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

Optimal. Leaf size=67 \[ \frac {a^2 c^2 \sqrt {c x^2} \log (a+b x)}{b^3 x}-\frac {a c^2 \sqrt {c x^2}}{b^2}+\frac {c^2 x \sqrt {c x^2}}{2 b} \]

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 67, 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 c^2 \sqrt {c x^2} \log (a+b x)}{b^3 x}-\frac {a c^2 \sqrt {c x^2}}{b^2}+\frac {c^2 x \sqrt {c x^2}}{2 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c*x^2)^(5/2)/(x^3*(a + b*x)),x]

[Out]

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

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

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 42, normalized size = 0.63 \begin {gather*} \frac {c^3 x \left (2 a^2 \log (a+b x)+b x (b x-2 a)\right )}{2 b^3 \sqrt {c x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.04, size = 44, normalized size = 0.66 \begin {gather*} \left (c x^2\right )^{5/2} \left (\frac {a^2 \log (a+b x)}{b^3 x^5}+\frac {b x-2 a}{2 b^2 x^4}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(c*x^2)^(5/2)*((-2*a + b*x)/(2*b^2*x^4) + (a^2*Log[a + b*x])/(b^3*x^5))

________________________________________________________________________________________

fricas [A]  time = 1.23, size = 48, normalized size = 0.72 \begin {gather*} \frac {{\left (b^{2} c^{2} x^{2} - 2 \, a b c^{2} x + 2 \, a^{2} c^{2} \log \left (b x + a\right )\right )} \sqrt {c x^{2}}}{2 \, b^{3} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(b^2*c^2*x^2 - 2*a*b*c^2*x + 2*a^2*c^2*log(b*x + a))*sqrt(c*x^2)/(b^3*x)

________________________________________________________________________________________

giac [A]  time = 1.13, size = 66, normalized size = 0.99 \begin {gather*} \frac {1}{2} \, {\left (\frac {2 \, a^{2} c^{2} \log \left ({\left | b x + a \right |}\right ) \mathrm {sgn}\relax (x)}{b^{3}} - \frac {2 \, a^{2} c^{2} \log \left ({\left | a \right |}\right ) \mathrm {sgn}\relax (x)}{b^{3}} + \frac {b c^{2} x^{2} \mathrm {sgn}\relax (x) - 2 \, a c^{2} x \mathrm {sgn}\relax (x)}{b^{2}}\right )} \sqrt {c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(2*a^2*c^2*log(abs(b*x + a))*sgn(x)/b^3 - 2*a^2*c^2*log(abs(a))*sgn(x)/b^3 + (b*c^2*x^2*sgn(x) - 2*a*c^2*x
*sgn(x))/b^2)*sqrt(c)

________________________________________________________________________________________

maple [A]  time = 0.01, size = 40, normalized size = 0.60 \begin {gather*} \frac {\left (c \,x^{2}\right )^{\frac {5}{2}} \left (b^{2} x^{2}+2 a^{2} \ln \left (b x +a \right )-2 a b x \right )}{2 b^{3} x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(c*x^2)^(5/2)*(b^2*x^2+2*a^2*ln(b*x+a)-2*a*b*x)/x^5/b^3

________________________________________________________________________________________

maxima [A]  time = 1.50, size = 97, normalized size = 1.45 \begin {gather*} \frac {\left (-1\right )^{\frac {2 \, c x}{b}} a^{2} c^{\frac {5}{2}} \log \left (\frac {2 \, c x}{b}\right )}{b^{3}} + \frac {\left (-1\right )^{\frac {2 \, a c x}{b}} a^{2} c^{\frac {5}{2}} \log \left (-\frac {2 \, a c x}{b {\left | b x + a \right |}}\right )}{b^{3}} + \frac {\sqrt {c x^{2}} c^{2} x}{2 \, b} - \frac {\sqrt {c x^{2}} a c^{2}}{b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-1)^(2*c*x/b)*a^2*c^(5/2)*log(2*c*x/b)/b^3 + (-1)^(2*a*c*x/b)*a^2*c^(5/2)*log(-2*a*c*x/(b*abs(b*x + a)))/b^3
+ 1/2*sqrt(c*x^2)*c^2*x/b - sqrt(c*x^2)*a*c^2/b^2

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (c\,x^2\right )}^{5/2}}{x^3\,\left (a+b\,x\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2)^(5/2)/(x^3*(a + b*x)),x)

[Out]

int((c*x^2)^(5/2)/(x^3*(a + b*x)), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________