\(\int (d+e x)^{-6-2 p} (a+c x^2)^p \, dx\) [745]

Optimal result

Integrand size = 21, antiderivative size = 559 \[ \int (d+e x)^{-6-2 p} \left (a+c x^2\right )^p \, dx=-\frac {e (d+e x)^{-5-2 p} \left (a+c x^2\right )^{1+p}}{\left (c d^2+a e^2\right ) (5+2 p)}+\frac {c e \left (3 a e^2 (2+p)-c d^2 \left (18+11 p+2 p^2\right )\right ) (d+e x)^{-3-2 p} \left (a+c x^2\right )^{1+p}}{\left (c d^2+a e^2\right )^3 (2+p) (3+2 p) (5+2 p)}+\frac {c^2 d e (3+p) \left (a e^2 (8+5 p)-c d^2 \left (8+7 p+2 p^2\right )\right ) (d+e x)^{-2 (1+p)} \left (a+c x^2\right )^{1+p}}{\left (c d^2+a e^2\right )^4 (1+p) (2+p) (3+2 p) (5+2 p)}-\frac {c d e (4+p) (d+e x)^{-2 (2+p)} \left (a+c x^2\right )^{1+p}}{\left (c d^2+a e^2\right )^2 (2+p) (5+2 p)}-\frac {c^2 \left (3 a^2 e^4-6 a c d^2 e^2 (5+2 p)+c^2 d^4 \left (15+16 p+4 p^2\right )\right ) \left (\sqrt {-a}-\sqrt {c} x\right ) \left (-\frac {\left (\sqrt {c} d+\sqrt {-a} e\right ) \left (\sqrt {-a}+\sqrt {c} x\right )}{\left (\sqrt {c} d-\sqrt {-a} e\right ) \left (\sqrt {-a}-\sqrt {c} x\right )}\right )^{-p} (d+e x)^{-1-2 p} \left (a+c x^2\right )^p \operatorname {Hypergeometric2F1}\left (-1-2 p,-p,-2 p,\frac {2 \sqrt {-a} \sqrt {c} (d+e x)}{\left (\sqrt {c} d-\sqrt {-a} e\right ) \left (\sqrt {-a}-\sqrt {c} x\right )}\right )}{\left (\sqrt {c} d+\sqrt {-a} e\right ) \left (c d^2+a e^2\right )^4 (1+2 p) (3+2 p) (5+2 p)} \]

[Out]

-e*(e*x+d)^(-5-2*p)*(c*x^2+a)^(p+1)/(a*e^2+c*d^2)/(5+2*p)+c*e*(3*a*e^2*(2+p)-c*d^2*(2*p^2+11*p+18))*(e*x+d)^(-
3-2*p)*(c*x^2+a)^(p+1)/(a*e^2+c*d^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*(2*p^2+7*p+8))
*(c*x^2+a)^(p+1)/(a*e^2+c*d^2)^4/(p+1)/(2+p)/(3+2*p)/(5+2*p)/((e*x+d)^(2+2*p))-c*d*e*(4+p)*(c*x^2+a)^(p+1)/(a*
e^2+c*d^2)^2/(2+p)/(5+2*p)/((e*x+d)^(4+2*p))-c^2*(3*a^2*e^4-6*a*c*d^2*e^2*(5+2*p)+c^2*d^4*(4*p^2+16*p+15))*(e*
x+d)^(-1-2*p)*(c*x^2+a)^p*hypergeom([-p, -1-2*p],[-2*p],2*(e*x+d)*(-a)^(1/2)*c^(1/2)/(-e*(-a)^(1/2)+d*c^(1/2))
/((-a)^(1/2)-x*c^(1/2)))*((-a)^(1/2)-x*c^(1/2))/(a*e^2+c*d^2)^4/(1+2*p)/(3+2*p)/(5+2*p)/(e*(-a)^(1/2)+d*c^(1/2
))/((-(e*(-a)^(1/2)+d*c^(1/2))*((-a)^(1/2)+x*c^(1/2))/(-e*(-a)^(1/2)+d*c^(1/2))/((-a)^(1/2)-x*c^(1/2)))^p)

Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 559, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {759, 851, 821, 741} \[ \int (d+e x)^{-6-2 p} \left (a+c x^2\right )^p \, dx=-\frac {c^2 \left (\sqrt {-a}-\sqrt {c} x\right ) \left (a+c x^2\right )^p (d+e x)^{-2 p-1} \left (3 a^2 e^4-6 a c d^2 e^2 (2 p+5)+c^2 d^4 \left (4 p^2+16 p+15\right )\right ) \left (-\frac {\left (\sqrt {-a}+\sqrt {c} x\right ) \left (\sqrt {-a} e+\sqrt {c} d\right )}{\left (\sqrt {-a}-\sqrt {c} x\right ) \left (\sqrt {c} d-\sqrt {-a} e\right )}\right )^{-p} \operatorname {Hypergeometric2F1}\left (-2 p-1,-p,-2 p,\frac {2 \sqrt {-a} \sqrt {c} (d+e x)}{\left (\sqrt {c} d-\sqrt {-a} e\right ) \left (\sqrt {-a}-\sqrt {c} x\right )}\right )}{(2 p+1) (2 p+3) (2 p+5) \left (\sqrt {-a} e+\sqrt {c} d\right ) \left (a e^2+c d^2\right )^4}+\frac {c^2 d e (p+3) \left (a+c x^2\right )^{p+1} (d+e x)^{-2 (p+1)} \left (a e^2 (5 p+8)-c d^2 \left (2 p^2+7 p+8\right )\right )}{(p+1) (p+2) (2 p+3) (2 p+5) \left (a e^2+c d^2\right )^4}+\frac {c e \left (a+c x^2\right )^{p+1} (d+e x)^{-2 p-3} \left (3 a e^2 (p+2)-c d^2 \left (2 p^2+11 p+18\right )\right )}{(p+2) (2 p+3) (2 p+5) \left (a e^2+c d^2\right )^3}-\frac {e \left (a+c x^2\right )^{p+1} (d+e x)^{-2 p-5}}{(2 p+5) \left (a e^2+c d^2\right )}-\frac {c d e (p+4) \left (a+c x^2\right )^{p+1} (d+e x)^{-2 (p+2)}}{(p+2) (2 p+5) \left (a e^2+c d^2\right )^2} \]

[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 741

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/((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))*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)))], 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]

Rule 759

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 821

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 851

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]

Rubi steps \begin{align*} \text {integral}& = -\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 [F]

\[ \int (d+e x)^{-6-2 p} \left (a+c x^2\right )^p \, dx=\int (d+e x)^{-6-2 p} \left (a+c x^2\right )^p \, dx \]

[In]

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

[Out]

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

Maple [F]

\[\int \left (e x +d \right )^{-6-2 p} \left (c \,x^{2}+a \right )^{p}d x\]

[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)

Fricas [F]

\[ \int (d+e x)^{-6-2 p} \left (a+c x^2\right )^p \, dx=\int { {\left (c x^{2} + a\right )}^{p} {\left (e x + d\right )}^{-2 \, p - 6} \,d x } \]

[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)

Sympy [F(-1)]

Timed out. \[ \int (d+e x)^{-6-2 p} \left (a+c x^2\right )^p \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

\[ \int (d+e x)^{-6-2 p} \left (a+c x^2\right )^p \, dx=\int { {\left (c x^{2} + a\right )}^{p} {\left (e x + d\right )}^{-2 \, p - 6} \,d x } \]

[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)

Giac [F]

\[ \int (d+e x)^{-6-2 p} \left (a+c x^2\right )^p \, dx=\int { {\left (c x^{2} + a\right )}^{p} {\left (e x + d\right )}^{-2 \, p - 6} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int (d+e x)^{-6-2 p} \left (a+c x^2\right )^p \, dx=\int \frac {{\left (c\,x^2+a\right )}^p}{{\left (d+e\,x\right )}^{2\,p+6}} \,d x \]

[In]

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

[Out]

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