3.9.8 \(\int \frac {(a+b x)^2}{x^3 (c x^2)^{5/2}} \, dx\)

Optimal. Leaf size=66 \[ -\frac {a^2}{7 c^2 x^6 \sqrt {c x^2}}-\frac {a b}{3 c^2 x^5 \sqrt {c x^2}}-\frac {b^2}{5 c^2 x^4 \sqrt {c x^2}} \]

________________________________________________________________________________________

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} \begin {gather*} -\frac {a^2}{7 c^2 x^6 \sqrt {c x^2}}-\frac {a b}{3 c^2 x^5 \sqrt {c x^2}}-\frac {b^2}{5 c^2 x^4 \sqrt {c x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^2/(x^3*(c*x^2)^(5/2)),x]

[Out]

-a^2/(7*c^2*x^6*Sqrt[c*x^2]) - (a*b)/(3*c^2*x^5*Sqrt[c*x^2]) - b^2/(5*c^2*x^4*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^3 \left (c x^2\right )^{5/2}} \, dx &=\frac {x \int \frac {(a+b x)^2}{x^8} \, dx}{c^2 \sqrt {c x^2}}\\ &=\frac {x \int \left (\frac {a^2}{x^8}+\frac {2 a b}{x^7}+\frac {b^2}{x^6}\right ) \, dx}{c^2 \sqrt {c x^2}}\\ &=-\frac {a^2}{7 c^2 x^6 \sqrt {c x^2}}-\frac {a b}{3 c^2 x^5 \sqrt {c x^2}}-\frac {b^2}{5 c^2 x^4 \sqrt {c x^2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 33, normalized size = 0.50 \begin {gather*} \frac {c \left (-15 a^2-35 a b x-21 b^2 x^2\right )}{105 \left (c x^2\right )^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^2/(x^3*(c*x^2)^(5/2)),x]

[Out]

(c*(-15*a^2 - 35*a*b*x - 21*b^2*x^2))/(105*(c*x^2)^(7/2))

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.03, size = 35, normalized size = 0.53 \begin {gather*} \frac {-15 a^2-35 a b x-21 b^2 x^2}{105 x^2 \left (c x^2\right )^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(a + b*x)^2/(x^3*(c*x^2)^(5/2)),x]

[Out]

(-15*a^2 - 35*a*b*x - 21*b^2*x^2)/(105*x^2*(c*x^2)^(5/2))

________________________________________________________________________________________

fricas [A]  time = 0.74, size = 34, normalized size = 0.52 \begin {gather*} -\frac {{\left (21 \, b^{2} x^{2} + 35 \, a b x + 15 \, a^{2}\right )} \sqrt {c x^{2}}}{105 \, c^{3} x^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^2/x^3/(c*x^2)^(5/2),x, algorithm="fricas")

[Out]

-1/105*(21*b^2*x^2 + 35*a*b*x + 15*a^2)*sqrt(c*x^2)/(c^3*x^8)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (b x + a\right )}^{2}}{\left (c x^{2}\right )^{\frac {5}{2}} x^{3}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^2/x^3/(c*x^2)^(5/2),x, algorithm="giac")

[Out]

integrate((b*x + a)^2/((c*x^2)^(5/2)*x^3), x)

________________________________________________________________________________________

maple [A]  time = 0.00, size = 32, normalized size = 0.48 \begin {gather*} -\frac {21 b^{2} x^{2}+35 a b x +15 a^{2}}{105 \left (c \,x^{2}\right )^{\frac {5}{2}} x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^2/x^3/(c*x^2)^(5/2),x)

[Out]

-1/105*(21*b^2*x^2+35*a*b*x+15*a^2)/x^2/(c*x^2)^(5/2)

________________________________________________________________________________________

maxima [A]  time = 1.37, size = 33, normalized size = 0.50 \begin {gather*} -\frac {b^{2}}{5 \, c^{\frac {5}{2}} x^{5}} - \frac {a b}{3 \, c^{\frac {5}{2}} x^{6}} - \frac {a^{2}}{7 \, c^{\frac {5}{2}} x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^2/x^3/(c*x^2)^(5/2),x, algorithm="maxima")

[Out]

-1/5*b^2/(c^(5/2)*x^5) - 1/3*a*b/(c^(5/2)*x^6) - 1/7*a^2/(c^(5/2)*x^7)

________________________________________________________________________________________

mupad [B]  time = 0.18, size = 42, normalized size = 0.64 \begin {gather*} -\frac {15\,a^2\,\sqrt {x^2}+21\,b^2\,x^2\,\sqrt {x^2}+35\,a\,b\,x\,\sqrt {x^2}}{105\,c^{5/2}\,x^8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^2/(x^3*(c*x^2)^(5/2)),x)

[Out]

-(15*a^2*(x^2)^(1/2) + 21*b^2*x^2*(x^2)^(1/2) + 35*a*b*x*(x^2)^(1/2))/(105*c^(5/2)*x^8)

________________________________________________________________________________________

sympy [A]  time = 1.70, size = 56, normalized size = 0.85 \begin {gather*} - \frac {a^{2}}{7 c^{\frac {5}{2}} x^{2} \left (x^{2}\right )^{\frac {5}{2}}} - \frac {a b}{3 c^{\frac {5}{2}} x \left (x^{2}\right )^{\frac {5}{2}}} - \frac {b^{2}}{5 c^{\frac {5}{2}} \left (x^{2}\right )^{\frac {5}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**2/x**3/(c*x**2)**(5/2),x)

[Out]

-a**2/(7*c**(5/2)*x**2*(x**2)**(5/2)) - a*b/(3*c**(5/2)*x*(x**2)**(5/2)) - b**2/(5*c**(5/2)*(x**2)**(5/2))

________________________________________________________________________________________