3.2560 \(\int (d+e x)^{3/2} \left (a+b x+c x^2\right )^p \, dx\)

Optimal. Leaf size=185 \[ \frac{2 (d+e x)^{5/2} \left (a+b x+c x^2\right )^p \left (1-\frac{2 c (d+e x)}{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}\right )^{-p} \left (1-\frac{2 c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}\right )^{-p} F_1\left (\frac{5}{2};-p,-p;\frac{7}{2};\frac{2 c (d+e x)}{2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e},\frac{2 c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{5 e} \]

[Out]

(2*(d + e*x)^(5/2)*(a + b*x + c*x^2)^p*AppellF1[5/2, -p, -p, 7/2, (2*c*(d + e*x)
)/(2*c*d - (b - Sqrt[b^2 - 4*a*c])*e), (2*c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 -
4*a*c])*e)])/(5*e*(1 - (2*c*(d + e*x))/(2*c*d - (b - Sqrt[b^2 - 4*a*c])*e))^p*(1
- (2*c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e))^p)

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Rubi [A]  time = 0.457288, antiderivative size = 185, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{2 (d+e x)^{5/2} \left (a+b x+c x^2\right )^p \left (1-\frac{2 c (d+e x)}{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}\right )^{-p} \left (1-\frac{2 c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}\right )^{-p} F_1\left (\frac{5}{2};-p,-p;\frac{7}{2};\frac{2 c (d+e x)}{2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e},\frac{2 c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{5 e} \]

Antiderivative was successfully verified.

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

[Out]

(2*(d + e*x)^(5/2)*(a + b*x + c*x^2)^p*AppellF1[5/2, -p, -p, 7/2, (2*c*(d + e*x)
)/(2*c*d - (b - Sqrt[b^2 - 4*a*c])*e), (2*c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 -
4*a*c])*e)])/(5*e*(1 - (2*c*(d + e*x))/(2*c*d - (b - Sqrt[b^2 - 4*a*c])*e))^p*(1
- (2*c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e))^p)

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Rubi in Sympy [A]  time = 36.3734, size = 167, normalized size = 0.9 \[ \frac{2 \left (d + e x\right )^{\frac{5}{2}} \left (\frac{c \left (- 2 d - 2 e x\right )}{2 c d - e \left (b + \sqrt{- 4 a c + b^{2}}\right )} + 1\right )^{- p} \left (\frac{c \left (2 d + 2 e x\right )}{b e - 2 c d - e \sqrt{- 4 a c + b^{2}}} + 1\right )^{- p} \left (a + b x + c x^{2}\right )^{p} \operatorname{appellf_{1}}{\left (\frac{5}{2},- p,- p,\frac{7}{2},\frac{c \left (- 2 d - 2 e x\right )}{b e - 2 c d - e \sqrt{- 4 a c + b^{2}}},\frac{c \left (2 d + 2 e x\right )}{2 c d - e \left (b + \sqrt{- 4 a c + b^{2}}\right )} \right )}}{5 e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**(3/2)*(c*x**2+b*x+a)**p,x)

[Out]

2*(d + e*x)**(5/2)*(c*(-2*d - 2*e*x)/(2*c*d - e*(b + sqrt(-4*a*c + b**2))) + 1)*
*(-p)*(c*(2*d + 2*e*x)/(b*e - 2*c*d - e*sqrt(-4*a*c + b**2)) + 1)**(-p)*(a + b*x
+ c*x**2)**p*appellf1(5/2, -p, -p, 7/2, c*(-2*d - 2*e*x)/(b*e - 2*c*d - e*sqrt(
-4*a*c + b**2)), c*(2*d + 2*e*x)/(2*c*d - e*(b + sqrt(-4*a*c + b**2))))/(5*e)

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Mathematica [B]  time = 12.3593, size = 590, normalized size = 3.19 \[ -\frac{7 (d+e x)^{5/2} \left (e \left (e \sqrt{\frac{b^2-4 a c}{e^2}}-b\right )+2 c d\right ) \left (2 c d-e \left (e \sqrt{\frac{b^2-4 a c}{e^2}}+b\right )\right ) \left (-e \sqrt{\frac{b^2-4 a c}{e^2}}+b+2 c x\right ) \left (e \sqrt{\frac{b^2-4 a c}{e^2}}+b+2 c x\right ) (a+x (b+c x))^{p-1} F_1\left (\frac{5}{2};-p,-p;\frac{7}{2};\frac{2 c (d+e x)}{2 c d-e \left (b+\sqrt{\frac{b^2-4 a c}{e^2}} e\right )},\frac{2 c (d+e x)}{2 c d+e \left (\sqrt{\frac{b^2-4 a c}{e^2}} e-b\right )}\right )}{40 c^2 e \left (p (d+e x) \left (\left (e \left (e \sqrt{\frac{b^2-4 a c}{e^2}}-b\right )+2 c d\right ) F_1\left (\frac{7}{2};1-p,-p;\frac{9}{2};\frac{2 c (d+e x)}{2 c d-e \left (b+\sqrt{\frac{b^2-4 a c}{e^2}} e\right )},\frac{2 c (d+e x)}{2 c d+e \left (\sqrt{\frac{b^2-4 a c}{e^2}} e-b\right )}\right )+\left (2 c d-e \left (e \sqrt{\frac{b^2-4 a c}{e^2}}+b\right )\right ) F_1\left (\frac{7}{2};-p,1-p;\frac{9}{2};\frac{2 c (d+e x)}{2 c d-e \left (b+\sqrt{\frac{b^2-4 a c}{e^2}} e\right )},\frac{2 c (d+e x)}{2 c d+e \left (\sqrt{\frac{b^2-4 a c}{e^2}} e-b\right )}\right )\right )-7 \left (e (a e-b d)+c d^2\right ) F_1\left (\frac{5}{2};-p,-p;\frac{7}{2};\frac{2 c (d+e x)}{2 c d-e \left (b+\sqrt{\frac{b^2-4 a c}{e^2}} e\right )},\frac{2 c (d+e x)}{2 c d+e \left (\sqrt{\frac{b^2-4 a c}{e^2}} e-b\right )}\right )\right )} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(d + e*x)^(3/2)*(a + b*x + c*x^2)^p,x]

[Out]

(-7*(2*c*d + e*(-b + Sqrt[(b^2 - 4*a*c)/e^2]*e))*(2*c*d - e*(b + Sqrt[(b^2 - 4*a
*c)/e^2]*e))*(b - Sqrt[(b^2 - 4*a*c)/e^2]*e + 2*c*x)*(b + Sqrt[(b^2 - 4*a*c)/e^2
]*e + 2*c*x)*(d + e*x)^(5/2)*(a + x*(b + c*x))^(-1 + p)*AppellF1[5/2, -p, -p, 7/
2, (2*c*(d + e*x))/(2*c*d - e*(b + Sqrt[(b^2 - 4*a*c)/e^2]*e)), (2*c*(d + e*x))/
(2*c*d + e*(-b + Sqrt[(b^2 - 4*a*c)/e^2]*e))])/(40*c^2*e*(-7*(c*d^2 + e*(-(b*d)
+ a*e))*AppellF1[5/2, -p, -p, 7/2, (2*c*(d + e*x))/(2*c*d - e*(b + Sqrt[(b^2 - 4
*a*c)/e^2]*e)), (2*c*(d + e*x))/(2*c*d + e*(-b + Sqrt[(b^2 - 4*a*c)/e^2]*e))] +
p*(d + e*x)*((2*c*d + e*(-b + Sqrt[(b^2 - 4*a*c)/e^2]*e))*AppellF1[7/2, 1 - p, -
p, 9/2, (2*c*(d + e*x))/(2*c*d - e*(b + Sqrt[(b^2 - 4*a*c)/e^2]*e)), (2*c*(d + e
*x))/(2*c*d + e*(-b + Sqrt[(b^2 - 4*a*c)/e^2]*e))] + (2*c*d - e*(b + Sqrt[(b^2 -
4*a*c)/e^2]*e))*AppellF1[7/2, -p, 1 - p, 9/2, (2*c*(d + e*x))/(2*c*d - e*(b + S
qrt[(b^2 - 4*a*c)/e^2]*e)), (2*c*(d + e*x))/(2*c*d + e*(-b + Sqrt[(b^2 - 4*a*c)/
e^2]*e))])))

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Maple [F]  time = 0.126, size = 0, normalized size = 0. \[ \int \left ( ex+d \right ) ^{{\frac{3}{2}}} \left ( c{x}^{2}+bx+a \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (e x + d\right )}^{\frac{3}{2}}{\left (c x^{2} + b x + a\right )}^{p}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(3/2)*(c*x^2 + b*x + a)^p,x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (e x + d\right )}^{\frac{3}{2}}{\left (c x^{2} + b x + a\right )}^{p}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(3/2)*(c*x^2 + b*x + a)^p,x, algorithm="fricas")

[Out]

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

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (e x + d\right )}^{\frac{3}{2}}{\left (c x^{2} + b x + a\right )}^{p}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(3/2)*(c*x^2 + b*x + a)^p,x, algorithm="giac")

[Out]

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