3.118 \(\int (d x)^m \left (b x+c x^2\right )^{5/2} \, dx\)

Optimal. Leaf size=73 \[ \frac{2 b^2 (b+c x) \left (b x+c x^2\right )^{5/2} (d x)^m \left (-\frac{c x}{b}\right )^{-m-\frac{1}{2}} \, _2F_1\left (\frac{7}{2},-m-\frac{5}{2};\frac{9}{2};\frac{c x}{b}+1\right )}{7 c^3 x^2} \]

[Out]

(2*b^2*(-((c*x)/b))^(-1/2 - m)*(d*x)^m*(b + c*x)*(b*x + c*x^2)^(5/2)*Hypergeomet
ric2F1[7/2, -5/2 - m, 9/2, 1 + (c*x)/b])/(7*c^3*x^2)

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Rubi [A]  time = 0.0986085, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ \frac{2 b^2 (b+c x) \left (b x+c x^2\right )^{5/2} (d x)^m \left (-\frac{c x}{b}\right )^{-m-\frac{1}{2}} \, _2F_1\left (\frac{7}{2},-m-\frac{5}{2};\frac{9}{2};\frac{c x}{b}+1\right )}{7 c^3 x^2} \]

Antiderivative was successfully verified.

[In]  Int[(d*x)^m*(b*x + c*x^2)^(5/2),x]

[Out]

(2*b^2*(-((c*x)/b))^(-1/2 - m)*(d*x)^m*(b + c*x)*(b*x + c*x^2)^(5/2)*Hypergeomet
ric2F1[7/2, -5/2 - m, 9/2, 1 + (c*x)/b])/(7*c^3*x^2)

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Rubi in Sympy [A]  time = 14.8196, size = 76, normalized size = 1.04 \[ \frac{2 b^{2} x^{- m - \frac{5}{2}} x^{m + \frac{1}{2}} \left (d x\right )^{m} \left (- \frac{c x}{b}\right )^{- m - \frac{1}{2}} \left (b + c x\right ) \left (b x + c x^{2}\right )^{\frac{5}{2}}{{}_{2}F_{1}\left (\begin{matrix} - m - \frac{5}{2}, \frac{7}{2} \\ \frac{9}{2} \end{matrix}\middle |{1 + \frac{c x}{b}} \right )}}{7 c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x)**m*(c*x**2+b*x)**(5/2),x)

[Out]

2*b**2*x**(-m - 5/2)*x**(m + 1/2)*(d*x)**m*(-c*x/b)**(-m - 1/2)*(b + c*x)*(b*x +
c*x**2)**(5/2)*hyper((-m - 5/2, 7/2), (9/2,), 1 + c*x/b)/(7*c**3)

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Mathematica [B]  time = 0.205919, size = 157, normalized size = 2.15 \[ \frac{2 x^3 \sqrt{x (b+c x)} (d x)^m \left (b^2 \left (4 m^2+40 m+99\right ) \, _2F_1\left (-\frac{1}{2},m+\frac{7}{2};m+\frac{9}{2};-\frac{c x}{b}\right )+c (2 m+7) x \left (2 b (2 m+11) \, _2F_1\left (-\frac{1}{2},m+\frac{9}{2};m+\frac{11}{2};-\frac{c x}{b}\right )+c (2 m+9) x \, _2F_1\left (-\frac{1}{2},m+\frac{11}{2};m+\frac{13}{2};-\frac{c x}{b}\right )\right )\right )}{(2 m+7) (2 m+9) (2 m+11) \sqrt{\frac{c x}{b}+1}} \]

Antiderivative was successfully verified.

[In]  Integrate[(d*x)^m*(b*x + c*x^2)^(5/2),x]

[Out]

(2*x^3*(d*x)^m*Sqrt[x*(b + c*x)]*(b^2*(99 + 40*m + 4*m^2)*Hypergeometric2F1[-1/2
, 7/2 + m, 9/2 + m, -((c*x)/b)] + c*(7 + 2*m)*x*(2*b*(11 + 2*m)*Hypergeometric2F
1[-1/2, 9/2 + m, 11/2 + m, -((c*x)/b)] + c*(9 + 2*m)*x*Hypergeometric2F1[-1/2, 1
1/2 + m, 13/2 + m, -((c*x)/b)])))/((7 + 2*m)*(9 + 2*m)*(11 + 2*m)*Sqrt[1 + (c*x)
/b])

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Maple [F]  time = 0.033, size = 0, normalized size = 0. \[ \int \left ( dx \right ) ^{m} \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x)^m*(c*x^2+b*x)^(5/2),x)

[Out]

int((d*x)^m*(c*x^2+b*x)^(5/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (c x^{2} + b x\right )}^{\frac{5}{2}} \left (d x\right )^{m}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(5/2)*(d*x)^m,x, algorithm="maxima")

[Out]

integrate((c*x^2 + b*x)^(5/2)*(d*x)^m, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (c^{2} x^{4} + 2 \, b c x^{3} + b^{2} x^{2}\right )} \sqrt{c x^{2} + b x} \left (d x\right )^{m}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(5/2)*(d*x)^m,x, algorithm="fricas")

[Out]

integral((c^2*x^4 + 2*b*c*x^3 + b^2*x^2)*sqrt(c*x^2 + b*x)*(d*x)^m, x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x)**m*(c*x**2+b*x)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (c x^{2} + b x\right )}^{\frac{5}{2}} \left (d x\right )^{m}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(5/2)*(d*x)^m,x, algorithm="giac")

[Out]

integrate((c*x^2 + b*x)^(5/2)*(d*x)^m, x)