3.592 \(\int \frac {a+c x^2}{\sqrt {f+g x}} \, dx\)

Optimal. Leaf size=61 \[ \frac {2 \sqrt {f+g x} \left (a g^2+c f^2\right )}{g^3}+\frac {2 c (f+g x)^{5/2}}{5 g^3}-\frac {4 c f (f+g x)^{3/2}}{3 g^3} \]

[Out]

-4/3*c*f*(g*x+f)^(3/2)/g^3+2/5*c*(g*x+f)^(5/2)/g^3+2*(a*g^2+c*f^2)*(g*x+f)^(1/2)/g^3

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Rubi [A]  time = 0.03, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {697} \[ \frac {2 \sqrt {f+g x} \left (a g^2+c f^2\right )}{g^3}+\frac {2 c (f+g x)^{5/2}}{5 g^3}-\frac {4 c f (f+g x)^{3/2}}{3 g^3} \]

Antiderivative was successfully verified.

[In]

Int[(a + c*x^2)/Sqrt[f + g*x],x]

[Out]

(2*(c*f^2 + a*g^2)*Sqrt[f + g*x])/g^3 - (4*c*f*(f + g*x)^(3/2))/(3*g^3) + (2*c*(f + g*x)^(5/2))/(5*g^3)

Rule 697

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

Rubi steps

\begin {align*} \int \frac {a+c x^2}{\sqrt {f+g x}} \, dx &=\int \left (\frac {c f^2+a g^2}{g^2 \sqrt {f+g x}}-\frac {2 c f \sqrt {f+g x}}{g^2}+\frac {c (f+g x)^{3/2}}{g^2}\right ) \, dx\\ &=\frac {2 \left (c f^2+a g^2\right ) \sqrt {f+g x}}{g^3}-\frac {4 c f (f+g x)^{3/2}}{3 g^3}+\frac {2 c (f+g x)^{5/2}}{5 g^3}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 44, normalized size = 0.72 \[ \frac {2 \sqrt {f+g x} \left (15 a g^2+c \left (8 f^2-4 f g x+3 g^2 x^2\right )\right )}{15 g^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + c*x^2)/Sqrt[f + g*x],x]

[Out]

(2*Sqrt[f + g*x]*(15*a*g^2 + c*(8*f^2 - 4*f*g*x + 3*g^2*x^2)))/(15*g^3)

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fricas [A]  time = 0.59, size = 40, normalized size = 0.66 \[ \frac {2 \, {\left (3 \, c g^{2} x^{2} - 4 \, c f g x + 8 \, c f^{2} + 15 \, a g^{2}\right )} \sqrt {g x + f}}{15 \, g^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)/(g*x+f)^(1/2),x, algorithm="fricas")

[Out]

2/15*(3*c*g^2*x^2 - 4*c*f*g*x + 8*c*f^2 + 15*a*g^2)*sqrt(g*x + f)/g^3

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giac [A]  time = 0.15, size = 53, normalized size = 0.87 \[ \frac {2 \, {\left (15 \, \sqrt {g x + f} a + \frac {{\left (3 \, {\left (g x + f\right )}^{\frac {5}{2}} - 10 \, {\left (g x + f\right )}^{\frac {3}{2}} f + 15 \, \sqrt {g x + f} f^{2}\right )} c}{g^{2}}\right )}}{15 \, g} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)/(g*x+f)^(1/2),x, algorithm="giac")

[Out]

2/15*(15*sqrt(g*x + f)*a + (3*(g*x + f)^(5/2) - 10*(g*x + f)^(3/2)*f + 15*sqrt(g*x + f)*f^2)*c/g^2)/g

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maple [A]  time = 0.00, size = 41, normalized size = 0.67 \[ \frac {2 \sqrt {g x +f}\, \left (3 c \,x^{2} g^{2}-4 c f x g +15 a \,g^{2}+8 c \,f^{2}\right )}{15 g^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+a)/(g*x+f)^(1/2),x)

[Out]

2/15*(g*x+f)^(1/2)*(3*c*g^2*x^2-4*c*f*g*x+15*a*g^2+8*c*f^2)/g^3

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maxima [A]  time = 0.43, size = 53, normalized size = 0.87 \[ \frac {2 \, {\left (15 \, \sqrt {g x + f} a + \frac {{\left (3 \, {\left (g x + f\right )}^{\frac {5}{2}} - 10 \, {\left (g x + f\right )}^{\frac {3}{2}} f + 15 \, \sqrt {g x + f} f^{2}\right )} c}{g^{2}}\right )}}{15 \, g} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)/(g*x+f)^(1/2),x, algorithm="maxima")

[Out]

2/15*(15*sqrt(g*x + f)*a + (3*(g*x + f)^(5/2) - 10*(g*x + f)^(3/2)*f + 15*sqrt(g*x + f)*f^2)*c/g^2)/g

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mupad [B]  time = 2.56, size = 44, normalized size = 0.72 \[ \frac {2\,\sqrt {f+g\,x}\,\left (3\,c\,{\left (f+g\,x\right )}^2+15\,a\,g^2+15\,c\,f^2-10\,c\,f\,\left (f+g\,x\right )\right )}{15\,g^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + c*x^2)/(f + g*x)^(1/2),x)

[Out]

(2*(f + g*x)^(1/2)*(3*c*(f + g*x)^2 + 15*a*g^2 + 15*c*f^2 - 10*c*f*(f + g*x)))/(15*g^3)

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sympy [A]  time = 13.10, size = 150, normalized size = 2.46 \[ \begin {cases} \frac {- \frac {2 a f}{\sqrt {f + g x}} - 2 a \left (- \frac {f}{\sqrt {f + g x}} - \sqrt {f + g x}\right ) - \frac {2 c f \left (\frac {f^{2}}{\sqrt {f + g x}} + 2 f \sqrt {f + g x} - \frac {\left (f + g x\right )^{\frac {3}{2}}}{3}\right )}{g^{2}} - \frac {2 c \left (- \frac {f^{3}}{\sqrt {f + g x}} - 3 f^{2} \sqrt {f + g x} + f \left (f + g x\right )^{\frac {3}{2}} - \frac {\left (f + g x\right )^{\frac {5}{2}}}{5}\right )}{g^{2}}}{g} & \text {for}\: g \neq 0 \\\frac {a x + \frac {c x^{3}}{3}}{\sqrt {f}} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+a)/(g*x+f)**(1/2),x)

[Out]

Piecewise(((-2*a*f/sqrt(f + g*x) - 2*a*(-f/sqrt(f + g*x) - sqrt(f + g*x)) - 2*c*f*(f**2/sqrt(f + g*x) + 2*f*sq
rt(f + g*x) - (f + g*x)**(3/2)/3)/g**2 - 2*c*(-f**3/sqrt(f + g*x) - 3*f**2*sqrt(f + g*x) + f*(f + g*x)**(3/2)
- (f + g*x)**(5/2)/5)/g**2)/g, Ne(g, 0)), ((a*x + c*x**3/3)/sqrt(f), True))

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