3.8.44 \(\int (d+e x)^{-5-2 p} (a+c x^2)^p \, dx\) [744]

Optimal. Leaf size=436 \[ -\frac {c d e (3+p) (d+e x)^{-3-2 p} \left (a+c x^2\right )^{1+p}}{\left (c d^2+a e^2\right )^2 (2+p) (3+2 p)}+\frac {c e \left (a e^2 (3+2 p)-c d^2 \left (9+8 p+2 p^2\right )\right ) (d+e x)^{-2 (1+p)} \left (a+c x^2\right )^{1+p}}{2 \left (c d^2+a e^2\right )^3 (1+p) (2+p) (3+2 p)}-\frac {e (d+e x)^{-2 (2+p)} \left (a+c x^2\right )^{1+p}}{2 \left (c d^2+a e^2\right ) (2+p)}+\frac {c^2 d \left (3 a e^2-c d^2 (3+2 p)\right ) \left (\sqrt {-a}-\sqrt {c} x\right ) \left (-\frac {\left (\sqrt {c} d+\sqrt {-a} e\right ) \left (\sqrt {-a}+\sqrt {c} x\right )}{\left (\sqrt {c} d-\sqrt {-a} e\right ) \left (\sqrt {-a}-\sqrt {c} x\right )}\right )^{-p} (d+e x)^{-1-2 p} \left (a+c x^2\right )^p \, _2F_1\left (-1-2 p,-p;-2 p;\frac {2 \sqrt {-a} \sqrt {c} (d+e x)}{\left (\sqrt {c} d-\sqrt {-a} e\right ) \left (\sqrt {-a}-\sqrt {c} x\right )}\right )}{\left (\sqrt {c} d+\sqrt {-a} e\right ) \left (c d^2+a e^2\right )^3 (1+2 p) (3+2 p)} \]

[Out]

-c*d*e*(3+p)*(e*x+d)^(-3-2*p)*(c*x^2+a)^(1+p)/(a*e^2+c*d^2)^2/(2+p)/(3+2*p)+1/2*c*e*(a*e^2*(3+2*p)-c*d^2*(2*p^
2+8*p+9))*(c*x^2+a)^(1+p)/(a*e^2+c*d^2)^3/(1+p)/(2+p)/(3+2*p)/((e*x+d)^(2+2*p))-1/2*e*(c*x^2+a)^(1+p)/(a*e^2+c
*d^2)/(2+p)/((e*x+d)^(4+2*p))+c^2*d*(3*a*e^2-c*d^2*(3+2*p))*(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)^3/(1+2*p)/(3+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 = 0.30, antiderivative size = 436, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {759, 851, 821, 741} \begin {gather*} \frac {c^2 d \left (\sqrt {-a}-\sqrt {c} x\right ) \left (a+c x^2\right )^p (d+e x)^{-2 p-1} \left (3 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 )^3}+\frac {c e \left (a+c x^2\right )^{p+1} (d+e x)^{-2 (p+1)} \left (a e^2 (2 p+3)-c d^2 \left (2 p^2+8 p+9\right )\right )}{2 (p+1) (p+2) (2 p+3) \left (a e^2+c d^2\right )^3}-\frac {c d e (p+3) \left (a+c x^2\right )^{p+1} (d+e x)^{-2 p-3}}{(p+2) (2 p+3) \left (a e^2+c d^2\right )^2}-\frac {e \left (a+c x^2\right )^{p+1} (d+e x)^{-2 (p+2)}}{2 (p+2) \left (a e^2+c d^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^(-5 - 2*p)*(a + c*x^2)^p,x]

[Out]

-((c*d*e*(3 + p)*(d + e*x)^(-3 - 2*p)*(a + c*x^2)^(1 + p))/((c*d^2 + a*e^2)^2*(2 + p)*(3 + 2*p))) + (c*e*(a*e^
2*(3 + 2*p) - c*d^2*(9 + 8*p + 2*p^2))*(a + c*x^2)^(1 + p))/(2*(c*d^2 + a*e^2)^3*(1 + p)*(2 + p)*(3 + 2*p)*(d
+ e*x)^(2*(1 + p))) - (e*(a + c*x^2)^(1 + p))/(2*(c*d^2 + a*e^2)*(2 + p)*(d + e*x)^(2*(2 + p))) + (c^2*d*(3*a*
e^2 - c*d^2*(3 + 2*p))*(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)^3*(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 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*} \int (d+e x)^{-5-2 p} \left (a+c x^2\right )^p \, dx &=-\frac {e (d+e x)^{-2 (2+p)} \left (a+c x^2\right )^{1+p}}{2 \left (c d^2+a e^2\right ) (2+p)}-\frac {c \int (d+e x)^{-4-2 p} (-2 d (2+p)+2 e x) \left (a+c x^2\right )^p \, dx}{2 \left (c d^2+a e^2\right ) (2+p)}\\ &=-\frac {c d e (3+p) (d+e x)^{-3-2 p} \left (a+c x^2\right )^{1+p}}{\left (c d^2+a e^2\right )^2 (2+p) (3+2 p)}-\frac {e (d+e x)^{-2 (2+p)} \left (a+c x^2\right )^{1+p}}{2 \left (c d^2+a e^2\right ) (2+p)}+\frac {c \int (d+e x)^{-3-2 p} \left (-2 (3+2 p) \left (a e^2-c d^2 (2+p)\right )-2 c d e (3+p) x\right ) \left (a+c x^2\right )^p \, dx}{2 \left (c d^2+a e^2\right )^2 (2+p) (3+2 p)}\\ &=-\frac {c d e (3+p) (d+e x)^{-3-2 p} \left (a+c x^2\right )^{1+p}}{\left (c d^2+a e^2\right )^2 (2+p) (3+2 p)}+\frac {c e \left (a e^2 (3+2 p)-c d^2 \left (9+8 p+2 p^2\right )\right ) (d+e x)^{-2 (1+p)} \left (a+c x^2\right )^{1+p}}{2 \left (c d^2+a e^2\right )^3 (1+p) (2+p) (3+2 p)}-\frac {e (d+e x)^{-2 (2+p)} \left (a+c x^2\right )^{1+p}}{2 \left (c d^2+a e^2\right ) (2+p)}-\frac {\left (c^2 d \left (3 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 )^3 (3+2 p)}\\ &=-\frac {c d e (3+p) (d+e x)^{-3-2 p} \left (a+c x^2\right )^{1+p}}{\left (c d^2+a e^2\right )^2 (2+p) (3+2 p)}+\frac {c e \left (a e^2 (3+2 p)-c d^2 \left (9+8 p+2 p^2\right )\right ) (d+e x)^{-2 (1+p)} \left (a+c x^2\right )^{1+p}}{2 \left (c d^2+a e^2\right )^3 (1+p) (2+p) (3+2 p)}-\frac {e (d+e x)^{-2 (2+p)} \left (a+c x^2\right )^{1+p}}{2 \left (c d^2+a e^2\right ) (2+p)}+\frac {c^2 d \left (3 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 )^3 (1+2 p) (3+2 p)}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(2500\) vs. \(2(436)=872\).
time = 49.49, size = 2500, normalized size = 5.73 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[(d + e*x)^(-5 - 2*p)*(a + c*x^2)^p,x]

[Out]

-1/4*((a + c*x^2)^p*(1 - (d + e*x)/(d + Sqrt[-(a/c)]*e))^(1 + p)*(6*(d + Sqrt[-(a/c)]*e)^4*p*(Sqrt[-(a/c)] + x
)*Gamma[-p]*Gamma[-2*(1 + p)]*Hypergeometric2F1[1, -p, -3 - 2*p, (2*Sqrt[-(a/c)]*(d + e*x))/((d + Sqrt[-(a/c)]
*e)*(Sqrt[-(a/c)] + x))] + 22*(d + Sqrt[-(a/c)]*e)^4*p^2*(Sqrt[-(a/c)] + x)*Gamma[-p]*Gamma[-2*(1 + p)]*Hyperg
eometric2F1[1, -p, -3 - 2*p, (2*Sqrt[-(a/c)]*(d + e*x))/((d + Sqrt[-(a/c)]*e)*(Sqrt[-(a/c)] + x))] + 24*(d + S
qrt[-(a/c)]*e)^4*p^3*(Sqrt[-(a/c)] + x)*Gamma[-p]*Gamma[-2*(1 + p)]*Hypergeometric2F1[1, -p, -3 - 2*p, (2*Sqrt
[-(a/c)]*(d + e*x))/((d + Sqrt[-(a/c)]*e)*(Sqrt[-(a/c)] + x))] + 8*(d + Sqrt[-(a/c)]*e)^4*p^4*(Sqrt[-(a/c)] +
x)*Gamma[-p]*Gamma[-2*(1 + p)]*Hypergeometric2F1[1, -p, -3 - 2*p, (2*Sqrt[-(a/c)]*(d + e*x))/((d + Sqrt[-(a/c)
]*e)*(Sqrt[-(a/c)] + x))] + 6*(d + Sqrt[-(a/c)]*e)^2*p*(Sqrt[-(a/c)] + x)*(d + e*x)^2*Gamma[-p]*Gamma[-2*(1 +
p)]*Hypergeometric2F1[1, -p, -1 - 2*p, (2*Sqrt[-(a/c)]*(d + e*x))/((d + Sqrt[-(a/c)]*e)*(Sqrt[-(a/c)] + x))] +
12*(d + Sqrt[-(a/c)]*e)^2*p^2*(Sqrt[-(a/c)] + x)*(d + e*x)^2*Gamma[-p]*Gamma[-2*(1 + p)]*Hypergeometric2F1[1,
-p, -1 - 2*p, (2*Sqrt[-(a/c)]*(d + e*x))/((d + Sqrt[-(a/c)]*e)*(Sqrt[-(a/c)] + x))] + 6*(d + Sqrt[-(a/c)]*e)*
p*(Sqrt[-(a/c)] + x)*(d + e*x)^3*Gamma[-p]*Gamma[-2*(1 + p)]*Hypergeometric2F1[1, -p, -2*p, (2*Sqrt[-(a/c)]*(d
+ e*x))/((d + Sqrt[-(a/c)]*e)*(Sqrt[-(a/c)] + x))] + 6*(d + Sqrt[-(a/c)]*e)^3*p*(Sqrt[-(a/c)] + x)*(d + e*x)*
Gamma[-p]*Gamma[-2*(1 + p)]*Hypergeometric2F1[1, -p, -2*(1 + p), (2*Sqrt[-(a/c)]*(d + e*x))/((d + Sqrt[-(a/c)]
*e)*(Sqrt[-(a/c)] + x))] + 18*(d + Sqrt[-(a/c)]*e)^3*p^2*(Sqrt[-(a/c)] + x)*(d + e*x)*Gamma[-p]*Gamma[-2*(1 +
p)]*Hypergeometric2F1[1, -p, -2*(1 + p), (2*Sqrt[-(a/c)]*(d + e*x))/((d + Sqrt[-(a/c)]*e)*(Sqrt[-(a/c)] + x))]
+ 12*(d + Sqrt[-(a/c)]*e)^3*p^3*(Sqrt[-(a/c)] + x)*(d + e*x)*Gamma[-p]*Gamma[-2*(1 + p)]*Hypergeometric2F1[1,
-p, -2*(1 + p), (2*Sqrt[-(a/c)]*(d + e*x))/((d + Sqrt[-(a/c)]*e)*(Sqrt[-(a/c)] + x))] + 18*Sqrt[-(a/c)]*(d +
Sqrt[-(a/c)]*e)^2*p*(d + e*x)^2*Gamma[-3 - 2*p]*Gamma[1 - p]*Hypergeometric2F1[2, 1 - p, -1 - 2*p, (2*Sqrt[-(a
/c)]*(d + e*x))/((d + Sqrt[-(a/c)]*e)*(Sqrt[-(a/c)] + x))] + 48*Sqrt[-(a/c)]*(d + Sqrt[-(a/c)]*e)^2*p^2*(d + e
*x)^2*Gamma[-3 - 2*p]*Gamma[1 - p]*Hypergeometric2F1[2, 1 - p, -1 - 2*p, (2*Sqrt[-(a/c)]*(d + e*x))/((d + Sqrt
[-(a/c)]*e)*(Sqrt[-(a/c)] + x))] + 24*Sqrt[-(a/c)]*(d + Sqrt[-(a/c)]*e)^2*p^3*(d + e*x)^2*Gamma[-3 - 2*p]*Gamm
a[1 - p]*Hypergeometric2F1[2, 1 - p, -1 - 2*p, (2*Sqrt[-(a/c)]*(d + e*x))/((d + Sqrt[-(a/c)]*e)*(Sqrt[-(a/c)]
+ x))] + 33*Sqrt[-(a/c)]*(d + e*x)^4*Gamma[-3 - 2*p]*Gamma[1 - p]*Hypergeometric2F1[2, 1 - p, 1 - 2*p, (2*Sqrt
[-(a/c)]*(d + e*x))/((d + Sqrt[-(a/c)]*e)*(Sqrt[-(a/c)] + x))] + 22*Sqrt[-(a/c)]*p*(d + e*x)^4*Gamma[-3 - 2*p]
*Gamma[1 - p]*Hypergeometric2F1[2, 1 - p, 1 - 2*p, (2*Sqrt[-(a/c)]*(d + e*x))/((d + Sqrt[-(a/c)]*e)*(Sqrt[-(a/
c)] + x))] + 54*Sqrt[-(a/c)]*(d + Sqrt[-(a/c)]*e)*p*(d + e*x)^3*Gamma[-3 - 2*p]*Gamma[1 - p]*Hypergeometric2F1
[2, 1 - p, -2*p, (2*Sqrt[-(a/c)]*(d + e*x))/((d + Sqrt[-(a/c)]*e)*(Sqrt[-(a/c)] + x))] + 36*Sqrt[-(a/c)]*(d +
Sqrt[-(a/c)]*e)*p^2*(d + e*x)^3*Gamma[-3 - 2*p]*Gamma[1 - p]*Hypergeometric2F1[2, 1 - p, -2*p, (2*Sqrt[-(a/c)]
*(d + e*x))/((d + Sqrt[-(a/c)]*e)*(Sqrt[-(a/c)] + x))] + 18*Sqrt[-(a/c)]*(d + e*x)^4*Gamma[-3 - 2*p]*Gamma[1 -
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))] + 12*Sqrt[-(a/c)]*p*(d + e*x)^4*Gamma[-3 - 2*p]*Gamma[1 - 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))] + 18*Sqrt[-(a/c)]*(d + Sqrt[-(a
/c)]*e)*p*(d + e*x)^3*Gamma[-3 - 2*p]*Gamma[1 - p]*HypergeometricPFQ[{2, 2, 1 - p}, {1, -2*p}, (2*Sqrt[-(a/c)]
*(d + e*x))/((d + Sqrt[-(a/c)]*e)*(Sqrt[-(a/c)] + x))] + 12*Sqrt[-(a/c)]*(d + Sqrt[-(a/c)]*e)*p^2*(d + e*x)^3*
Gamma[-3 - 2*p]*Gamma[1 - p]*HypergeometricPFQ[{2, 2, 1 - p}, {1, -2*p}, (2*Sqrt[-(a/c)]*(d + e*x))/((d + Sqrt
[-(a/c)]*e)*(Sqrt[-(a/c)] + x))] + 3*Sqrt[-(a/c)]*(d + e*x)^4*Gamma[-3 - 2*p]*Gamma[1 - p]*HypergeometricPFQ[{
2, 2, 2, 1 - p}, {1, 1, 1 - 2*p}, (2*Sqrt[-(a/c)]*(d + e*x))/((d + Sqrt[-(a/c)]*e)*(Sqrt[-(a/c)] + x))] + 2*Sq
rt[-(a/c)]*p*(d + e*x)^4*Gamma[-3 - 2*p]*Gamma[1 - p]*HypergeometricPFQ[{2, 2, 2, 1 - p}, {1, 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)^4*p*(1 + p)*(2 +
p)*(1 + 2*p)*(3 + 2*p)*((e*(Sqrt[-(a/c)] - x))/(d + Sqrt[-(a/c)]*e))^p*(Sqrt[-(a/c)] + x)*(d + e*x)^(2*(2 + p)
)*Gamma[-p]*Gamma[-2*(1 + p)])

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Maple [F]
time = 0.14, size = 0, normalized size = 0.00 \[\int \left (e x +d \right )^{-5-2 p} \left (c \,x^{2}+a \right )^{p}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(-5-2*p)*(c*x^2+a)^p,x)

[Out]

int((e*x+d)^(-5-2*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)^(-5-2*p)*(c*x^2+a)^p,x, algorithm="maxima")

[Out]

integrate((c*x^2 + a)^p*(x*e + d)^(-2*p - 5), 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)^(-5-2*p)*(c*x^2+a)^p,x, algorithm="fricas")

[Out]

integral((c*x^2 + a)^p*(x*e + d)^(-2*p - 5), 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)**(-5-2*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)^(-5-2*p)*(c*x^2+a)^p,x, algorithm="giac")

[Out]

integrate((c*x^2 + a)^p*(x*e + d)^(-2*p - 5), 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+5}} \,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 + 5),x)

[Out]

int((a + c*x^2)^p/(d + e*x)^(2*p + 5), x)

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