3.2563 \(\int \frac{\left (a+b x+c x^2\right )^p}{(d+e x)^{3/2}} \, dx\)

Optimal. Leaf size=183 \[ -\frac{2 \left (a+b x+c x^2\right )^p \left (1-\frac{2 c (d+e x)}{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}\right )^{-p} \left (1-\frac{2 c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}\right )^{-p} F_1\left (-\frac{1}{2};-p,-p;\frac{1}{2};\frac{2 c (d+e x)}{2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e},\frac{2 c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{e \sqrt{d+e x}} \]

[Out]

(-2*(a + b*x + c*x^2)^p*AppellF1[-1/2, -p, -p, 1/2, (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)])/(
e*Sqrt[d + e*x]*(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 = 0.393924, antiderivative size = 183, 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 \[ -\frac{2 \left (a+b x+c x^2\right )^p \left (1-\frac{2 c (d+e x)}{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}\right )^{-p} \left (1-\frac{2 c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}\right )^{-p} F_1\left (-\frac{1}{2};-p,-p;\frac{1}{2};\frac{2 c (d+e x)}{2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e},\frac{2 c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{e \sqrt{d+e x}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(a + b*x + c*x^2)^p*AppellF1[-1/2, -p, -p, 1/2, (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)])/(
e*Sqrt[d + e*x]*(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 = 36.5931, size = 168, normalized size = 0.92 \[ - \frac{2 \left (\frac{c \left (- 2 d - 2 e x\right )}{2 c d - e \left (b + \sqrt{- 4 a c + b^{2}}\right )} + 1\right )^{- p} \left (\frac{c \left (2 d + 2 e x\right )}{b e - 2 c d - e \sqrt{- 4 a c + b^{2}}} + 1\right )^{- p} \left (a + b x + c x^{2}\right )^{p} \operatorname{appellf_{1}}{\left (- \frac{1}{2},- p,- p,\frac{1}{2},\frac{c \left (- 2 d - 2 e x\right )}{b e - 2 c d - e \sqrt{- 4 a c + b^{2}}},\frac{c \left (2 d + 2 e x\right )}{2 c d - e \left (b + \sqrt{- 4 a c + b^{2}}\right )} \right )}}{e \sqrt{d + e x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-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*ap
pellf1(-1/2, -p, -p, 1/2, 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))))/(e*sqrt(d + e*x))

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Mathematica [A]  time = 0.746491, size = 209, normalized size = 1.14 \[ -\frac{2^{1-2 p} (a+x (b+c x))^p \left (\frac{e \left (\sqrt{b^2-4 a c}-b-2 c x\right )}{4 e \left (\sqrt{b^2-4 a c}-b\right )+8 c d}\right )^{-p} \left (\frac{e \left (\sqrt{b^2-4 a c}+b+2 c x\right )}{e \left (\sqrt{b^2-4 a c}+b\right )-2 c d}\right )^{-p} F_1\left (-\frac{1}{2};-p,-p;\frac{1}{2};\frac{2 c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e},\frac{2 c (d+e x)}{2 c d+\left (\sqrt{b^2-4 a c}-b\right ) e}\right )}{e \sqrt{d+e x}} \]

Warning: Unable to verify antiderivative.

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

[Out]

-((2^(1 - 2*p)*(a + x*(b + c*x))^p*AppellF1[-1/2, -p, -p, 1/2, (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)])/(e*((e*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x))/(8*c*d + 4*(-b + Sqrt[b^2 -
4*a*c])*e))^p*((e*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/(-2*c*d + (b + Sqrt[b^2 - 4*a
*c])*e))^p*Sqrt[d + e*x]))

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Maple [F]  time = 0.131, size = 0, normalized size = 0. \[ \int{ \left ( c{x}^{2}+bx+a \right ) ^{p} \left ( ex+d \right ) ^{-{\frac{3}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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