3.4.72 \(\int \frac {x^2 (d+e x)^n}{(a+c x^2)^2} \, dx\) [372]

Optimal. Leaf size=306 \[ -\frac {(a e+c d x) (d+e x)^{1+n}}{2 c \left (c d^2+a e^2\right ) \left (a+c x^2\right )}+\frac {\left (c d^2-\sqrt {-a} \sqrt {c} d e n+a e^2 (1+n)\right ) (d+e x)^{1+n} \, _2F_1\left (1,1+n;2+n;\frac {\sqrt {c} (d+e x)}{\sqrt {c} d-\sqrt {-a} e}\right )}{4 \sqrt {-a} c \left (\sqrt {c} d-\sqrt {-a} e\right ) \left (c d^2+a e^2\right ) (1+n)}-\frac {\left (c d^2+\sqrt {-a} \sqrt {c} d e n+a e^2 (1+n)\right ) (d+e x)^{1+n} \, _2F_1\left (1,1+n;2+n;\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}\right )}{4 \sqrt {-a} c \left (\sqrt {c} d+\sqrt {-a} e\right ) \left (c d^2+a e^2\right ) (1+n)} \]

[Out]

-1/2*(c*d*x+a*e)*(e*x+d)^(1+n)/c/(a*e^2+c*d^2)/(c*x^2+a)+1/4*(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)))*(c*d^2+a*e^2*(1+n)-d*e*n*(-a)^(1/2)*c^(1/2))/c/(a*e^2+c*d^2)/(1+n)/(-a)^(1/2)
/(-e*(-a)^(1/2)+d*c^(1/2))-1/4*(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))
)*(c*d^2+a*e^2*(1+n)+d*e*n*(-a)^(1/2)*c^(1/2))/c/(a*e^2+c*d^2)/(1+n)/(-a)^(1/2)/(e*(-a)^(1/2)+d*c^(1/2))

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Rubi [A]
time = 0.35, antiderivative size = 306, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {1663, 845, 70} \begin {gather*} \frac {(d+e x)^{n+1} \left (-\sqrt {-a} \sqrt {c} d e n+a e^2 (n+1)+c d^2\right ) \, _2F_1\left (1,n+1;n+2;\frac {\sqrt {c} (d+e x)}{\sqrt {c} d-\sqrt {-a} e}\right )}{4 \sqrt {-a} c (n+1) \left (\sqrt {c} d-\sqrt {-a} e\right ) \left (a e^2+c d^2\right )}-\frac {(d+e x)^{n+1} \left (\sqrt {-a} \sqrt {c} d e n+a e^2 (n+1)+c d^2\right ) \, _2F_1\left (1,n+1;n+2;\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}\right )}{4 \sqrt {-a} c (n+1) \left (\sqrt {-a} e+\sqrt {c} d\right ) \left (a e^2+c d^2\right )}-\frac {(d+e x)^{n+1} (a e+c d x)}{2 c \left (a+c x^2\right ) \left (a e^2+c d^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*((a*e + c*d*x)*(d + e*x)^(1 + n))/(c*(c*d^2 + a*e^2)*(a + c*x^2)) + ((c*d^2 - Sqrt[-a]*Sqrt[c]*d*e*n + a*
e^2*(1 + n))*(d + e*x)^(1 + n)*Hypergeometric2F1[1, 1 + n, 2 + n, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d - Sqrt[-a]*e)
])/(4*Sqrt[-a]*c*(Sqrt[c]*d - Sqrt[-a]*e)*(c*d^2 + a*e^2)*(1 + n)) - ((c*d^2 + Sqrt[-a]*Sqrt[c]*d*e*n + a*e^2*
(1 + n))*(d + e*x)^(1 + n)*Hypergeometric2F1[1, 1 + n, 2 + n, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)])/(
4*Sqrt[-a]*c*(Sqrt[c]*d + Sqrt[-a]*e)*(c*d^2 + a*e^2)*(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 845

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 1663

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq,
a + c*x^2, x], f = Coeff[PolynomialRemainder[Pq, a + c*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + c
*x^2, x], x, 1]}, Simp[(-(d + e*x)^(m + 1))*(a + c*x^2)^(p + 1)*((a*(e*f - d*g) + (c*d*f + a*e*g)*x)/(2*a*(p +
1)*(c*d^2 + a*e^2))), x] + Dist[1/(2*a*(p + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*ExpandTo
Sum[2*a*(p + 1)*(c*d^2 + a*e^2)*Q + c*d^2*f*(2*p + 3) - a*e*(d*g*m - e*f*(m + 2*p + 3)) + e*(c*d*f + a*e*g)*(m
+ 2*p + 4)*x, x], x], x]] /; FreeQ[{a, c, d, e, m}, x] && PolyQ[Pq, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1]
&&  !(IGtQ[m, 0] && RationalQ[a, c, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0]))

Rubi steps

\begin {align*} \int \frac {x^2 (d+e x)^n}{\left (a+c x^2\right )^2} \, dx &=-\frac {(a e+c d x) (d+e x)^{1+n}}{2 c \left (c d^2+a e^2\right ) \left (a+c x^2\right )}-\frac {\int \frac {(d+e x)^n \left (-\frac {a \left (c d^2+a e^2 (1+n)\right )}{c}-a d e n x\right )}{a+c x^2} \, dx}{2 a \left (c d^2+a e^2\right )}\\ &=-\frac {(a e+c d x) (d+e x)^{1+n}}{2 c \left (c d^2+a e^2\right ) \left (a+c x^2\right )}-\frac {\int \left (\frac {\left (\frac {a^2 d e n}{\sqrt {c}}-\frac {\sqrt {-a} a \left (c d^2+a e^2 (1+n)\right )}{c}\right ) (d+e x)^n}{2 a \left (\sqrt {-a}-\sqrt {c} x\right )}+\frac {\left (-\frac {a^2 d e n}{\sqrt {c}}-\frac {\sqrt {-a} a \left (c d^2+a e^2 (1+n)\right )}{c}\right ) (d+e x)^n}{2 a \left (\sqrt {-a}+\sqrt {c} x\right )}\right ) \, dx}{2 a \left (c d^2+a e^2\right )}\\ &=-\frac {(a e+c d x) (d+e x)^{1+n}}{2 c \left (c d^2+a e^2\right ) \left (a+c x^2\right )}-\frac {\left (c d^2-\sqrt {-a} \sqrt {c} d e n+a e^2 (1+n)\right ) \int \frac {(d+e x)^n}{\sqrt {-a}+\sqrt {c} x} \, dx}{4 \sqrt {-a} c \left (c d^2+a e^2\right )}-\frac {\left (c d^2+\sqrt {-a} \sqrt {c} d e n+a e^2 (1+n)\right ) \int \frac {(d+e x)^n}{\sqrt {-a}-\sqrt {c} x} \, dx}{4 \sqrt {-a} c \left (c d^2+a e^2\right )}\\ &=-\frac {(a e+c d x) (d+e x)^{1+n}}{2 c \left (c d^2+a e^2\right ) \left (a+c x^2\right )}+\frac {\left (c d^2-\sqrt {-a} \sqrt {c} d e n+a e^2 (1+n)\right ) (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 )}{4 \sqrt {-a} c \left (\sqrt {c} d-\sqrt {-a} e\right ) \left (c d^2+a e^2\right ) (1+n)}-\frac {\left (c d^2+\sqrt {-a} \sqrt {c} d e n+a e^2 (1+n)\right ) (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 )}{4 \sqrt {-a} c \left (\sqrt {c} d+\sqrt {-a} e\right ) \left (c d^2+a e^2\right ) (1+n)}\\ \end {align*}

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Mathematica [A]
time = 0.59, size = 403, normalized size = 1.32 \begin {gather*} \frac {(d+e x)^{1+n} \left (-\frac {2 (a e+c d x)}{\left (c d^2+a e^2\right ) \left (a+c x^2\right )}+\frac {2 \, _2F_1\left (1,1+n;2+n;\frac {\sqrt {c} (d+e x)}{\sqrt {c} d-\sqrt {-a} e}\right )}{\sqrt {-a} \left (\sqrt {c} d-\sqrt {-a} e\right ) (1+n)}-\frac {2 \, _2F_1\left (1,1+n;2+n;\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}\right )}{\sqrt {-a} \left (\sqrt {c} d+\sqrt {-a} e\right ) (1+n)}+\frac {a \left (\frac {\left (c d^2-a e^2 (-1+n)+\sqrt {-a} \sqrt {c} d e n\right ) \, _2F_1\left (1,1+n;2+n;\frac {\sqrt {c} (d+e x)}{\sqrt {c} d-\sqrt {-a} e}\right )}{\sqrt {c} d-\sqrt {-a} e}-\frac {\left (c d^2-a e^2 (-1+n)-\sqrt {-a} \sqrt {c} d e n\right ) \, _2F_1\left (1,1+n;2+n;\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}\right )}{\sqrt {c} d+\sqrt {-a} e}\right )}{(-a)^{3/2} \left (c d^2+a e^2\right ) (1+n)}\right )}{4 c} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((d + e*x)^(1 + n)*((-2*(a*e + c*d*x))/((c*d^2 + a*e^2)*(a + c*x^2)) + (2*Hypergeometric2F1[1, 1 + n, 2 + n, (
Sqrt[c]*(d + e*x))/(Sqrt[c]*d - Sqrt[-a]*e)])/(Sqrt[-a]*(Sqrt[c]*d - Sqrt[-a]*e)*(1 + n)) - (2*Hypergeometric2
F1[1, 1 + n, 2 + n, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)])/(Sqrt[-a]*(Sqrt[c]*d + Sqrt[-a]*e)*(1 + n))
+ (a*(((c*d^2 - a*e^2*(-1 + n) + Sqrt[-a]*Sqrt[c]*d*e*n)*Hypergeometric2F1[1, 1 + n, 2 + n, (Sqrt[c]*(d + e*x
))/(Sqrt[c]*d - Sqrt[-a]*e)])/(Sqrt[c]*d - Sqrt[-a]*e) - ((c*d^2 - a*e^2*(-1 + n) - Sqrt[-a]*Sqrt[c]*d*e*n)*Hy
pergeometric2F1[1, 1 + n, 2 + n, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)])/(Sqrt[c]*d + Sqrt[-a]*e)))/((-
a)^(3/2)*(c*d^2 + a*e^2)*(1 + n))))/(4*c)

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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {x^{2} \left (e x +d \right )^{n}}{\left (c \,x^{2}+a \right )^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

[Out]

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

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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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)^2,x, algorithm="giac")

[Out]

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

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

[Out]

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

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