3.2080 \(\int (d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^p \, dx\)

Optimal. Leaf size=100 \[ -\frac{(d+e x)^{m+1} (a e+c d x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^p \, _2F_1\left (1,m+2 p+2;m+p+2;\frac{c d (d+e x)}{c d^2-a e^2}\right )}{(m+p+1) \left (c d^2-a e^2\right )} \]

[Out]

-(((a*e + c*d*x)*(d + e*x)^(1 + m)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^p*Hyp
ergeometric2F1[1, 2 + m + 2*p, 2 + m + p, (c*d*(d + e*x))/(c*d^2 - a*e^2)])/((c*
d^2 - a*e^2)*(1 + m + p)))

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Rubi [A]  time = 0.256739, antiderivative size = 123, normalized size of antiderivative = 1.23, number of steps used = 4, number of rules used = 4, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.114 \[ \frac{(d+e x)^m (a e+c d x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^p \left (\frac{c d (d+e x)}{c d^2-a e^2}\right )^{-m-p} \, _2F_1\left (-m-p,p+1;p+2;-\frac{e (a e+c d x)}{c d^2-a e^2}\right )}{c d (p+1)} \]

Antiderivative was successfully verified.

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

[Out]

((a*e + c*d*x)*(d + e*x)^m*((c*d*(d + e*x))/(c*d^2 - a*e^2))^(-m - p)*(a*d*e + (
c*d^2 + a*e^2)*x + c*d*e*x^2)^p*Hypergeometric2F1[-m - p, 1 + p, 2 + p, -((e*(a*
e + c*d*x))/(c*d^2 - a*e^2))])/(c*d*(1 + p))

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Rubi in Sympy [A]  time = 61.7356, size = 124, normalized size = 1.24 \[ \frac{\left (\frac{c d \left (- d - e x\right )}{a e^{2} - c d^{2}}\right )^{- m - p} \left (d + e x\right )^{- p} \left (d + e x\right )^{m + p} \left (a e + c d x\right )^{- p} \left (a e + c d x\right )^{p + 1} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - m - p, p + 1 \\ p + 2 \end{matrix}\middle |{\frac{e \left (a e + c d x\right )}{a e^{2} - c d^{2}}} \right )}}{c d \left (p + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.181414, size = 97, normalized size = 0.97 \[ \frac{(d+e x)^{m+1} \left (\frac{e (a e+c d x)}{a e^2-c d^2}\right )^{-p} ((d+e x) (a e+c d x))^p \, _2F_1\left (-p,m+p+1;m+p+2;\frac{c d (d+e x)}{c d^2-a e^2}\right )}{e (m+p+1)} \]

Antiderivative was successfully verified.

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

[Out]

((d + e*x)^(1 + m)*((a*e + c*d*x)*(d + e*x))^p*Hypergeometric2F1[-p, 1 + m + p,
2 + m + p, (c*d*(d + e*x))/(c*d^2 - a*e^2)])/(e*(1 + m + p)*((e*(a*e + c*d*x))/(
-(c*d^2) + a*e^2))^p)

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Maple [F]  time = 0.264, size = 0, normalized size = 0. \[ \int \left ( ex+d \right ) ^{m} \left ( aed+ \left ( a{e}^{2}+c{d}^{2} \right ) x+cde{x}^{2} \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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