### 3.742 $$\int (d+e x)^{-3-2 p} (a+c x^2)^p \, dx$$

Optimal. Leaf size=270 $-\frac{c d \left (\sqrt{-a}-\sqrt{c} x\right ) \left (a+c x^2\right )^p (d+e x)^{-2 p-1} \left (-\frac{\left (\sqrt{-a}+\sqrt{c} x\right ) \left (\sqrt{-a} e+\sqrt{c} d\right )}{\left (\sqrt{-a}-\sqrt{c} x\right ) \left (\sqrt{c} d-\sqrt{-a} e\right )}\right )^{-p} \, _2F_1\left (-2 p-1,-p;-2 p;\frac{2 \sqrt{-a} \sqrt{c} (d+e x)}{\left (\sqrt{c} d-\sqrt{-a} e\right ) \left (\sqrt{-a}-\sqrt{c} x\right )}\right )}{(2 p+1) \left (\sqrt{-a} e+\sqrt{c} d\right ) \left (a e^2+c d^2\right )}-\frac{e \left (a+c x^2\right )^{p+1} (d+e x)^{-2 (p+1)}}{2 (p+1) \left (a e^2+c d^2\right )}$

[Out]

-(e*(a + c*x^2)^(1 + p))/(2*(c*d^2 + a*e^2)*(1 + p)*(d + e*x)^(2*(1 + p))) - (c*d*(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)*(1 + 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 = 0.0857186, antiderivative size = 270, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.095, Rules used = {731, 727} $-\frac{c d \left (\sqrt{-a}-\sqrt{c} x\right ) \left (a+c x^2\right )^p (d+e x)^{-2 p-1} \left (-\frac{\left (\sqrt{-a}+\sqrt{c} x\right ) \left (\sqrt{-a} e+\sqrt{c} d\right )}{\left (\sqrt{-a}-\sqrt{c} x\right ) \left (\sqrt{c} d-\sqrt{-a} e\right )}\right )^{-p} \, _2F_1\left (-2 p-1,-p;-2 p;\frac{2 \sqrt{-a} \sqrt{c} (d+e x)}{\left (\sqrt{c} d-\sqrt{-a} e\right ) \left (\sqrt{-a}-\sqrt{c} x\right )}\right )}{(2 p+1) \left (\sqrt{-a} e+\sqrt{c} d\right ) \left (a e^2+c d^2\right )}-\frac{e \left (a+c x^2\right )^{p+1} (d+e x)^{-2 (p+1)}}{2 (p+1) \left (a e^2+c d^2\right )}$

Antiderivative was successfully veriﬁed.

[In]

Int[(d + e*x)^(-3 - 2*p)*(a + c*x^2)^p,x]

[Out]

-(e*(a + c*x^2)^(1 + p))/(2*(c*d^2 + a*e^2)*(1 + p)*(d + e*x)^(2*(1 + p))) - (c*d*(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)*(1 + 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 731

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*d)/(c*d^2 + a*e^2), Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x]
/; FreeQ[{a, c, d, e, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] && EqQ[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)^{-3-2 p} \left (a+c x^2\right )^p \, dx &=-\frac{e (d+e x)^{-2 (1+p)} \left (a+c x^2\right )^{1+p}}{2 \left (c d^2+a e^2\right ) (1+p)}+\frac{(c d) \int (d+e x)^{-2-2 p} \left (a+c x^2\right )^p \, dx}{c d^2+a e^2}\\ &=-\frac{e (d+e x)^{-2 (1+p)} \left (a+c x^2\right )^{1+p}}{2 \left (c d^2+a e^2\right ) (1+p)}-\frac{c d \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 ) (1+2 p)}\\ \end{align*}

Mathematica [A]  time = 49.557, size = 368, normalized size = 1.36 $\frac{2^{-2 p-3} \text{Gamma}\left (-p-\frac{1}{2}\right ) \left (a+c x^2\right )^p (d+e x)^{-2 (p+1)} \left (\frac{e \left (\sqrt{-\frac{a}{c}}-x\right )}{e \sqrt{-\frac{a}{c}}+d}\right )^{-p} \left (1-\frac{d+e x}{e \sqrt{-\frac{a}{c}}+d}\right )^{p+1} \left (\text{Gamma}(1-2 p) \text{Gamma}(-p) \left (e \sqrt{-\frac{a}{c}}+d\right ) \left (e \left (2 p \sqrt{-\frac{a}{c}}+\sqrt{-\frac{a}{c}}+x\right )+2 d (p+1)\right ) \, _2F_1\left (1,-p;-2 p;\frac{2 \sqrt{-\frac{a}{c}} (d+e x)}{\left (d+\sqrt{-\frac{a}{c}} e\right ) \left (x+\sqrt{-\frac{a}{c}}\right )}\right )+\frac{2 e \text{Gamma}(1-p) \text{Gamma}(-2 p) \left (c x \sqrt{-\frac{a}{c}}+a\right ) (d+e x) \, _2F_1\left (2,1-p;1-2 p;\frac{2 \sqrt{-\frac{a}{c}} (d+e x)}{\left (d+\sqrt{-\frac{a}{c}} e\right ) \left (x+\sqrt{-\frac{a}{c}}\right )}\right )}{c \left (\sqrt{-\frac{a}{c}}+x\right )}\right )}{\sqrt{\pi } e (p+1) \text{Gamma}(1-2 p) \text{Gamma}(-2 p) \left (e \sqrt{-\frac{a}{c}}+d\right )^2}$

Warning: Unable to verify antiderivative.

[In]

Integrate[(d + e*x)^(-3 - 2*p)*(a + c*x^2)^p,x]

[Out]

(2^(-3 - 2*p)*(a + c*x^2)^p*(1 - (d + e*x)/(d + Sqrt[-(a/c)]*e))^(1 + p)*Gamma[-1/2 - p]*((d + Sqrt[-(a/c)]*e)
*(2*d*(1 + p) + e*(Sqrt[-(a/c)] + 2*Sqrt[-(a/c)]*p + 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*e*(a + Sqrt[-(a/c)]*c*x)*(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))])/(c*(Sqrt[-(a/c)] + x))))/(e*(d + Sqrt[-(a/c)]*e)^2*(1 + p)*Sqrt[Pi]*((e*(Sqrt[-(a/c)
] - x))/(d + Sqrt[-(a/c)]*e))^p*(d + e*x)^(2*(1 + p))*Gamma[1 - 2*p]*Gamma[-2*p])

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Maple [F]  time = 0.614, size = 0, normalized size = 0. \begin{align*} \int \left ( ex+d \right ) ^{-3-2\,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)^(-3-2*p)*(c*x^2+a)^p,x)

[Out]

int((e*x+d)^(-3-2*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 - 3}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(-3-2*p)*(c*x^2+a)^p,x, algorithm="maxima")

[Out]

integrate((c*x^2 + a)^p*(e*x + d)^(-2*p - 3), 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 - 3}, x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(-3-2*p)*(c*x^2+a)^p,x, algorithm="fricas")

[Out]

integral((c*x^2 + a)^p*(e*x + d)^(-2*p - 3), 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)**(-3-2*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 - 3}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(-3-2*p)*(c*x^2+a)^p,x, algorithm="giac")

[Out]

integrate((c*x^2 + a)^p*(e*x + d)^(-2*p - 3), x)