3.720 \(\int (d+e x)^m \left (a+c x^2\right )^p \, dx\)

Optimal. Leaf size=152 \[ \frac{\left (a+c x^2\right )^p (d+e x)^{m+1} \left (1-\frac{d+e x}{d-\frac{\sqrt{-a} e}{\sqrt{c}}}\right )^{-p} \left (1-\frac{d+e x}{\frac{\sqrt{-a} e}{\sqrt{c}}+d}\right )^{-p} F_1\left (m+1;-p,-p;m+2;\frac{d+e x}{d-\frac{\sqrt{-a} e}{\sqrt{c}}},\frac{d+e x}{d+\frac{\sqrt{-a} e}{\sqrt{c}}}\right )}{e (m+1)} \]

[Out]

((d + e*x)^(1 + m)*(a + c*x^2)^p*AppellF1[1 + m, -p, -p, 2 + m, (d + e*x)/(d - (
Sqrt[-a]*e)/Sqrt[c]), (d + e*x)/(d + (Sqrt[-a]*e)/Sqrt[c])])/(e*(1 + m)*(1 - (d
+ e*x)/(d - (Sqrt[-a]*e)/Sqrt[c]))^p*(1 - (d + e*x)/(d + (Sqrt[-a]*e)/Sqrt[c]))^
p)

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Rubi [A]  time = 0.221301, antiderivative size = 152, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \frac{\left (a+c x^2\right )^p (d+e x)^{m+1} \left (1-\frac{d+e x}{d-\frac{\sqrt{-a} e}{\sqrt{c}}}\right )^{-p} \left (1-\frac{d+e x}{\frac{\sqrt{-a} e}{\sqrt{c}}+d}\right )^{-p} F_1\left (m+1;-p,-p;m+2;\frac{d+e x}{d-\frac{\sqrt{-a} e}{\sqrt{c}}},\frac{d+e x}{d+\frac{\sqrt{-a} e}{\sqrt{c}}}\right )}{e (m+1)} \]

Antiderivative was successfully verified.

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

[Out]

((d + e*x)^(1 + m)*(a + c*x^2)^p*AppellF1[1 + m, -p, -p, 2 + m, (d + e*x)/(d - (
Sqrt[-a]*e)/Sqrt[c]), (d + e*x)/(d + (Sqrt[-a]*e)/Sqrt[c])])/(e*(1 + m)*(1 - (d
+ e*x)/(d - (Sqrt[-a]*e)/Sqrt[c]))^p*(1 - (d + e*x)/(d + (Sqrt[-a]*e)/Sqrt[c]))^
p)

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Rubi in Sympy [A]  time = 31.1283, size = 139, normalized size = 0.91 \[ \frac{\left (a + c x^{2}\right )^{p} \left (d + e x\right )^{m + 1} \left (\frac{\sqrt{c} \left (- d - e x\right )}{\sqrt{c} d - e \sqrt{- a}} + 1\right )^{- p} \left (\frac{\sqrt{c} \left (- d - e x\right )}{\sqrt{c} d + e \sqrt{- a}} + 1\right )^{- p} \operatorname{appellf_{1}}{\left (m + 1,- p,- p,m + 2,\frac{\sqrt{c} \left (d + e x\right )}{\sqrt{c} d - e \sqrt{- a}},\frac{\sqrt{c} \left (d + e x\right )}{\sqrt{c} d + e \sqrt{- a}} \right )}}{e \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**m*(c*x**2+a)**p,x)

[Out]

(a + c*x**2)**p*(d + e*x)**(m + 1)*(sqrt(c)*(-d - e*x)/(sqrt(c)*d - e*sqrt(-a))
+ 1)**(-p)*(sqrt(c)*(-d - e*x)/(sqrt(c)*d + e*sqrt(-a)) + 1)**(-p)*appellf1(m +
1, -p, -p, m + 2, sqrt(c)*(d + e*x)/(sqrt(c)*d - e*sqrt(-a)), sqrt(c)*(d + e*x)/
(sqrt(c)*d + e*sqrt(-a)))/(e*(m + 1))

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Mathematica [A]  time = 0.24176, size = 157, normalized size = 1.03 \[ \frac{\left (a+c x^2\right )^p (d+e x)^{m+1} \left (\frac{e \left (\sqrt{-\frac{a}{c}}-x\right )}{e \sqrt{-\frac{a}{c}}+d}\right )^{-p} \left (\frac{e \left (\sqrt{-\frac{a}{c}}+x\right )}{e \sqrt{-\frac{a}{c}}-d}\right )^{-p} F_1\left (m+1;-p,-p;m+2;\frac{d+e x}{d-\sqrt{-\frac{a}{c}} e},\frac{d+e x}{d+\sqrt{-\frac{a}{c}} e}\right )}{e (m+1)} \]

Warning: Unable to verify antiderivative.

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

[Out]

((d + e*x)^(1 + m)*(a + c*x^2)^p*AppellF1[1 + m, -p, -p, 2 + m, (d + e*x)/(d - S
qrt[-(a/c)]*e), (d + e*x)/(d + Sqrt[-(a/c)]*e)])/(e*(1 + m)*((e*(Sqrt[-(a/c)] -
x))/(d + Sqrt[-(a/c)]*e))^p*((e*(Sqrt[-(a/c)] + x))/(-d + Sqrt[-(a/c)]*e))^p)

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Maple [F]  time = 0.122, size = 0, normalized size = 0. \[ \int \left ( ex+d \right ) ^{m} \left ( c{x}^{2}+a \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^m*(c*x^2+a)^p,x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (c x^{2} + a\right )}^{p}{\left (e x + d\right )}^{m}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^p*(e*x + d)^m,x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (c x^{2} + a\right )}^{p}{\left (e x + d\right )}^{m}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^p*(e*x + d)^m,x, algorithm="fricas")

[Out]

integral((c*x^2 + a)^p*(e*x + d)^m, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**m*(c*x**2+a)**p,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (c x^{2} + a\right )}^{p}{\left (e x + d\right )}^{m}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^p*(e*x + d)^m,x, algorithm="giac")

[Out]

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