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

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

[Out]

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

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Rubi [A]  time = 0.242762, 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 \[ \frac{\sqrt{b x+c x^2} (d+e x)^{m+1} F_1\left (m+1;-\frac{1}{2},-\frac{1}{2};m+2;\frac{d+e x}{d},\frac{c (d+e x)}{c d-b e}\right )}{e (m+1) \sqrt{-\frac{e x}{d}} \sqrt{1-\frac{c (d+e x)}{c d-b e}}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 20.8862, size = 85, normalized size = 0.81 \[ \frac{\left (d + e x\right )^{m + 1} \sqrt{b x + c x^{2}} \operatorname{appellf_{1}}{\left (m + 1,- \frac{1}{2},- \frac{1}{2},m + 2,\frac{d + e x}{d},\frac{c \left (- d - e x\right )}{b e - c d} \right )}}{e \sqrt{- \frac{e x}{d}} \left (m + 1\right ) \sqrt{\frac{c \left (d + e x\right )}{b e - c d} + 1}} \]

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]

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

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Mathematica [A]  time = 0.369731, size = 151, normalized size = 1.44 \[ \frac{10 b d x \sqrt{x (b+c x)} (d+e x)^m F_1\left (\frac{3}{2};-\frac{1}{2},-m;\frac{5}{2};-\frac{c x}{b},-\frac{e x}{d}\right )}{15 b d F_1\left (\frac{3}{2};-\frac{1}{2},-m;\frac{5}{2};-\frac{c x}{b},-\frac{e x}{d}\right )+6 b e m x F_1\left (\frac{5}{2};-\frac{1}{2},1-m;\frac{7}{2};-\frac{c x}{b},-\frac{e x}{d}\right )+3 c d x F_1\left (\frac{5}{2};\frac{1}{2},-m;\frac{7}{2};-\frac{c x}{b},-\frac{e x}{d}\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

(10*b*d*x*Sqrt[x*(b + c*x)]*(d + e*x)^m*AppellF1[3/2, -1/2, -m, 5/2, -((c*x)/b),
-((e*x)/d)])/(15*b*d*AppellF1[3/2, -1/2, -m, 5/2, -((c*x)/b), -((e*x)/d)] + 6*b
*e*m*x*AppellF1[5/2, -1/2, 1 - m, 7/2, -((c*x)/b), -((e*x)/d)] + 3*c*d*x*AppellF
1[5/2, 1/2, -m, 7/2, -((c*x)/b), -((e*x)/d)])

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Maple [F]  time = 0.049, size = 0, normalized size = 0. \[ \int \left ( ex+d \right ) ^{m}\sqrt{c{x}^{2}+bx}\, dx \]

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. \[ \int \sqrt{c x^{2} + b x}{\left (e x + d\right )}^{m}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

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(sqrt(x*(b + c*x))*(d + e*x)**m, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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