3.448 \(\int \frac{(d+e x)^m}{\sqrt{b x+c x^2}} \, dx\)

Optimal. Leaf size=105 \[ \frac{\sqrt{-\frac{e x}{d}} (d+e x)^{m+1} \sqrt{1-\frac{c (d+e x)}{c d-b e}} F_1\left (m+1;\frac{1}{2},\frac{1}{2};m+2;\frac{d+e x}{d},\frac{c (d+e x)}{c d-b e}\right )}{e (m+1) \sqrt{b x+c x^2}} \]

[Out]

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

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Rubi [A]  time = 0.238752, antiderivative size = 105, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095 \[ \frac{\sqrt{-\frac{e x}{d}} (d+e x)^{m+1} \sqrt{1-\frac{c (d+e x)}{c d-b e}} F_1\left (m+1;\frac{1}{2},\frac{1}{2};m+2;\frac{d+e x}{d},\frac{c (d+e x)}{c d-b e}\right )}{e (m+1) \sqrt{b x+c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 21.3347, size = 82, normalized size = 0.78 \[ \frac{\sqrt{- \frac{e x}{d}} \left (d + e x\right )^{m + 1} \sqrt{\frac{c \left (d + e x\right )}{b e - c d} + 1} \operatorname{appellf_{1}}{\left (m + 1,\frac{1}{2},\frac{1}{2},m + 2,\frac{d + e x}{d},\frac{c \left (- d - e x\right )}{b e - c d} \right )}}{e \left (m + 1\right ) \sqrt{b x + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**m/(c*x**2+b*x)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.427587, size = 151, normalized size = 1.44 \[ \frac{6 b d x (d+e x)^m F_1\left (\frac{1}{2};\frac{1}{2},-m;\frac{3}{2};-\frac{c x}{b},-\frac{e x}{d}\right )}{\sqrt{x (b+c x)} \left (3 b d F_1\left (\frac{1}{2};\frac{1}{2},-m;\frac{3}{2};-\frac{c x}{b},-\frac{e x}{d}\right )+2 b e m x F_1\left (\frac{3}{2};\frac{1}{2},1-m;\frac{5}{2};-\frac{c x}{b},-\frac{e x}{d}\right )-c d x F_1\left (\frac{3}{2};\frac{3}{2},-m;\frac{5}{2};-\frac{c x}{b},-\frac{e x}{d}\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

(6*b*d*x*(d + e*x)^m*AppellF1[1/2, 1/2, -m, 3/2, -((c*x)/b), -((e*x)/d)])/(Sqrt[
x*(b + c*x)]*(3*b*d*AppellF1[1/2, 1/2, -m, 3/2, -((c*x)/b), -((e*x)/d)] + 2*b*e*
m*x*AppellF1[3/2, 1/2, 1 - m, 5/2, -((c*x)/b), -((e*x)/d)] - c*d*x*AppellF1[3/2,
3/2, -m, 5/2, -((c*x)/b), -((e*x)/d)]))

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Maple [F]  time = 0.051, size = 0, normalized size = 0. \[ \int{ \left ( ex+d \right ) ^{m}{\frac{1}{\sqrt{c{x}^{2}+bx}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^m/(c*x^2+b*x)^(1/2),x)

[Out]

int((e*x+d)^m/(c*x^2+b*x)^(1/2),x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**m/(c*x**2+b*x)**(1/2),x)

[Out]

Integral((d + e*x)**m/sqrt(x*(b + c*x)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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