\(\int \frac {x^2 \sqrt {c x^2}}{a+b x} \, dx\) [853]

Optimal result

Integrand size = 20, antiderivative size = 80 \[ \int \frac {x^2 \sqrt {c x^2}}{a+b x} \, dx=\frac {a^2 \sqrt {c x^2}}{b^3}-\frac {a x \sqrt {c x^2}}{2 b^2}+\frac {x^2 \sqrt {c x^2}}{3 b}-\frac {a^3 \sqrt {c x^2} \log (a+b x)}{b^4 x} \]

[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] (verified)

Time = 0.02 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {15, 45} \[ \int \frac {x^2 \sqrt {c x^2}}{a+b x} \, dx=-\frac {a^3 \sqrt {c x^2} \log (a+b x)}{b^4 x}+\frac {a^2 \sqrt {c x^2}}{b^3}-\frac {a x \sqrt {c x^2}}{2 b^2}+\frac {x^2 \sqrt {c x^2}}{3 b} \]

[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 45

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*} \text {integral}& = \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] (verified)

Time = 0.04 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.68 \[ \int \frac {x^2 \sqrt {c x^2}}{a+b x} \, dx=\sqrt {c x^2} \left (\frac {6 a^2-3 a b x+2 b^2 x^2}{6 b^3}-\frac {a^3 \log (a+b x)}{b^4 x}\right ) \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.23 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.65

method	result	size
default	\(-\frac {\sqrt {c \,x^{2}}\, \left (-2 b^{3} x^{3}+3 a \,b^{2} x^{2}+6 a^{3} \ln \left (b x +a \right )-6 a^{2} b x \right )}{6 x \,b^{4}}\)	\(52\)
risch	\(\frac {\sqrt {c \,x^{2}}\, \left (\frac {1}{3} b^{2} x^{3}-\frac {1}{2} a b \,x^{2}+a^{2} x \right )}{x \,b^{3}}-\frac {a^{3} \ln \left (b x +a \right ) \sqrt {c \,x^{2}}}{b^{4} x}\)	\(61\)

[In]

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

[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

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.64 \[ \int \frac {x^2 \sqrt {c x^2}}{a+b x} \, dx=\frac {{\left (2 \, b^{3} x^{3} - 3 \, a b^{2} x^{2} + 6 \, a^{2} b x - 6 \, a^{3} \log \left (b x + a\right )\right )} \sqrt {c x^{2}}}{6 \, b^{4} x} \]

[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)

Sympy [F]

\[ \int \frac {x^2 \sqrt {c x^2}}{a+b x} \, dx=\int \frac {x^{2} \sqrt {c x^{2}}}{a + b x}\, dx \]

[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)

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.38 \[ \int \frac {x^2 \sqrt {c x^2}}{a+b x} \, dx=-\frac {\left (-1\right )^{\frac {2 \, c x}{b}} a^{3} \sqrt {c} \log \left (\frac {2 \, c x}{b}\right )}{b^{4}} - \frac {\left (-1\right )^{\frac {2 \, a c x}{b}} a^{3} \sqrt {c} \log \left (-\frac {2 \, a c x}{b {\left | b x + a \right |}}\right )}{b^{4}} - \frac {\sqrt {c x^{2}} a x}{2 \, b^{2}} + \frac {\sqrt {c x^{2}} a^{2}}{b^{3}} + \frac {\left (c x^{2}\right )^{\frac {3}{2}}}{3 \, b c} \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.38 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.86 \[ \int \frac {x^2 \sqrt {c x^2}}{a+b x} \, dx=-\frac {1}{6} \, \sqrt {c} {\left (\frac {6 \, a^{3} \log \left ({\left | b x + a \right |}\right ) \mathrm {sgn}\left (x\right )}{b^{4}} - \frac {6 \, a^{3} \log \left ({\left | a \right |}\right ) \mathrm {sgn}\left (x\right )}{b^{4}} - \frac {2 \, b^{2} x^{3} \mathrm {sgn}\left (x\right ) - 3 \, a b x^{2} \mathrm {sgn}\left (x\right ) + 6 \, a^{2} x \mathrm {sgn}\left (x\right )}{b^{3}}\right )} \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {x^2 \sqrt {c x^2}}{a+b x} \, dx=\int \frac {x^2\,\sqrt {c\,x^2}}{a+b\,x} \,d x \]

[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)