3.2272 \(\int \frac{a+b x+c x^2}{(d+e x)^{3/2}} \, dx\)

Optimal. Leaf size=71 \[ -\frac{2 \left (a e^2-b d e+c d^2\right )}{e^3 \sqrt{d+e x}}-\frac{2 \sqrt{d+e x} (2 c d-b e)}{e^3}+\frac{2 c (d+e x)^{3/2}}{3 e^3} \]

[Out]

(-2*(c*d^2 - b*d*e + a*e^2))/(e^3*Sqrt[d + e*x]) - (2*(2*c*d - b*e)*Sqrt[d + e*x])/e^3 + (2*c*(d + e*x)^(3/2))
/(3*e^3)

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Rubi [A]  time = 0.0306339, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {698} \[ -\frac{2 \left (a e^2-b d e+c d^2\right )}{e^3 \sqrt{d+e x}}-\frac{2 \sqrt{d+e x} (2 c d-b e)}{e^3}+\frac{2 c (d+e x)^{3/2}}{3 e^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(c*d^2 - b*d*e + a*e^2))/(e^3*Sqrt[d + e*x]) - (2*(2*c*d - b*e)*Sqrt[d + e*x])/e^3 + (2*c*(d + e*x)^(3/2))
/(3*e^3)

Rule 698

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*
e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

\begin{align*} \int \frac{a+b x+c x^2}{(d+e x)^{3/2}} \, dx &=\int \left (\frac{c d^2-b d e+a e^2}{e^2 (d+e x)^{3/2}}+\frac{-2 c d+b e}{e^2 \sqrt{d+e x}}+\frac{c \sqrt{d+e x}}{e^2}\right ) \, dx\\ &=-\frac{2 \left (c d^2-b d e+a e^2\right )}{e^3 \sqrt{d+e x}}-\frac{2 (2 c d-b e) \sqrt{d+e x}}{e^3}+\frac{2 c (d+e x)^{3/2}}{3 e^3}\\ \end{align*}

Mathematica [A]  time = 0.050236, size = 54, normalized size = 0.76 \[ \frac{6 e (-a e+2 b d+b e x)+2 c \left (-8 d^2-4 d e x+e^2 x^2\right )}{3 e^3 \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(6*e*(2*b*d - a*e + b*e*x) + 2*c*(-8*d^2 - 4*d*e*x + e^2*x^2))/(3*e^3*Sqrt[d + e*x])

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Maple [A]  time = 0.042, size = 53, normalized size = 0.8 \begin{align*} -{\frac{-2\,c{e}^{2}{x}^{2}-6\,b{e}^{2}x+8\,cdex+6\,a{e}^{2}-12\,bde+16\,c{d}^{2}}{3\,{e}^{3}}{\frac{1}{\sqrt{ex+d}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3/(e*x+d)^(1/2)*(-c*e^2*x^2-3*b*e^2*x+4*c*d*e*x+3*a*e^2-6*b*d*e+8*c*d^2)/e^3

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Maxima [A]  time = 0.975398, size = 89, normalized size = 1.25 \begin{align*} \frac{2 \,{\left (\frac{{\left (e x + d\right )}^{\frac{3}{2}} c - 3 \,{\left (2 \, c d - b e\right )} \sqrt{e x + d}}{e^{2}} - \frac{3 \,{\left (c d^{2} - b d e + a e^{2}\right )}}{\sqrt{e x + d} e^{2}}\right )}}{3 \, e} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)/(e*x+d)^(3/2),x, algorithm="maxima")

[Out]

2/3*(((e*x + d)^(3/2)*c - 3*(2*c*d - b*e)*sqrt(e*x + d))/e^2 - 3*(c*d^2 - b*d*e + a*e^2)/(sqrt(e*x + d)*e^2))/
e

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Fricas [A]  time = 2.18648, size = 136, normalized size = 1.92 \begin{align*} \frac{2 \,{\left (c e^{2} x^{2} - 8 \, c d^{2} + 6 \, b d e - 3 \, a e^{2} -{\left (4 \, c d e - 3 \, b e^{2}\right )} x\right )} \sqrt{e x + d}}{3 \,{\left (e^{4} x + d e^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)/(e*x+d)^(3/2),x, algorithm="fricas")

[Out]

2/3*(c*e^2*x^2 - 8*c*d^2 + 6*b*d*e - 3*a*e^2 - (4*c*d*e - 3*b*e^2)*x)*sqrt(e*x + d)/(e^4*x + d*e^3)

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Sympy [A]  time = 9.0996, size = 70, normalized size = 0.99 \begin{align*} \frac{2 c \left (d + e x\right )^{\frac{3}{2}}}{3 e^{3}} + \frac{\sqrt{d + e x} \left (2 b e - 4 c d\right )}{e^{3}} - \frac{2 \left (a e^{2} - b d e + c d^{2}\right )}{e^{3} \sqrt{d + e x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*c*(d + e*x)**(3/2)/(3*e**3) + sqrt(d + e*x)*(2*b*e - 4*c*d)/e**3 - 2*(a*e**2 - b*d*e + c*d**2)/(e**3*sqrt(d
+ e*x))

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Giac [A]  time = 1.1096, size = 99, normalized size = 1.39 \begin{align*} \frac{2}{3} \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} c e^{6} - 6 \, \sqrt{x e + d} c d e^{6} + 3 \, \sqrt{x e + d} b e^{7}\right )} e^{\left (-9\right )} - \frac{2 \,{\left (c d^{2} - b d e + a e^{2}\right )} e^{\left (-3\right )}}{\sqrt{x e + d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)/(e*x+d)^(3/2),x, algorithm="giac")

[Out]

2/3*((x*e + d)^(3/2)*c*e^6 - 6*sqrt(x*e + d)*c*d*e^6 + 3*sqrt(x*e + d)*b*e^7)*e^(-9) - 2*(c*d^2 - b*d*e + a*e^
2)*e^(-3)/sqrt(x*e + d)