3.853 \(\int \frac {x^2 \sqrt {c x^2}}{a+b x} \, dx\)

Optimal. Leaf size=80 \[ -\frac {a^3 \sqrt {c x^2} \log (a+b x)}{b^4 x}+\frac {a^2 \sqrt {c x^2}}{b^3}-\frac {a x \sqrt {c x^2}}{2 b^2}+\frac {x^2 \sqrt {c x^2}}{3 b} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 80, 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} \[ \frac {a^2 \sqrt {c x^2}}{b^3}-\frac {a^3 \sqrt {c x^2} \log (a+b x)}{b^4 x}-\frac {a x \sqrt {c x^2}}{2 b^2}+\frac {x^2 \sqrt {c x^2}}{3 b} \]

Antiderivative was successfully verified.

[In]

Int[(x^2*Sqrt[c*x^2])/(a + b*x),x]

[Out]

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

________________________________________________________________________________________

Mathematica [A]  time = 0.02, size = 52, normalized size = 0.65 \[ \frac {c x \left (b x \left (6 a^2-3 a b x+2 b^2 x^2\right )-6 a^3 \log (a+b x)\right )}{6 b^4 \sqrt {c x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(c*x*(b*x*(6*a^2 - 3*a*b*x + 2*b^2*x^2) - 6*a^3*Log[a + b*x]))/(6*b^4*Sqrt[c*x^2])

________________________________________________________________________________________

fricas [A]  time = 0.42, size = 51, normalized size = 0.64 \[ \frac {{\left (2 \, b^{3} x^{3} - 3 \, a b^{2} x^{2} + 6 \, a^{2} b x - 6 \, a^{3} \log \left (b x + a\right )\right )} \sqrt {c x^{2}}}{6 \, b^{4} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(2*b^3*x^3 - 3*a*b^2*x^2 + 6*a^2*b*x - 6*a^3*log(b*x + a))*sqrt(c*x^2)/(b^4*x)

________________________________________________________________________________________

giac [A]  time = 0.94, size = 69, normalized size = 0.86 \[ -\frac {1}{6} \, \sqrt {c} {\left (\frac {6 \, a^{3} \log \left ({\left | b x + a \right |}\right ) \mathrm {sgn}\relax (x)}{b^{4}} - \frac {6 \, a^{3} \log \left ({\left | a \right |}\right ) \mathrm {sgn}\relax (x)}{b^{4}} - \frac {2 \, b^{2} x^{3} \mathrm {sgn}\relax (x) - 3 \, a b x^{2} \mathrm {sgn}\relax (x) + 6 \, a^{2} x \mathrm {sgn}\relax (x)}{b^{3}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*sqrt(c)*(6*a^3*log(abs(b*x + a))*sgn(x)/b^4 - 6*a^3*log(abs(a))*sgn(x)/b^4 - (2*b^2*x^3*sgn(x) - 3*a*b*x^
2*sgn(x) + 6*a^2*x*sgn(x))/b^3)

________________________________________________________________________________________

maple [A]  time = 0.01, size = 52, normalized size = 0.65 \[ -\frac {\sqrt {c \,x^{2}}\, \left (-2 b^{3} x^{3}+3 a \,b^{2} x^{2}+6 a^{3} \ln \left (b x +a \right )-6 a^{2} b x \right )}{6 b^{4} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(c*x^2)^(1/2)*(-2*b^3*x^3+3*a*b^2*x^2+6*a^3*ln(b*x+a)-6*a^2*b*x)/x/b^4

________________________________________________________________________________________

maxima [A]  time = 1.56, size = 110, normalized size = 1.38 \[ -\frac {\left (-1\right )^{\frac {2 \, c x}{b}} a^{3} \sqrt {c} \log \left (\frac {2 \, c x}{b}\right )}{b^{4}} - \frac {\left (-1\right )^{\frac {2 \, a c x}{b}} a^{3} \sqrt {c} \log \left (-\frac {2 \, a c x}{b {\left | b x + a \right |}}\right )}{b^{4}} - \frac {\sqrt {c x^{2}} a x}{2 \, b^{2}} + \frac {\sqrt {c x^{2}} a^{2}}{b^{3}} + \frac {\left (c x^{2}\right )^{\frac {3}{2}}}{3 \, b c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(-1)^(2*c*x/b)*a^3*sqrt(c)*log(2*c*x/b)/b^4 - (-1)^(2*a*c*x/b)*a^3*sqrt(c)*log(-2*a*c*x/(b*abs(b*x + a)))/b^4
- 1/2*sqrt(c*x^2)*a*x/b^2 + sqrt(c*x^2)*a^2/b^3 + 1/3*(c*x^2)^(3/2)/(b*c)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^2\,\sqrt {c\,x^2}}{a+b\,x} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2*(c*x^2)^(1/2))/(a + b*x),x)

[Out]

int((x^2*(c*x^2)^(1/2))/(a + b*x), x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2} \sqrt {c x^{2}}}{a + b x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________