3.1421 \(\int \frac{(b d+2 c d x)^m}{\sqrt{a+b x+c x^2}} \, dx\)

Optimal. Leaf size=96 \[ -\frac{\sqrt{4 a-\frac{b^2}{c}+\frac{(b+2 c x)^2}{c}} (b d+2 c d x)^{m+1} \, _2F_1\left (1,\frac{m+2}{2};\frac{m+3}{2};\frac{(b+2 c x)^2}{b^2-4 a c}\right )}{d (m+1) \left (b^2-4 a c\right )} \]

[Out]

-(((b*d + 2*c*d*x)^(1 + m)*Sqrt[4*a - b^2/c + (b + 2*c*x)^2/c]*Hypergeometric2F1
[1, (2 + m)/2, (3 + m)/2, (b + 2*c*x)^2/(b^2 - 4*a*c)])/((b^2 - 4*a*c)*d*(1 + m)
))

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Rubi [A]  time = 0.274671, antiderivative size = 104, normalized size of antiderivative = 1.08, number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ \frac{\sqrt{1-\frac{(b+2 c x)^2}{b^2-4 a c}} (d (b+2 c x))^{m+1} \, _2F_1\left (\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\frac{(b+2 c x)^2}{b^2-4 a c}\right )}{2 c d (m+1) \sqrt{a+b x+c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

((d*(b + 2*c*x))^(1 + m)*Sqrt[1 - (b + 2*c*x)^2/(b^2 - 4*a*c)]*Hypergeometric2F1
[1/2, (1 + m)/2, (3 + m)/2, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(2*c*d*(1 + m)*Sqrt[a
+ b*x + c*x^2])

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Rubi in Sympy [A]  time = 34.8061, size = 107, normalized size = 1.11 \[ - \frac{2 \left (b d + 2 c d x\right )^{m + 1} \sqrt{a - \frac{b^{2}}{4 c} + \frac{\left (b + 2 c x\right )^{2}}{4 c}}{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{m}{2} + \frac{1}{2} \\ \frac{m}{2} + \frac{3}{2} \end{matrix}\middle |{- \frac{\left (b + 2 c x\right )^{2}}{4 a c - b^{2}}} \right )}}{d \left (m + 1\right ) \left (- 4 a c + b^{2}\right ) \sqrt{\frac{\left (b + 2 c x\right )^{2}}{4 a c - b^{2}} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.152413, size = 107, normalized size = 1.11 \[ \frac{(b+2 c x) \sqrt{-\frac{c (a+x (b+c x))}{b^2-4 a c}} (d (b+2 c x))^m \, _2F_1\left (\frac{1}{2},\frac{m}{2}+\frac{1}{2};\frac{m}{2}+\frac{3}{2};\frac{(b+2 c x)^2}{b^2-4 a c}\right )}{c (m+1) \sqrt{a+x (b+c x)}} \]

Antiderivative was successfully verified.

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

[Out]

((b + 2*c*x)*(d*(b + 2*c*x))^m*Sqrt[-((c*(a + x*(b + c*x)))/(b^2 - 4*a*c))]*Hype
rgeometricPFQ[{1/2, 1/2 + m/2}, {3/2 + m/2}, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(c*(1
+ m)*Sqrt[a + x*(b + c*x)])

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Maple [F]  time = 0.124, size = 0, normalized size = 0. \[ \int{ \left ( 2\,cdx+bd \right ) ^{m}{\frac{1}{\sqrt{c{x}^{2}+bx+a}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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