3.1420 \(\int (b d+2 c d x)^m \sqrt{a+b x+c x^2} \, dx\)

Optimal. Leaf size=98 \[ -\frac{\left (4 a-\frac{b^2}{c}+\frac{(b+2 c x)^2}{c}\right )^{3/2} (b d+2 c d x)^{m+1} \, _2F_1\left (1,\frac{m+4}{2};\frac{m+3}{2};\frac{(b+2 c x)^2}{b^2-4 a c}\right )}{4 d (m+1) \left (b^2-4 a c\right )} \]

[Out]

-((b*d + 2*c*d*x)^(1 + m)*(4*a - b^2/c + (b + 2*c*x)^2/c)^(3/2)*Hypergeometric2F
1[1, (4 + m)/2, (3 + m)/2, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(4*(b^2 - 4*a*c)*d*(1 +
m))

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Rubi [A]  time = 0.279025, antiderivative size = 104, normalized size of antiderivative = 1.06, number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ \frac{\sqrt{a+b x+c x^2} (d (b+2 c x))^{m+1} \, _2F_1\left (-\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\frac{(b+2 c x)^2}{b^2-4 a c}\right )}{2 c d (m+1) \sqrt{1-\frac{(b+2 c x)^2}{b^2-4 a c}}} \]

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[a + b*x + c*x^2]*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[1 - (b + 2*c*x)^2
/(b^2 - 4*a*c)])

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Rubi in Sympy [A]  time = 32.4416, size = 100, normalized size = 1.02 \[ \frac{\left (b d + 2 c d x\right )^{m + 1} \sqrt{a - \frac{b^{2}}{4 c} + \frac{\left (b + 2 c x\right )^{2}}{4 c}}{{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{m}{2} + \frac{1}{2} \\ \frac{m}{2} + \frac{3}{2} \end{matrix}\middle |{- \frac{\left (b + 2 c x\right )^{2}}{4 a c - b^{2}}} \right )}}{2 c d \left (m + 1\right ) \sqrt{\frac{\left (b + 2 c x\right )^{2}}{4 a c - b^{2}} + 1}} \]

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]

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

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Mathematica [A]  time = 0.148054, size = 110, normalized size = 1.12 \[ \frac{(b+2 c x) \sqrt{a+x (b+c x)} (d (b+2 c x))^m \, _2F_1\left (-\frac{1}{2},\frac{m}{2}+\frac{1}{2};\frac{m}{2}+\frac{3}{2};\frac{(b+2 c x)^2}{b^2-4 a c}\right )}{4 c (m+1) \sqrt{-\frac{c (a+x (b+c x))}{b^2-4 a c}}} \]

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

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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