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

Optimal. Leaf size=66 \[ -\frac {a^2}{4 c x^3 \sqrt {c x^2}}-\frac {2 a b}{3 c x^2 \sqrt {c x^2}}-\frac {b^2}{2 c x \sqrt {c x^2}} \]

[Out]

-1/4*a^2/c/x^3/(c*x^2)^(1/2)-2/3*a*b/c/x^2/(c*x^2)^(1/2)-1/2*b^2/c/x/(c*x^2)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {15, 43} \[ -\frac {a^2}{4 c x^3 \sqrt {c x^2}}-\frac {2 a b}{3 c x^2 \sqrt {c x^2}}-\frac {b^2}{2 c x \sqrt {c x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-a^2/(4*c*x^3*Sqrt[c*x^2]) - (2*a*b)/(3*c*x^2*Sqrt[c*x^2]) - b^2/(2*c*x*Sqrt[c*x^2])

Rule 15

Int[(u_.)*((a_.)*(x_)^(n_))^(m_), x_Symbol] :> Dist[(a^IntPart[m]*(a*x^n)^FracPart[m])/x^(n*FracPart[m]), Int[
u*x^(m*n), x], x] /; FreeQ[{a, m, n}, x] &&  !IntegerQ[m]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 38, normalized size = 0.58 \[ -\frac {\sqrt {c x^2} \left (3 a^2+8 a b x+6 b^2 x^2\right )}{12 c^2 x^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/12*(Sqrt[c*x^2]*(3*a^2 + 8*a*b*x + 6*b^2*x^2))/(c^2*x^5)

________________________________________________________________________________________

fricas [A]  time = 0.42, size = 34, normalized size = 0.52 \[ -\frac {{\left (6 \, b^{2} x^{2} + 8 \, a b x + 3 \, a^{2}\right )} \sqrt {c x^{2}}}{12 \, c^{2} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*(6*b^2*x^2 + 8*a*b*x + 3*a^2)*sqrt(c*x^2)/(c^2*x^5)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sage0*x

________________________________________________________________________________________

maple [A]  time = 0.01, size = 32, normalized size = 0.48 \[ -\frac {6 b^{2} x^{2}+8 a b x +3 a^{2}}{12 \left (c \,x^{2}\right )^{\frac {3}{2}} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*(6*b^2*x^2+8*a*b*x+3*a^2)/x/(c*x^2)^(3/2)

________________________________________________________________________________________

maxima [A]  time = 1.31, size = 33, normalized size = 0.50 \[ -\frac {b^{2}}{2 \, c^{\frac {3}{2}} x^{2}} - \frac {2 \, a b}{3 \, c^{\frac {3}{2}} x^{3}} - \frac {a^{2}}{4 \, c^{\frac {3}{2}} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*b^2/(c^(3/2)*x^2) - 2/3*a*b/(c^(3/2)*x^3) - 1/4*a^2/(c^(3/2)*x^4)

________________________________________________________________________________________

mupad [B]  time = 0.19, size = 42, normalized size = 0.64 \[ -\frac {3\,a^2\,\sqrt {x^2}+6\,b^2\,x^2\,\sqrt {x^2}+8\,a\,b\,x\,\sqrt {x^2}}{12\,c^{3/2}\,x^5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(3*a^2*(x^2)^(1/2) + 6*b^2*x^2*(x^2)^(1/2) + 8*a*b*x*(x^2)^(1/2))/(12*c^(3/2)*x^5)

________________________________________________________________________________________

sympy [A]  time = 0.81, size = 56, normalized size = 0.85 \[ - \frac {a^{2}}{4 c^{\frac {3}{2}} x \left (x^{2}\right )^{\frac {3}{2}}} - \frac {2 a b}{3 c^{\frac {3}{2}} \left (x^{2}\right )^{\frac {3}{2}}} - \frac {b^{2} x}{2 c^{\frac {3}{2}} \left (x^{2}\right )^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a**2/(4*c**(3/2)*x*(x**2)**(3/2)) - 2*a*b/(3*c**(3/2)*(x**2)**(3/2)) - b**2*x/(2*c**(3/2)*(x**2)**(3/2))

________________________________________________________________________________________