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

Optimal. Leaf size=61 \[ \frac{2 x (d x)^{3/2} \left (\frac{c x}{b}+1\right )^{-p} \left (b x+c x^2\right )^p \, _2F_1\left (-p,p+\frac{5}{2};p+\frac{7}{2};-\frac{c x}{b}\right )}{2 p+5} \]

[Out]

(2*x*(d*x)^(3/2)*(b*x + c*x^2)^p*Hypergeometric2F1[-p, 5/2 + p, 7/2 + p, -((c*x)
/b)])/((5 + 2*p)*(1 + (c*x)/b)^p)

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Rubi [A]  time = 0.0723971, antiderivative size = 61, 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 \[ \frac{2 x (d x)^{3/2} \left (\frac{c x}{b}+1\right )^{-p} \left (b x+c x^2\right )^p \, _2F_1\left (-p,p+\frac{5}{2};p+\frac{7}{2};-\frac{c x}{b}\right )}{2 p+5} \]

Antiderivative was successfully verified.

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

[Out]

(2*x*(d*x)^(3/2)*(b*x + c*x^2)^p*Hypergeometric2F1[-p, 5/2 + p, 7/2 + p, -((c*x)
/b)])/((5 + 2*p)*(1 + (c*x)/b)^p)

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Rubi in Sympy [A]  time = 11.9447, size = 63, normalized size = 1.03 \[ \frac{2 x^{- p - \frac{3}{2}} x^{p + \frac{5}{2}} \left (d x\right )^{\frac{3}{2}} \left (1 + \frac{c x}{b}\right )^{- p} \left (b x + c x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, p + \frac{5}{2} \\ p + \frac{7}{2} \end{matrix}\middle |{- \frac{c x}{b}} \right )}}{2 p + 5} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*x**(-p - 3/2)*x**(p + 5/2)*(d*x)**(3/2)*(1 + c*x/b)**(-p)*(b*x + c*x**2)**p*hy
per((-p, p + 5/2), (p + 7/2,), -c*x/b)/(2*p + 5)

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Mathematica [A]  time = 0.0599264, size = 59, normalized size = 0.97 \[ \frac{2 x (d x)^{3/2} (x (b+c x))^p \left (\frac{c x}{b}+1\right )^{-p} \, _2F_1\left (-p,p+\frac{5}{2};p+\frac{7}{2};-\frac{c x}{b}\right )}{2 p+5} \]

Antiderivative was successfully verified.

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

[Out]

(2*x*(d*x)^(3/2)*(x*(b + c*x))^p*Hypergeometric2F1[-p, 5/2 + p, 7/2 + p, -((c*x)
/b)])/((5 + 2*p)*(1 + (c*x)/b)^p)

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Maple [F]  time = 0.033, size = 0, normalized size = 0. \[ \int \left ( dx \right ) ^{{\frac{3}{2}}} \left ( c{x}^{2}+bx \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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