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]

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

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Rubi [A]  time = 0.09, antiderivative size = 270, normalized size of antiderivative = 1.00, 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 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]

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]

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*}

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Mathematica [A]  time = 37.08, size = 368, normalized size = 1.36 \[ \frac {2^{-2 p-3} \Gamma \left (-p-\frac {1}{2}\right ) \left (a+c x^2\right )^p (d+e x)^{-2 (p+1)} \left (\frac {e \left (\sqrt {-\frac {a}{c}}-x\right )}{e \sqrt {-\frac {a}{c}}+d}\right )^{-p} \left (1-\frac {d+e x}{e \sqrt {-\frac {a}{c}}+d}\right )^{p+1} \left (\Gamma (1-2 p) \Gamma (-p) \left (e \sqrt {-\frac {a}{c}}+d\right ) \left (e \left (2 p \sqrt {-\frac {a}{c}}+\sqrt {-\frac {a}{c}}+x\right )+2 d (p+1)\right ) \, _2F_1\left (1,-p;-2 p;\frac {2 \sqrt {-\frac {a}{c}} (d+e x)}{\left (d+\sqrt {-\frac {a}{c}} e\right ) \left (x+\sqrt {-\frac {a}{c}}\right )}\right )+\frac {2 e \Gamma (1-p) \Gamma (-2 p) \left (c x \sqrt {-\frac {a}{c}}+a\right ) (d+e x) \, _2F_1\left (2,1-p;1-2 p;\frac {2 \sqrt {-\frac {a}{c}} (d+e x)}{\left (d+\sqrt {-\frac {a}{c}} e\right ) \left (x+\sqrt {-\frac {a}{c}}\right )}\right )}{c \left (\sqrt {-\frac {a}{c}}+x\right )}\right )}{\sqrt {\pi } e (p+1) \Gamma (1-2 p) \Gamma (-2 p) \left (e \sqrt {-\frac {a}{c}}+d\right )^2} \]

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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fricas [F]  time = 1.09, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (c x^{2} + a\right )}^{p} {\left (e x + d\right )}^{-2 \, p - 3}, x\right ) \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (c x^{2} + a\right )}^{p} {\left (e x + d\right )}^{-2 \, p - 3}\,{d x} \]

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)

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maple [F]  time = 0.68, size = 0, normalized size = 0.00 \[ \int \left (c \,x^{2}+a \right )^{p} \left (e x +d \right )^{-2 p -3}\, dx \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (c x^{2} + a\right )}^{p} {\left (e x + d\right )}^{-2 \, p - 3}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (c\,x^2+a\right )}^p}{{\left (d+e\,x\right )}^{2\,p+3}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + c*x^2)^p/(d + e*x)^(2*p + 3),x)

[Out]

int((a + c*x^2)^p/(d + e*x)^(2*p + 3), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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