### 3.743 $$\int (d+e x)^{-4-2 p} (a+c x^2)^p \, dx$$

Optimal. Leaf size=347 $\frac{c \left (\sqrt{-a}-\sqrt{c} x\right ) \left (a+c x^2\right )^p (d+e x)^{-2 p-1} \left (a e^2-c d^2 (2 p+3)\right ) \left (-\frac{\left (\sqrt{-a}+\sqrt{c} x\right ) \left (\sqrt{-a} e+\sqrt{c} d\right )}{\left (\sqrt{-a}-\sqrt{c} x\right ) \left (\sqrt{c} d-\sqrt{-a} e\right )}\right )^{-p} \, _2F_1\left (-2 p-1,-p;-2 p;\frac{2 \sqrt{-a} \sqrt{c} (d+e x)}{\left (\sqrt{c} d-\sqrt{-a} e\right ) \left (\sqrt{-a}-\sqrt{c} x\right )}\right )}{(2 p+1) (2 p+3) \left (\sqrt{-a} e+\sqrt{c} d\right ) \left (a e^2+c d^2\right )^2}-\frac{e \left (a+c x^2\right )^{p+1} (d+e x)^{-2 p-3}}{(2 p+3) \left (a e^2+c d^2\right )}-\frac{c d e (p+2) \left (a+c x^2\right )^{p+1} (d+e x)^{-2 (p+1)}}{(p+1) (2 p+3) \left (a e^2+c d^2\right )^2}$

[Out]

-((e*(d + e*x)^(-3 - 2*p)*(a + c*x^2)^(1 + p))/((c*d^2 + a*e^2)*(3 + 2*p))) - (c*d*e*(2 + p)*(a + c*x^2)^(1 +
p))/((c*d^2 + a*e^2)^2*(1 + p)*(3 + 2*p)*(d + e*x)^(2*(1 + p))) + (c*(a*e^2 - c*d^2*(3 + 2*p))*(Sqrt[-a] - Sqr
t[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)^2*(1 + 2*p)*(3
+ 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 = 0.182588, antiderivative size = 347, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.143, Rules used = {745, 807, 727} $\frac{c \left (\sqrt{-a}-\sqrt{c} x\right ) \left (a+c x^2\right )^p (d+e x)^{-2 p-1} \left (a e^2-c d^2 (2 p+3)\right ) \left (-\frac{\left (\sqrt{-a}+\sqrt{c} x\right ) \left (\sqrt{-a} e+\sqrt{c} d\right )}{\left (\sqrt{-a}-\sqrt{c} x\right ) \left (\sqrt{c} d-\sqrt{-a} e\right )}\right )^{-p} \, _2F_1\left (-2 p-1,-p;-2 p;\frac{2 \sqrt{-a} \sqrt{c} (d+e x)}{\left (\sqrt{c} d-\sqrt{-a} e\right ) \left (\sqrt{-a}-\sqrt{c} x\right )}\right )}{(2 p+1) (2 p+3) \left (\sqrt{-a} e+\sqrt{c} d\right ) \left (a e^2+c d^2\right )^2}-\frac{e \left (a+c x^2\right )^{p+1} (d+e x)^{-2 p-3}}{(2 p+3) \left (a e^2+c d^2\right )}-\frac{c d e (p+2) \left (a+c x^2\right )^{p+1} (d+e x)^{-2 (p+1)}}{(p+1) (2 p+3) \left (a e^2+c d^2\right )^2}$

Antiderivative was successfully veriﬁed.

[In]

Int[(d + e*x)^(-4 - 2*p)*(a + c*x^2)^p,x]

[Out]

-((e*(d + e*x)^(-3 - 2*p)*(a + c*x^2)^(1 + p))/((c*d^2 + a*e^2)*(3 + 2*p))) - (c*d*e*(2 + p)*(a + c*x^2)^(1 +
p))/((c*d^2 + a*e^2)^2*(1 + p)*(3 + 2*p)*(d + e*x)^(2*(1 + p))) + (c*(a*e^2 - c*d^2*(3 + 2*p))*(Sqrt[-a] - Sqr
t[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)^2*(1 + 2*p)*(3
+ 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 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)^{-4-2 p} \left (a+c x^2\right )^p \, dx &=-\frac{e (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)}-\frac{c \int (d+e x)^{-3-2 p} (-d (3+2 p)+e x) \left (a+c x^2\right )^p \, dx}{\left (c d^2+a e^2\right ) (3+2 p)}\\ &=-\frac{e (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)}-\frac{c d e (2+p) (d+e x)^{-2 (1+p)} \left (a+c x^2\right )^{1+p}}{\left (c d^2+a e^2\right )^2 (1+p) (3+2 p)}-\frac{\left (c \left (a e^2-c d^2 (3+2 p)\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 )^2 (3+2 p)}\\ &=-\frac{e (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)}-\frac{c d e (2+p) (d+e x)^{-2 (1+p)} \left (a+c x^2\right )^{1+p}}{\left (c d^2+a e^2\right )^2 (1+p) (3+2 p)}+\frac{c \left (a e^2-c d^2 (3+2 p)\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 )^2 (1+2 p) (3+2 p)}\\ \end{align*}

Mathematica [B]  time = 25.074, size = 1439, normalized size = 4.15 $\text{result too large to display}$

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-2*(d + e*x)^(-3 - 2*p)*(a + c*x^2)^p*(1 - (d + e*x)/(d + Sqrt[-(a/c)]*e))^(1 + p)*Gamma[-2*(1 + p)]*((d + Sq
rt[-(a/c)]*e)^3*(Sqrt[-(a/c)] + x)*Gamma[1 - 2*p]*Gamma[-p]*Hypergeometric2F1[1, -p, -2*p, (2*Sqrt[-(a/c)]*(d
+ e*x))/((d + Sqrt[-(a/c)]*e)*(Sqrt[-(a/c)] + x))] + 3*(d + Sqrt[-(a/c)]*e)^3*p*(Sqrt[-(a/c)] + x)*Gamma[1 - 2
*p]*Gamma[-p]*Hypergeometric2F1[1, -p, -2*p, (2*Sqrt[-(a/c)]*(d + e*x))/((d + Sqrt[-(a/c)]*e)*(Sqrt[-(a/c)] +
x))] + 2*(d + Sqrt[-(a/c)]*e)^3*p^2*(Sqrt[-(a/c)] + x)*Gamma[1 - 2*p]*Gamma[-p]*Hypergeometric2F1[1, -p, -2*p,
(2*Sqrt[-(a/c)]*(d + e*x))/((d + Sqrt[-(a/c)]*e)*(Sqrt[-(a/c)] + x))] + (d + Sqrt[-(a/c)]*e)^2*(Sqrt[-(a/c)]
+ x)*(d + e*x)*Gamma[1 - 2*p]*Gamma[-p]*Hypergeometric2F1[1, -p, -2*p, (2*Sqrt[-(a/c)]*(d + e*x))/((d + Sqrt[-
(a/c)]*e)*(Sqrt[-(a/c)] + x))] + 2*(d + Sqrt[-(a/c)]*e)^2*p*(Sqrt[-(a/c)] + x)*(d + e*x)*Gamma[1 - 2*p]*Gamma[
-p]*Hypergeometric2F1[1, -p, -2*p, (2*Sqrt[-(a/c)]*(d + e*x))/((d + Sqrt[-(a/c)]*e)*(Sqrt[-(a/c)] + x))] + (d
+ Sqrt[-(a/c)]*e)*(Sqrt[-(a/c)] + x)*(d + e*x)^2*Gamma[1 - 2*p]*Gamma[-p]*Hypergeometric2F1[1, -p, -2*p, (2*Sq
rt[-(a/c)]*(d + e*x))/((d + Sqrt[-(a/c)]*e)*(Sqrt[-(a/c)] + x))] - 3*Sqrt[-(a/c)]*(d + Sqrt[-(a/c)]*e)^2*(d +
e*x)*Gamma[1 - p]*Gamma[-2*p]*Hypergeometric2F1[2, 1 - p, 1 - 2*p, (2*Sqrt[-(a/c)]*(d + e*x))/((d + Sqrt[-(a/c
)]*e)*(Sqrt[-(a/c)] + x))] - 4*Sqrt[-(a/c)]*(d + Sqrt[-(a/c)]*e)^2*p*(d + e*x)*Gamma[1 - p]*Gamma[-2*p]*Hyperg
eometric2F1[2, 1 - p, 1 - 2*p, (2*Sqrt[-(a/c)]*(d + e*x))/((d + Sqrt[-(a/c)]*e)*(Sqrt[-(a/c)] + x))] + 4*Sqrt[
-(a/c)]*(d + Sqrt[-(a/c)]*e)*p*(d + e*x)^2*Gamma[1 - p]*Gamma[-2*p]*Hypergeometric2F1[2, 1 - p, 1 - 2*p, (2*Sq
rt[-(a/c)]*(d + e*x))/((d + Sqrt[-(a/c)]*e)*(Sqrt[-(a/c)] + x))] + 3*Sqrt[-(a/c)]*(d + e*x)^3*Gamma[1 - p]*Gam
ma[-2*p]*Hypergeometric2F1[2, 1 - p, 1 - 2*p, (2*Sqrt[-(a/c)]*(d + e*x))/((d + Sqrt[-(a/c)]*e)*(Sqrt[-(a/c)] +
x))] + Sqrt[-(a/c)]*(d + Sqrt[-(a/c)]*e)^2*(d + e*x)*Gamma[1 - p]*Gamma[-2*p]*HypergeometricPFQ[{2, 2, 1 - p}
, {1, 1 - 2*p}, (2*Sqrt[-(a/c)]*(d + e*x))/((d + Sqrt[-(a/c)]*e)*(Sqrt[-(a/c)] + x))] - 2*Sqrt[-(a/c)]*(d + Sq
rt[-(a/c)]*e)*(d + e*x)^2*Gamma[1 - p]*Gamma[-2*p]*HypergeometricPFQ[{2, 2, 1 - p}, {1, 1 - 2*p}, (2*Sqrt[-(a/
c)]*(d + e*x))/((d + Sqrt[-(a/c)]*e)*(Sqrt[-(a/c)] + x))] + Sqrt[-(a/c)]*(d + e*x)^3*Gamma[1 - p]*Gamma[-2*p]*
HypergeometricPFQ[{2, 2, 1 - p}, {1, 1 - 2*p}, (2*Sqrt[-(a/c)]*(d + e*x))/((d + Sqrt[-(a/c)]*e)*(Sqrt[-(a/c)]
+ x))]))/(e*(d + Sqrt[-(a/c)]*e)^3*(3 + 2*p)*((e*(Sqrt[-(a/c)] - x))/(d + Sqrt[-(a/c)]*e))^p*(Sqrt[-(a/c)] + x
)*Gamma[1 - 2*p]*Gamma[-2*p]*Gamma[-p])

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Maple [F]  time = 0.594, size = 0, normalized size = 0. \begin{align*} \int \left ( ex+d \right ) ^{-4-2\,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)^(-4-2*p)*(c*x^2+a)^p,x)

[Out]

int((e*x+d)^(-4-2*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 - 4}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((c*x^2 + a)^p*(e*x + d)^(-2*p - 4), 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)**(-4-2*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 - 4}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((c*x^2 + a)^p*(e*x + d)^(-2*p - 4), x)