3.5.1 \(\int \frac {a+c x^2}{(f+g x)^{3/2}} \, dx\)

Optimal. Leaf size=59 \[ -\frac {2 \left (a g^2+c f^2\right )}{g^3 \sqrt {f+g x}}+\frac {2 c (f+g x)^{3/2}}{3 g^3}-\frac {4 c f \sqrt {f+g x}}{g^3} \]

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Rubi [A]  time = 0.03, antiderivative size = 59, 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} \begin {gather*} -\frac {2 \left (a g^2+c f^2\right )}{g^3 \sqrt {f+g x}}+\frac {2 c (f+g x)^{3/2}}{3 g^3}-\frac {4 c f \sqrt {f+g x}}{g^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + c*x^2)/(f + g*x)^(3/2),x]

[Out]

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

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Mathematica [A]  time = 0.03, size = 43, normalized size = 0.73 \begin {gather*} \frac {2 \left (c \left (-8 f^2-4 f g x+g^2 x^2\right )-3 a g^2\right )}{3 g^3 \sqrt {f+g x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + c*x^2)/(f + g*x)^(3/2),x]

[Out]

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

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IntegrateAlgebraic [A]  time = 0.03, size = 47, normalized size = 0.80 \begin {gather*} \frac {2 \left (-3 a g^2-3 c f^2-6 c f (f+g x)+c (f+g x)^2\right )}{3 g^3 \sqrt {f+g x}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(a + c*x^2)/(f + g*x)^(3/2),x]

[Out]

(2*(-3*c*f^2 - 3*a*g^2 - 6*c*f*(f + g*x) + c*(f + g*x)^2))/(3*g^3*Sqrt[f + g*x])

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fricas [A]  time = 0.38, size = 49, normalized size = 0.83 \begin {gather*} \frac {2 \, {\left (c g^{2} x^{2} - 4 \, c f g x - 8 \, c f^{2} - 3 \, a g^{2}\right )} \sqrt {g x + f}}{3 \, {\left (g^{4} x + f g^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)/(g*x+f)^(3/2),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.18, size = 56, normalized size = 0.95 \begin {gather*} -\frac {2 \, {\left (c f^{2} + a g^{2}\right )}}{\sqrt {g x + f} g^{3}} + \frac {2 \, {\left ({\left (g x + f\right )}^{\frac {3}{2}} c g^{6} - 6 \, \sqrt {g x + f} c f g^{6}\right )}}{3 \, g^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)/(g*x+f)^(3/2),x, algorithm="giac")

[Out]

-2*(c*f^2 + a*g^2)/(sqrt(g*x + f)*g^3) + 2/3*((g*x + f)^(3/2)*c*g^6 - 6*sqrt(g*x + f)*c*f*g^6)/g^9

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maple [A]  time = 0.00, size = 41, normalized size = 0.69 \begin {gather*} -\frac {2 \left (-c \,x^{2} g^{2}+4 c f x g +3 a \,g^{2}+8 c \,f^{2}\right )}{3 \sqrt {g x +f}\, g^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+a)/(g*x+f)^(3/2),x)

[Out]

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

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maxima [A]  time = 0.44, size = 54, normalized size = 0.92 \begin {gather*} \frac {2 \, {\left (\frac {{\left (g x + f\right )}^{\frac {3}{2}} c - 6 \, \sqrt {g x + f} c f}{g^{2}} - \frac {3 \, {\left (c f^{2} + a g^{2}\right )}}{\sqrt {g x + f} g^{2}}\right )}}{3 \, g} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)/(g*x+f)^(3/2),x, algorithm="maxima")

[Out]

2/3*(((g*x + f)^(3/2)*c - 6*sqrt(g*x + f)*c*f)/g^2 - 3*(c*f^2 + a*g^2)/(sqrt(g*x + f)*g^2))/g

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mupad [B]  time = 0.05, size = 44, normalized size = 0.75 \begin {gather*} -\frac {6\,a\,g^2-2\,c\,{\left (f+g\,x\right )}^2+6\,c\,f^2+12\,c\,f\,\left (f+g\,x\right )}{3\,g^3\,\sqrt {f+g\,x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + c*x^2)/(f + g*x)^(3/2),x)

[Out]

-(6*a*g^2 - 2*c*(f + g*x)^2 + 6*c*f^2 + 12*c*f*(f + g*x))/(3*g^3*(f + g*x)^(1/2))

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sympy [A]  time = 10.14, size = 58, normalized size = 0.98 \begin {gather*} - \frac {4 c f \sqrt {f + g x}}{g^{3}} + \frac {2 c \left (f + g x\right )^{\frac {3}{2}}}{3 g^{3}} - \frac {2 \left (a g^{2} + c f^{2}\right )}{g^{3} \sqrt {f + g x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+a)/(g*x+f)**(3/2),x)

[Out]

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

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