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

Optimal. Leaf size=71 \[ -\frac{2 b (b+c x) \left (b x+c x^2\right )^{3/2} (d x)^m \left (-\frac{c x}{b}\right )^{-m-\frac{1}{2}} \, _2F_1\left (\frac{5}{2},-m-\frac{3}{2};\frac{7}{2};\frac{c x}{b}+1\right )}{5 c^2 x} \]

[Out]

(-2*b*(-((c*x)/b))^(-1/2 - m)*(d*x)^m*(b + c*x)*(b*x + c*x^2)^(3/2)*Hypergeometr
ic2F1[5/2, -3/2 - m, 7/2, 1 + (c*x)/b])/(5*c^2*x)

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Rubi [A]  time = 0.0962461, antiderivative size = 71, 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 \[ -\frac{2 b (b+c x) \left (b x+c x^2\right )^{3/2} (d x)^m \left (-\frac{c x}{b}\right )^{-m-\frac{1}{2}} \, _2F_1\left (\frac{5}{2},-m-\frac{3}{2};\frac{7}{2};\frac{c x}{b}+1\right )}{5 c^2 x} \]

Antiderivative was successfully verified.

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

[Out]

(-2*b*(-((c*x)/b))^(-1/2 - m)*(d*x)^m*(b + c*x)*(b*x + c*x^2)^(3/2)*Hypergeometr
ic2F1[5/2, -3/2 - m, 7/2, 1 + (c*x)/b])/(5*c^2*x)

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Rubi in Sympy [A]  time = 14.5602, size = 76, normalized size = 1.07 \[ - \frac{2 b x^{- m - \frac{3}{2}} x^{m + \frac{1}{2}} \left (d x\right )^{m} \left (- \frac{c x}{b}\right )^{- m - \frac{1}{2}} \left (b + c x\right ) \left (b x + c x^{2}\right )^{\frac{3}{2}}{{}_{2}F_{1}\left (\begin{matrix} - m - \frac{3}{2}, \frac{5}{2} \\ \frac{7}{2} \end{matrix}\middle |{1 + \frac{c x}{b}} \right )}}{5 c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.112482, size = 105, normalized size = 1.48 \[ \frac{2 x^2 \sqrt{x (b+c x)} (d x)^m \left (b (2 m+7) \, _2F_1\left (-\frac{1}{2},m+\frac{5}{2};m+\frac{7}{2};-\frac{c x}{b}\right )+c (2 m+5) x \, _2F_1\left (-\frac{1}{2},m+\frac{7}{2};m+\frac{9}{2};-\frac{c x}{b}\right )\right )}{(2 m+5) (2 m+7) \sqrt{\frac{c x}{b}+1}} \]

Antiderivative was successfully verified.

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

[Out]

(2*x^2*(d*x)^m*Sqrt[x*(b + c*x)]*(b*(7 + 2*m)*Hypergeometric2F1[-1/2, 5/2 + m, 7
/2 + m, -((c*x)/b)] + c*(5 + 2*m)*x*Hypergeometric2F1[-1/2, 7/2 + m, 9/2 + m, -(
(c*x)/b)]))/((5 + 2*m)*(7 + 2*m)*Sqrt[1 + (c*x)/b])

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Maple [F]  time = 0.03, size = 0, normalized size = 0. \[ \int \left ( dx \right ) ^{m} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((d*x)**m*(x*(b + c*x))**(3/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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