3.8.42 \(\int (d+e x)^{-3-2 p} (a+c x^2)^p \, dx\) [742]

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

[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 = 0.06, 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 = {745, 741} \begin {gather*} -\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 )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*(e*(a + c*x^2)^(1 + p))/((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 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 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*(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 = 25.51, size = 368, normalized size = 1.36 \begin {gather*} \frac {2^{-3-2 p} \left (\frac {e \left (\sqrt {-\frac {a}{c}}-x\right )}{d+\sqrt {-\frac {a}{c}} e}\right )^{-p} (d+e x)^{-2 (1+p)} \left (a+c x^2\right )^p \left (1-\frac {d+e x}{d+\sqrt {-\frac {a}{c}} e}\right )^{1+p} \Gamma \left (-\frac {1}{2}-p\right ) \left (\left (d+\sqrt {-\frac {a}{c}} e\right ) \left (2 d (1+p)+e \left (\sqrt {-\frac {a}{c}}+2 \sqrt {-\frac {a}{c}} p+x\right )\right ) \Gamma (1-2 p) \Gamma (-p) \, _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 (\sqrt {-\frac {a}{c}}+x\right )}\right )+\frac {2 e \left (a+\sqrt {-\frac {a}{c}} c x\right ) (d+e x) \Gamma (1-p) \Gamma (-2 p) \, _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 (\sqrt {-\frac {a}{c}}+x\right )}\right )}{c \left (\sqrt {-\frac {a}{c}}+x\right )}\right )}{e \left (d+\sqrt {-\frac {a}{c}} e\right )^2 (1+p) \sqrt {\pi } \Gamma (1-2 p) \Gamma (-2 p)} \end {gather*}

Antiderivative was successfully verified.

[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 = 0.15, size = 0, normalized size = 0.00 \[\int \left (e x +d \right )^{-3-2 p} \left (c \,x^{2}+a \right )^{p}\, dx\]

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 \begin {gather*} \text {Failed to integrate} \end {gather*}

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*(x*e + d)^(-2*p - 3), x)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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*(x*e + d)^(-2*p - 3), x)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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*(x*e + d)^(-2*p - 3), x)

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

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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