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

Optimal. Leaf size=72 \[ \frac{2 \sqrt{a+b x} \left (a^2 e-a b d+b^2 c\right )}{b^3}+\frac{2 (a+b x)^{3/2} (b d-2 a e)}{3 b^3}+\frac{2 e (a+b x)^{5/2}}{5 b^3} \]

[Out]

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

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Rubi [A]  time = 0.0330131, antiderivative size = 72, 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} \[ \frac{2 \sqrt{a+b x} \left (a^2 e-a b d+b^2 c\right )}{b^3}+\frac{2 (a+b x)^{3/2} (b d-2 a e)}{3 b^3}+\frac{2 e (a+b x)^{5/2}}{5 b^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0710323, size = 53, normalized size = 0.74 \[ \frac{2 \sqrt{a+b x} \left (8 a^2 e-2 a b (5 d+2 e x)+b^2 (15 c+x (5 d+3 e x))\right )}{15 b^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.023, size = 53, normalized size = 0.7 \begin{align*}{\frac{6\,e{x}^{2}{b}^{2}-8\,abex+10\,{b}^{2}dx+16\,{a}^{2}e-20\,abd+30\,{b}^{2}c}{15\,{b}^{3}}\sqrt{bx+a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.954447, size = 104, normalized size = 1.44 \begin{align*} \frac{2 \,{\left (15 \, \sqrt{b x + a} c + \frac{5 \,{\left ({\left (b x + a\right )}^{\frac{3}{2}} - 3 \, \sqrt{b x + a} a\right )} d}{b} + \frac{{\left (3 \,{\left (b x + a\right )}^{\frac{5}{2}} - 10 \,{\left (b x + a\right )}^{\frac{3}{2}} a + 15 \, \sqrt{b x + a} a^{2}\right )} e}{b^{2}}\right )}}{15 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.26161, size = 127, normalized size = 1.76 \begin{align*} \frac{2 \,{\left (3 \, b^{2} e x^{2} + 15 \, b^{2} c - 10 \, a b d + 8 \, a^{2} e +{\left (5 \, b^{2} d - 4 \, a b e\right )} x\right )} \sqrt{b x + a}}{15 \, b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 7.77256, size = 223, normalized size = 3.1 \begin{align*} \begin{cases} - \frac{\frac{2 a c}{\sqrt{a + b x}} + \frac{2 a d \left (- \frac{a}{\sqrt{a + b x}} - \sqrt{a + b x}\right )}{b} + \frac{2 a e \left (\frac{a^{2}}{\sqrt{a + b x}} + 2 a \sqrt{a + b x} - \frac{\left (a + b x\right )^{\frac{3}{2}}}{3}\right )}{b^{2}} + 2 c \left (- \frac{a}{\sqrt{a + b x}} - \sqrt{a + b x}\right ) + \frac{2 d \left (\frac{a^{2}}{\sqrt{a + b x}} + 2 a \sqrt{a + b x} - \frac{\left (a + b x\right )^{\frac{3}{2}}}{3}\right )}{b} + \frac{2 e \left (- \frac{a^{3}}{\sqrt{a + b x}} - 3 a^{2} \sqrt{a + b x} + a \left (a + b x\right )^{\frac{3}{2}} - \frac{\left (a + b x\right )^{\frac{5}{2}}}{5}\right )}{b^{2}}}{b} & \text{for}\: b \neq 0 \\\frac{c x + \frac{d x^{2}}{2} + \frac{e x^{3}}{3}}{\sqrt{a}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.1084, size = 105, normalized size = 1.46 \begin{align*} \frac{2 \,{\left (15 \, \sqrt{b x + a} c + \frac{5 \,{\left ({\left (b x + a\right )}^{\frac{3}{2}} - 3 \, \sqrt{b x + a} a\right )} d}{b} + \frac{{\left (3 \,{\left (b x + a\right )}^{\frac{5}{2}} - 10 \,{\left (b x + a\right )}^{\frac{3}{2}} a + 15 \, \sqrt{b x + a} a^{2}\right )} e}{b^{2}}\right )}}{15 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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