3.4.65 \(\int \frac {x^2 (d+e x)^n}{a+c x^2} \, dx\) [365]

Optimal. Leaf size=194 \[ \frac {(d+e x)^{1+n}}{c e (1+n)}+\frac {\sqrt {-a} (d+e x)^{1+n} \, _2F_1\left (1,1+n;2+n;\frac {\sqrt {c} (d+e x)}{\sqrt {c} d-\sqrt {-a} e}\right )}{2 c \left (\sqrt {c} d-\sqrt {-a} e\right ) (1+n)}-\frac {\sqrt {-a} (d+e x)^{1+n} \, _2F_1\left (1,1+n;2+n;\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}\right )}{2 c \left (\sqrt {c} d+\sqrt {-a} e\right ) (1+n)} \]

[Out]

(e*x+d)^(1+n)/c/e/(1+n)+1/2*(e*x+d)^(1+n)*hypergeom([1, 1+n],[2+n],(e*x+d)*c^(1/2)/(-e*(-a)^(1/2)+d*c^(1/2)))*
(-a)^(1/2)/c/(1+n)/(-e*(-a)^(1/2)+d*c^(1/2))-1/2*(e*x+d)^(1+n)*hypergeom([1, 1+n],[2+n],(e*x+d)*c^(1/2)/(e*(-a
)^(1/2)+d*c^(1/2)))*(-a)^(1/2)/c/(1+n)/(e*(-a)^(1/2)+d*c^(1/2))

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Rubi [A]
time = 0.16, antiderivative size = 194, 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 = {1643, 726, 70} \begin {gather*} \frac {\sqrt {-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 (n+1) \left (\sqrt {c} d-\sqrt {-a} e\right )}-\frac {\sqrt {-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 (n+1) \left (\sqrt {-a} e+\sqrt {c} d\right )}+\frac {(d+e x)^{n+1}}{c e (n+1)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^2*(d + e*x)^n)/(a + c*x^2),x]

[Out]

(d + e*x)^(1 + n)/(c*e*(1 + n)) + (Sqrt[-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*(Sqrt[c]*d - Sqrt[-a]*e)*(1 + n)) - (Sqrt[-a]*(d + e*x)^(1 + n)*Hyperge
ometric2F1[1, 1 + n, 2 + n, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)])/(2*c*(Sqrt[c]*d + Sqrt[-a]*e)*(1 +
n))

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 726

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 1643

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]

Rubi steps

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

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Mathematica [A]
time = 0.14, size = 170, normalized size = 0.88 \begin {gather*} \frac {(d+e x)^{1+n} \left (2 \left (c d^2+a e^2\right )+e \left (\sqrt {-a} \sqrt {c} d-a e\right ) \, _2F_1\left (1,1+n;2+n;\frac {\sqrt {c} (d+e x)}{\sqrt {c} d-\sqrt {-a} e}\right )-e \left (\sqrt {-a} \sqrt {c} d+a e\right ) \, _2F_1\left (1,1+n;2+n;\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}\right )\right )}{2 c e \left (c d^2+a e^2\right ) (1+n)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {x^{2} \left (e x +d \right )^{n}}{c \,x^{2}+a}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2} \left (d + e x\right )^{n}}{a + c x^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^2\,{\left (d+e\,x\right )}^n}{c\,x^2+a} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2*(d + e*x)^n)/(a + c*x^2),x)

[Out]

int((x^2*(d + e*x)^n)/(a + c*x^2), x)

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