[1, 1 + n, 2 + n, (Sqrt[c]*(d + e*x))/(Sqrt[c]
*d + Sqrt[-a]*e)])/((Sqrt[c]*d + Sqrt[-a]*e)*(1 + n))))/(2*c^2)

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Maple [F]  time = 0.692, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( ex+d \right ) ^{n}{x}^{4}}{c{x}^{2}+a}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(e*x+d)^n/(c*x^2+a),x)

[Out]

int(x^4*(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^{4}}{c x^{2} + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(e*x+d)^n/(c*x^2+a),x, algorithm="maxima")

[Out]

integrate((e*x + d)^n*x^4/(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^{4}}{c x^{2} + a}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(e*x+d)^n/(c*x^2+a),x, algorithm="fricas")

[Out]

integral((e*x + d)^n*x^4/(c*x^2 + a), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4} \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**4*(e*x+d)**n/(c*x**2+a),x)

[Out]

Integral(x**4*(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^{4}}{c x^{2} + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(e*x+d)^n/(c*x^2+a),x, algorithm="giac")

[Out]

integrate((e*x + d)^n*x^4/(c*x^2 + a), x)