3.2549 \(\int \frac{(d+e x)^m}{\left (a+b x+c x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=189 \[ \frac{(d+e x)^{m+1} \left (1-\frac{2 c (d+e x)}{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}\right )^{3/2} \left (1-\frac{2 c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}\right )^{3/2} F_1\left (m+1;\frac{3}{2},\frac{3}{2};m+2;\frac{2 c (d+e x)}{2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e},\frac{2 c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{e (m+1) \left (a+b x+c x^2\right )^{3/2}} \]

[Out]

((d + e*x)^(1 + m)*(1 - (2*c*(d + e*x))/(2*c*d - (b - Sqrt[b^2 - 4*a*c])*e))^(3/
2)*(1 - (2*c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e))^(3/2)*AppellF1[1 +
m, 3/2, 3/2, 2 + m, (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*(1 + m)*(a + b*x + c*x^2)^(3/2
))

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Rubi [A]  time = 0.661462, antiderivative size = 189, 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 \[ \frac{(d+e x)^{m+1} \left (1-\frac{2 c (d+e x)}{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}\right )^{3/2} \left (1-\frac{2 c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}\right )^{3/2} F_1\left (m+1;\frac{3}{2},\frac{3}{2};m+2;\frac{2 c (d+e x)}{2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e},\frac{2 c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{e (m+1) \left (a+b x+c x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

((d + e*x)^(1 + m)*(1 - (2*c*(d + e*x))/(2*c*d - (b - Sqrt[b^2 - 4*a*c])*e))^(3/
2)*(1 - (2*c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e))^(3/2)*AppellF1[1 +
m, 3/2, 3/2, 2 + m, (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*(1 + m)*(a + b*x + c*x^2)^(3/2
))

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Rubi in Sympy [A]  time = 36.7331, size = 172, normalized size = 0.91 \[ \frac{\left (d + e x\right )^{m + 1} \left (\frac{c \left (- 2 d - 2 e x\right )}{2 c d - e \left (b + \sqrt{- 4 a c + b^{2}}\right )} + 1\right )^{\frac{3}{2}} \left (\frac{c \left (2 d + 2 e x\right )}{b e - 2 c d - e \sqrt{- 4 a c + b^{2}}} + 1\right )^{\frac{3}{2}} \operatorname{appellf_{1}}{\left (m + 1,\frac{3}{2},\frac{3}{2},m + 2,\frac{c \left (- 2 d - 2 e x\right )}{b e - 2 c d - e \sqrt{- 4 a c + b^{2}}},\frac{c \left (2 d + 2 e x\right )}{2 c d - e \left (b + \sqrt{- 4 a c + b^{2}}\right )} \right )}}{e \left (m + 1\right ) \left (a + b x + c x^{2}\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(d + e*x)**(m + 1)*(c*(-2*d - 2*e*x)/(2*c*d - e*(b + sqrt(-4*a*c + b**2))) + 1)*
*(3/2)*(c*(2*d + 2*e*x)/(b*e - 2*c*d - e*sqrt(-4*a*c + b**2)) + 1)**(3/2)*appell
f1(m + 1, 3/2, 3/2, m + 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*(m + 1)*(a + b*x +
c*x**2)**(3/2))

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Mathematica [A]  time = 0.806816, size = 267, normalized size = 1.41 \[ -\frac{e \left (\sqrt{b^2-4 a c}-b-2 c x\right ) \left (\sqrt{b^2-4 a c}+b+2 c x\right ) (d+e x)^{m+1} \sqrt{\frac{e \left (\sqrt{b^2-4 a c}-b-2 c x\right )}{e \left (\sqrt{b^2-4 a c}-b\right )+2 c d}} \sqrt{\frac{e \left (\sqrt{b^2-4 a c}+b+2 c x\right )}{e \left (\sqrt{b^2-4 a c}+b\right )-2 c d}} F_1\left (m+1;\frac{3}{2},\frac{3}{2};m+2;\frac{2 c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e},\frac{2 c (d+e x)}{2 c d+\left (\sqrt{b^2-4 a c}-b\right ) e}\right )}{4 c (m+1) (a+x (b+c x))^{3/2} \left (e (a e-b d)+c d^2\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

-(e*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x)*Sqrt[(e*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x))/(
2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)]*(b + Sqrt[b^2 - 4*a*c] + 2*c*x)*Sqrt[(e*(b
+ Sqrt[b^2 - 4*a*c] + 2*c*x))/(-2*c*d + (b + Sqrt[b^2 - 4*a*c])*e)]*(d + e*x)^(1
+ m)*AppellF1[1 + m, 3/2, 3/2, 2 + m, (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)])/(4*c*(c*d^2 +
e*(-(b*d) + a*e))*(1 + m)*(a + x*(b + c*x))^(3/2))

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Maple [F]  time = 0.125, size = 0, normalized size = 0. \[ \int{ \left ( ex+d \right ) ^{m} \left ( c{x}^{2}+bx+a \right ) ^{-{\frac{3}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((d + e*x)**m/(a + b*x + c*x**2)**(3/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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