3.745 \(\int (d+e x)^{-6-2 p} (a+c x^2)^p \, dx\)

Optimal. Leaf size=559 \[ -\frac{c^2 \left (\sqrt{-a}-\sqrt{c} x\right ) \left (a+c x^2\right )^p (d+e x)^{-2 p-1} \left (3 a^2 e^4-6 a c d^2 e^2 (2 p+5)+c^2 d^4 \left (4 p^2+16 p+15\right )\right ) \left (-\frac{\left (\sqrt{-a}+\sqrt{c} x\right ) \left (\sqrt{-a} e+\sqrt{c} d\right )}{\left (\sqrt{-a}-\sqrt{c} x\right ) \left (\sqrt{c} d-\sqrt{-a} e\right )}\right )^{-p} \, _2F_1\left (-2 p-1,-p;-2 p;\frac{2 \sqrt{-a} \sqrt{c} (d+e x)}{\left (\sqrt{c} d-\sqrt{-a} e\right ) \left (\sqrt{-a}-\sqrt{c} x\right )}\right )}{(2 p+1) (2 p+3) (2 p+5) \left (\sqrt{-a} e+\sqrt{c} d\right ) \left (a e^2+c d^2\right )^4}+\frac{c^2 d e (p+3) \left (a+c x^2\right )^{p+1} (d+e x)^{-2 (p+1)} \left (a e^2 (5 p+8)-c d^2 \left (2 p^2+7 p+8\right )\right )}{(p+1) (p+2) (2 p+3) (2 p+5) \left (a e^2+c d^2\right )^4}+\frac{c e \left (a+c x^2\right )^{p+1} (d+e x)^{-2 p-3} \left (3 a e^2 (p+2)-c d^2 \left (2 p^2+11 p+18\right )\right )}{(p+2) (2 p+3) (2 p+5) \left (a e^2+c d^2\right )^3}-\frac{e \left (a+c x^2\right )^{p+1} (d+e x)^{-2 p-5}}{(2 p+5) \left (a e^2+c d^2\right )}-\frac{c d e (p+4) \left (a+c x^2\right )^{p+1} (d+e x)^{-2 (p+2)}}{(p+2) (2 p+5) \left (a e^2+c d^2\right )^2} \]

[Out]

-((e*(d + e*x)^(-5 - 2*p)*(a + c*x^2)^(1 + p))/((c*d^2 + a*e^2)*(5 + 2*p))) + (c*e*(3*a*e^2*(2 + p) - c*d^2*(1
8 + 11*p + 2*p^2))*(d + e*x)^(-3 - 2*p)*(a + c*x^2)^(1 + p))/((c*d^2 + a*e^2)^3*(2 + p)*(3 + 2*p)*(5 + 2*p)) +
(c^2*d*e*(3 + p)*(a*e^2*(8 + 5*p) - c*d^2*(8 + 7*p + 2*p^2))*(a + c*x^2)^(1 + p))/((c*d^2 + a*e^2)^4*(1 + p)*
(2 + p)*(3 + 2*p)*(5 + 2*p)*(d + e*x)^(2*(1 + p))) - (c*d*e*(4 + p)*(a + c*x^2)^(1 + p))/((c*d^2 + a*e^2)^2*(2
+ p)*(5 + 2*p)*(d + e*x)^(2*(2 + p))) - (c^2*(3*a^2*e^4 - 6*a*c*d^2*e^2*(5 + 2*p) + c^2*d^4*(15 + 16*p + 4*p^
2))*(Sqrt[-a] - Sqrt[c]*x)*(d + e*x)^(-1 - 2*p)*(a + c*x^2)^p*Hypergeometric2F1[-1 - 2*p, -p, -2*p, (2*Sqrt[-a
]*Sqrt[c]*(d + e*x))/((Sqrt[c]*d - Sqrt[-a]*e)*(Sqrt[-a] - Sqrt[c]*x))])/((Sqrt[c]*d + Sqrt[-a]*e)*(c*d^2 + a*
e^2)^4*(1 + 2*p)*(3 + 2*p)*(5 + 2*p)*(-(((Sqrt[c]*d + Sqrt[-a]*e)*(Sqrt[-a] + Sqrt[c]*x))/((Sqrt[c]*d - Sqrt[-
a]*e)*(Sqrt[-a] - Sqrt[c]*x))))^p)

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Rubi [A]  time = 0.731034, antiderivative size = 559, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {745, 837, 807, 727} \[ -\frac{c^2 \left (\sqrt{-a}-\sqrt{c} x\right ) \left (a+c x^2\right )^p (d+e x)^{-2 p-1} \left (3 a^2 e^4-6 a c d^2 e^2 (2 p+5)+c^2 d^4 \left (4 p^2+16 p+15\right )\right ) \left (-\frac{\left (\sqrt{-a}+\sqrt{c} x\right ) \left (\sqrt{-a} e+\sqrt{c} d\right )}{\left (\sqrt{-a}-\sqrt{c} x\right ) \left (\sqrt{c} d-\sqrt{-a} e\right )}\right )^{-p} \, _2F_1\left (-2 p-1,-p;-2 p;\frac{2 \sqrt{-a} \sqrt{c} (d+e x)}{\left (\sqrt{c} d-\sqrt{-a} e\right ) \left (\sqrt{-a}-\sqrt{c} x\right )}\right )}{(2 p+1) (2 p+3) (2 p+5) \left (\sqrt{-a} e+\sqrt{c} d\right ) \left (a e^2+c d^2\right )^4}+\frac{c^2 d e (p+3) \left (a+c x^2\right )^{p+1} (d+e x)^{-2 (p+1)} \left (a e^2 (5 p+8)-c d^2 \left (2 p^2+7 p+8\right )\right )}{(p+1) (p+2) (2 p+3) (2 p+5) \left (a e^2+c d^2\right )^4}+\frac{c e \left (a+c x^2\right )^{p+1} (d+e x)^{-2 p-3} \left (3 a e^2 (p+2)-c d^2 \left (2 p^2+11 p+18\right )\right )}{(p+2) (2 p+3) (2 p+5) \left (a e^2+c d^2\right )^3}-\frac{e \left (a+c x^2\right )^{p+1} (d+e x)^{-2 p-5}}{(2 p+5) \left (a e^2+c d^2\right )}-\frac{c d e (p+4) \left (a+c x^2\right )^{p+1} (d+e x)^{-2 (p+2)}}{(p+2) (2 p+5) \left (a e^2+c d^2\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^(-6 - 2*p)*(a + c*x^2)^p,x]

[Out]

-((e*(d + e*x)^(-5 - 2*p)*(a + c*x^2)^(1 + p))/((c*d^2 + a*e^2)*(5 + 2*p))) + (c*e*(3*a*e^2*(2 + p) - c*d^2*(1
8 + 11*p + 2*p^2))*(d + e*x)^(-3 - 2*p)*(a + c*x^2)^(1 + p))/((c*d^2 + a*e^2)^3*(2 + p)*(3 + 2*p)*(5 + 2*p)) +
(c^2*d*e*(3 + p)*(a*e^2*(8 + 5*p) - c*d^2*(8 + 7*p + 2*p^2))*(a + c*x^2)^(1 + p))/((c*d^2 + a*e^2)^4*(1 + p)*
(2 + p)*(3 + 2*p)*(5 + 2*p)*(d + e*x)^(2*(1 + p))) - (c*d*e*(4 + p)*(a + c*x^2)^(1 + p))/((c*d^2 + a*e^2)^2*(2
+ p)*(5 + 2*p)*(d + e*x)^(2*(2 + p))) - (c^2*(3*a^2*e^4 - 6*a*c*d^2*e^2*(5 + 2*p) + c^2*d^4*(15 + 16*p + 4*p^
2))*(Sqrt[-a] - Sqrt[c]*x)*(d + e*x)^(-1 - 2*p)*(a + c*x^2)^p*Hypergeometric2F1[-1 - 2*p, -p, -2*p, (2*Sqrt[-a
]*Sqrt[c]*(d + e*x))/((Sqrt[c]*d - Sqrt[-a]*e)*(Sqrt[-a] - Sqrt[c]*x))])/((Sqrt[c]*d + Sqrt[-a]*e)*(c*d^2 + a*
e^2)^4*(1 + 2*p)*(3 + 2*p)*(5 + 2*p)*(-(((Sqrt[c]*d + Sqrt[-a]*e)*(Sqrt[-a] + Sqrt[c]*x))/((Sqrt[c]*d - Sqrt[-
a]*e)*(Sqrt[-a] - Sqrt[c]*x))))^p)

Rule 745

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^(m + 1)*(a + c*x^2)^(p
+ 1))/((m + 1)*(c*d^2 + a*e^2)), x] + Dist[c/((m + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^(m + 1)*Simp[d*(m + 1)
- e*(m + 2*p + 3)*x, x]*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] && NeQ[
m, -1] && ((LtQ[m, -1] && IntQuadraticQ[a, 0, c, d, e, m, p, x]) || (SumSimplerQ[m, 1] && IntegerQ[p]) || ILtQ
[Simplify[m + 2*p + 3], 0])

Rule 837

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((e*f - d*g)
*(d + e*x)^(m + 1)*(a + c*x^2)^(p + 1))/((m + 1)*(c*d^2 + a*e^2)), x] + Dist[1/((m + 1)*(c*d^2 + a*e^2)), Int[
(d + e*x)^(m + 1)*(a + c*x^2)^p*Simp[(c*d*f + a*e*g)*(m + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; Fr
eeQ[{a, c, d, e, f, g, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] && ILtQ[Simplify[m + 2*p + 3], 0] && NeQ[m, -1]

Rule 807

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Simp[((e*f - d*g
)*(d + e*x)^(m + 1)*(a + c*x^2)^(p + 1))/(2*(p + 1)*(c*d^2 + a*e^2)), x] + Dist[(c*d*f + a*e*g)/(c*d^2 + a*e^2
), Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && NeQ[c*d^2 + a*e^2, 0]
&& EqQ[Simplify[m + 2*p + 3], 0]

Rule 727

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((Rt[-(a*c), 2] - c*x)*(d + e*x)^(m
+ 1)*(a + c*x^2)^p*Hypergeometric2F1[m + 1, -p, m + 2, (2*c*Rt[-(a*c), 2]*(d + e*x))/((c*d - e*Rt[-(a*c), 2])
*(Rt[-(a*c), 2] - c*x))])/((m + 1)*(c*d + e*Rt[-(a*c), 2])*(((c*d + e*Rt[-(a*c), 2])*(Rt[-(a*c), 2] + c*x))/((
c*d - e*Rt[-(a*c), 2])*(-Rt[-(a*c), 2] + c*x)))^p), x] /; FreeQ[{a, c, d, e, m, p}, x] && NeQ[c*d^2 + a*e^2, 0
] &&  !IntegerQ[p] && EqQ[m + 2*p + 2, 0]

Rubi steps

\begin{align*} \int (d+e x)^{-6-2 p} \left (a+c x^2\right )^p \, dx &=-\frac{e (d+e x)^{-5-2 p} \left (a+c x^2\right )^{1+p}}{\left (c d^2+a e^2\right ) (5+2 p)}-\frac{c \int (d+e x)^{-5-2 p} (-d (5+2 p)+3 e x) \left (a+c x^2\right )^p \, dx}{\left (c d^2+a e^2\right ) (5+2 p)}\\ &=-\frac{e (d+e x)^{-5-2 p} \left (a+c x^2\right )^{1+p}}{\left (c d^2+a e^2\right ) (5+2 p)}-\frac{c d e (4+p) (d+e x)^{-2 (2+p)} \left (a+c x^2\right )^{1+p}}{\left (c d^2+a e^2\right )^2 (2+p) (5+2 p)}+\frac{c \int (d+e x)^{-4-2 p} \left (-2 (2+p) \left (3 a e^2-c d^2 (5+2 p)\right )-4 c d e (4+p) x\right ) \left (a+c x^2\right )^p \, dx}{2 \left (c d^2+a e^2\right )^2 (2+p) (5+2 p)}\\ &=-\frac{e (d+e x)^{-5-2 p} \left (a+c x^2\right )^{1+p}}{\left (c d^2+a e^2\right ) (5+2 p)}+\frac{c e \left (3 a e^2 (2+p)-c d^2 \left (18+11 p+2 p^2\right )\right ) (d+e x)^{-3-2 p} \left (a+c x^2\right )^{1+p}}{\left (c d^2+a e^2\right )^3 (2+p) (3+2 p) (5+2 p)}-\frac{c d e (4+p) (d+e x)^{-2 (2+p)} \left (a+c x^2\right )^{1+p}}{\left (c d^2+a e^2\right )^2 (2+p) (5+2 p)}-\frac{c \int (d+e x)^{-3-2 p} \left (2 c d (3+2 p) \left (a e^2 (14+5 p)-c d^2 \left (10+9 p+2 p^2\right )\right )-2 c e \left (3 a e^2 (2+p)-c d^2 \left (18+11 p+2 p^2\right )\right ) x\right ) \left (a+c x^2\right )^p \, dx}{2 \left (c d^2+a e^2\right )^3 (2+p) (3+2 p) (5+2 p)}\\ &=-\frac{e (d+e x)^{-5-2 p} \left (a+c x^2\right )^{1+p}}{\left (c d^2+a e^2\right ) (5+2 p)}+\frac{c e \left (3 a e^2 (2+p)-c d^2 \left (18+11 p+2 p^2\right )\right ) (d+e x)^{-3-2 p} \left (a+c x^2\right )^{1+p}}{\left (c d^2+a e^2\right )^3 (2+p) (3+2 p) (5+2 p)}+\frac{c^2 d e (3+p) \left (a e^2 (8+5 p)-c d^2 \left (8+7 p+2 p^2\right )\right ) (d+e x)^{-2 (1+p)} \left (a+c x^2\right )^{1+p}}{\left (c d^2+a e^2\right )^4 (1+p) (2+p) (3+2 p) (5+2 p)}-\frac{c d e (4+p) (d+e x)^{-2 (2+p)} \left (a+c x^2\right )^{1+p}}{\left (c d^2+a e^2\right )^2 (2+p) (5+2 p)}+\frac{\left (c^2 \left (3 a^2 e^4-6 a c d^2 e^2 (5+2 p)+c^2 d^4 \left (15+16 p+4 p^2\right )\right )\right ) \int (d+e x)^{-2-2 p} \left (a+c x^2\right )^p \, dx}{\left (c d^2+a e^2\right )^4 (3+2 p) (5+2 p)}\\ &=-\frac{e (d+e x)^{-5-2 p} \left (a+c x^2\right )^{1+p}}{\left (c d^2+a e^2\right ) (5+2 p)}+\frac{c e \left (3 a e^2 (2+p)-c d^2 \left (18+11 p+2 p^2\right )\right ) (d+e x)^{-3-2 p} \left (a+c x^2\right )^{1+p}}{\left (c d^2+a e^2\right )^3 (2+p) (3+2 p) (5+2 p)}+\frac{c^2 d e (3+p) \left (a e^2 (8+5 p)-c d^2 \left (8+7 p+2 p^2\right )\right ) (d+e x)^{-2 (1+p)} \left (a+c x^2\right )^{1+p}}{\left (c d^2+a e^2\right )^4 (1+p) (2+p) (3+2 p) (5+2 p)}-\frac{c d e (4+p) (d+e x)^{-2 (2+p)} \left (a+c x^2\right )^{1+p}}{\left (c d^2+a e^2\right )^2 (2+p) (5+2 p)}-\frac{c^2 \left (3 a^2 e^4-6 a c d^2 e^2 (5+2 p)+c^2 d^4 \left (15+16 p+4 p^2\right )\right ) \left (\sqrt{-a}-\sqrt{c} x\right ) \left (-\frac{\left (\sqrt{c} d+\sqrt{-a} e\right ) \left (\sqrt{-a}+\sqrt{c} x\right )}{\left (\sqrt{c} d-\sqrt{-a} e\right ) \left (\sqrt{-a}-\sqrt{c} x\right )}\right )^{-p} (d+e x)^{-1-2 p} \left (a+c x^2\right )^p \, _2F_1\left (-1-2 p,-p;-2 p;\frac{2 \sqrt{-a} \sqrt{c} (d+e x)}{\left (\sqrt{c} d-\sqrt{-a} e\right ) \left (\sqrt{-a}-\sqrt{c} x\right )}\right )}{\left (\sqrt{c} d+\sqrt{-a} e\right ) \left (c d^2+a e^2\right )^4 (1+2 p) (3+2 p) (5+2 p)}\\ \end{align*}

Mathematica [B]  time = 53.7932, size = 5685, normalized size = 10.17 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(d + e*x)^(-6 - 2*p)*(a + c*x^2)^p,x]

[Out]

Result too large to show

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Maple [F]  time = 0.599, size = 0, normalized size = 0. \begin{align*} \int \left ( ex+d \right ) ^{-6-2\,p} \left ( c{x}^{2}+a \right ) ^{p}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(-6-2*p)*(c*x^2+a)^p,x)

[Out]

int((e*x+d)^(-6-2*p)*(c*x^2+a)^p,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (c x^{2} + a\right )}^{p}{\left (e x + d\right )}^{-2 \, p - 6}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(-6-2*p)*(c*x^2+a)^p,x, algorithm="maxima")

[Out]

integrate((c*x^2 + a)^p*(e*x + d)^(-2*p - 6), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (c x^{2} + a\right )}^{p}{\left (e x + d\right )}^{-2 \, p - 6}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(-6-2*p)*(c*x^2+a)^p,x, algorithm="fricas")

[Out]

integral((c*x^2 + a)^p*(e*x + d)^(-2*p - 6), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**(-6-2*p)*(c*x**2+a)**p,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (c x^{2} + a\right )}^{p}{\left (e x + d\right )}^{-2 \, p - 6}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(-6-2*p)*(c*x^2+a)^p,x, algorithm="giac")

[Out]

integrate((c*x^2 + a)^p*(e*x + d)^(-2*p - 6), x)