3.1419 \(\int (b d+2 c d x)^m \left (a+b x+c x^2\right )^{3/2} \, dx\)

Optimal. Leaf size=98 \[ -\frac{\left (4 a-\frac{b^2}{c}+\frac{(b+2 c x)^2}{c}\right )^{5/2} (b d+2 c d x)^{m+1} \, _2F_1\left (1,\frac{m+6}{2};\frac{m+3}{2};\frac{(b+2 c x)^2}{b^2-4 a c}\right )}{16 d (m+1) \left (b^2-4 a c\right )} \]

[Out]

-((b*d + 2*c*d*x)^(1 + m)*(4*a - b^2/c + (b + 2*c*x)^2/c)^(5/2)*Hypergeometric2F
1[1, (6 + m)/2, (3 + m)/2, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(16*(b^2 - 4*a*c)*d*(1
+ m))

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Rubi [A]  time = 0.299232, antiderivative size = 112, normalized size of antiderivative = 1.14, number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ -\frac{\left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} (d (b+2 c x))^{m+1} \, _2F_1\left (-\frac{3}{2},\frac{m+1}{2};\frac{m+3}{2};\frac{(b+2 c x)^2}{b^2-4 a c}\right )}{8 c^2 d (m+1) \sqrt{1-\frac{(b+2 c x)^2}{b^2-4 a c}}} \]

Antiderivative was successfully verified.

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

[Out]

-((b^2 - 4*a*c)*(d*(b + 2*c*x))^(1 + m)*Sqrt[a + b*x + c*x^2]*Hypergeometric2F1[
-3/2, (1 + m)/2, (3 + m)/2, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(8*c^2*d*(1 + m)*Sqrt[
1 - (b + 2*c*x)^2/(b^2 - 4*a*c)])

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Rubi in Sympy [A]  time = 34.1589, size = 112, normalized size = 1.14 \[ - \frac{\left (- a c + \frac{b^{2}}{4}\right ) \left (b d + 2 c d x\right )^{m + 1} \sqrt{a - \frac{b^{2}}{4 c} + \frac{\left (b + 2 c x\right )^{2}}{4 c}}{{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{2}, \frac{m}{2} + \frac{1}{2} \\ \frac{m}{2} + \frac{3}{2} \end{matrix}\middle |{- \frac{\left (b + 2 c x\right )^{2}}{4 a c - b^{2}}} \right )}}{2 c^{2} d \left (m + 1\right ) \sqrt{\frac{\left (b + 2 c x\right )^{2}}{4 a c - b^{2}} + 1}} \]

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)**(3/2),x)

[Out]

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

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Mathematica [A]  time = 0.106603, size = 0, normalized size = 0. \[ \int (b d+2 c d x)^m \left (a+b x+c x^2\right )^{3/2} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [F]  time = 0.124, size = 0, normalized size = 0. \[ \int \left ( 2\,cdx+bd \right ) ^{m} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

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)**(3/2),x)

[Out]

Integral((d*(b + 2*c*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. \[ \int{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}}{\left (2 \, c d x + b d\right )}^{m}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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