3.2271 \(\int \frac{a+b x+c x^2}{\sqrt{d+e x}} \, dx\)

Optimal. Leaf size=73 \[ \frac{2 \sqrt{d+e x} \left (a e^2-b d e+c d^2\right )}{e^3}-\frac{2 (d+e x)^{3/2} (2 c d-b e)}{3 e^3}+\frac{2 c (d+e x)^{5/2}}{5 e^3} \]

[Out]

(2*(c*d^2 - b*d*e + a*e^2)*Sqrt[d + e*x])/e^3 - (2*(2*c*d - b*e)*(d + e*x)^(3/2))/(3*e^3) + (2*c*(d + e*x)^(5/
2))/(5*e^3)

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Rubi [A]  time = 0.0299421, antiderivative size = 73, 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} \[ \frac{2 \sqrt{d+e x} \left (a e^2-b d e+c d^2\right )}{e^3}-\frac{2 (d+e x)^{3/2} (2 c d-b e)}{3 e^3}+\frac{2 c (d+e x)^{5/2}}{5 e^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(c*d^2 - b*d*e + a*e^2)*Sqrt[d + e*x])/e^3 - (2*(2*c*d - b*e)*(d + e*x)^(3/2))/(3*e^3) + (2*c*(d + e*x)^(5/
2))/(5*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}{\sqrt{d+e x}} \, dx &=\int \left (\frac{c d^2-b d e+a e^2}{e^2 \sqrt{d+e x}}+\frac{(-2 c d+b e) \sqrt{d+e x}}{e^2}+\frac{c (d+e x)^{3/2}}{e^2}\right ) \, dx\\ &=\frac{2 \left (c d^2-b d e+a e^2\right ) \sqrt{d+e x}}{e^3}-\frac{2 (2 c d-b e) (d+e x)^{3/2}}{3 e^3}+\frac{2 c (d+e x)^{5/2}}{5 e^3}\\ \end{align*}

Mathematica [A]  time = 0.0505959, size = 54, normalized size = 0.74 \[ \frac{2 \sqrt{d+e x} \left (5 e (3 a e-2 b d+b e x)+c \left (8 d^2-4 d e x+3 e^2 x^2\right )\right )}{15 e^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[d + e*x]*(5*e*(-2*b*d + 3*a*e + b*e*x) + c*(8*d^2 - 4*d*e*x + 3*e^2*x^2)))/(15*e^3)

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Maple [A]  time = 0.042, size = 53, normalized size = 0.7 \begin{align*}{\frac{6\,c{e}^{2}{x}^{2}+10\,b{e}^{2}x-8\,cdex+30\,a{e}^{2}-20\,bde+16\,c{d}^{2}}{15\,{e}^{3}}\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)^(1/2),x)

[Out]

2/15*(e*x+d)^(1/2)*(3*c*e^2*x^2+5*b*e^2*x-4*c*d*e*x+15*a*e^2-10*b*d*e+8*c*d^2)/e^3

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Maxima [A]  time = 0.978035, size = 104, normalized size = 1.42 \begin{align*} \frac{2 \,{\left (15 \, \sqrt{e x + d} a + \frac{5 \,{\left ({\left (e x + d\right )}^{\frac{3}{2}} - 3 \, \sqrt{e x + d} d\right )} b}{e} + \frac{{\left (3 \,{\left (e x + d\right )}^{\frac{5}{2}} - 10 \,{\left (e x + d\right )}^{\frac{3}{2}} d + 15 \, \sqrt{e x + d} d^{2}\right )} c}{e^{2}}\right )}}{15 \, e} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)/(e*x+d)^(1/2),x, algorithm="maxima")

[Out]

2/15*(15*sqrt(e*x + d)*a + 5*((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d)*b/e + (3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/
2)*d + 15*sqrt(e*x + d)*d^2)*c/e^2)/e

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Fricas [A]  time = 2.27172, size = 127, normalized size = 1.74 \begin{align*} \frac{2 \,{\left (3 \, c e^{2} x^{2} + 8 \, c d^{2} - 10 \, b d e + 15 \, a e^{2} -{\left (4 \, c d e - 5 \, b e^{2}\right )} x\right )} \sqrt{e x + d}}{15 \, e^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)/(e*x+d)^(1/2),x, algorithm="fricas")

[Out]

2/15*(3*c*e^2*x^2 + 8*c*d^2 - 10*b*d*e + 15*a*e^2 - (4*c*d*e - 5*b*e^2)*x)*sqrt(e*x + d)/e^3

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Sympy [A]  time = 8.2473, size = 223, normalized size = 3.05 \begin{align*} \begin{cases} - \frac{\frac{2 a d}{\sqrt{d + e x}} + 2 a \left (- \frac{d}{\sqrt{d + e x}} - \sqrt{d + e x}\right ) + \frac{2 b d \left (- \frac{d}{\sqrt{d + e x}} - \sqrt{d + e x}\right )}{e} + \frac{2 b \left (\frac{d^{2}}{\sqrt{d + e x}} + 2 d \sqrt{d + e x} - \frac{\left (d + e x\right )^{\frac{3}{2}}}{3}\right )}{e} + \frac{2 c d \left (\frac{d^{2}}{\sqrt{d + e x}} + 2 d \sqrt{d + e x} - \frac{\left (d + e x\right )^{\frac{3}{2}}}{3}\right )}{e^{2}} + \frac{2 c \left (- \frac{d^{3}}{\sqrt{d + e x}} - 3 d^{2} \sqrt{d + e x} + d \left (d + e x\right )^{\frac{3}{2}} - \frac{\left (d + e x\right )^{\frac{5}{2}}}{5}\right )}{e^{2}}}{e} & \text{for}\: e \neq 0 \\\frac{a x + \frac{b x^{2}}{2} + \frac{c x^{3}}{3}}{\sqrt{d}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+b*x+a)/(e*x+d)**(1/2),x)

[Out]

Piecewise((-(2*a*d/sqrt(d + e*x) + 2*a*(-d/sqrt(d + e*x) - sqrt(d + e*x)) + 2*b*d*(-d/sqrt(d + e*x) - sqrt(d +
e*x))/e + 2*b*(d**2/sqrt(d + e*x) + 2*d*sqrt(d + e*x) - (d + e*x)**(3/2)/3)/e + 2*c*d*(d**2/sqrt(d + e*x) + 2
*d*sqrt(d + e*x) - (d + e*x)**(3/2)/3)/e**2 + 2*c*(-d**3/sqrt(d + e*x) - 3*d**2*sqrt(d + e*x) + d*(d + e*x)**(
3/2) - (d + e*x)**(5/2)/5)/e**2)/e, Ne(e, 0)), ((a*x + b*x**2/2 + c*x**3/3)/sqrt(d), True))

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Giac [A]  time = 1.10022, size = 108, normalized size = 1.48 \begin{align*} \frac{2}{15} \,{\left (5 \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} - 3 \, \sqrt{x e + d} d\right )} b e^{\left (-1\right )} +{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 10 \,{\left (x e + d\right )}^{\frac{3}{2}} d + 15 \, \sqrt{x e + d} d^{2}\right )} c e^{\left (-2\right )} + 15 \, \sqrt{x e + d} a\right )} e^{\left (-1\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)/(e*x+d)^(1/2),x, algorithm="giac")

[Out]

2/15*(5*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*b*e^(-1) + (3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x
*e + d)*d^2)*c*e^(-2) + 15*sqrt(x*e + d)*a)*e^(-1)