\(\int \frac {(c x^2)^{5/2}}{x^6 (a+b x)} \, dx\) [875]

Optimal result

Integrand size = 20, antiderivative size = 48 \[ \int \frac {\left (c x^2\right )^{5/2}}{x^6 (a+b x)} \, dx=\frac {c^2 \sqrt {c x^2} \log (x)}{a x}-\frac {c^2 \sqrt {c x^2} \log (a+b x)}{a x} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {15, 36, 29, 31} \[ \int \frac {\left (c x^2\right )^{5/2}}{x^6 (a+b x)} \, dx=\frac {c^2 \sqrt {c x^2} \log (x)}{a x}-\frac {c^2 \sqrt {c x^2} \log (a+b x)}{a x} \]

[In]

Int[(c*x^2)^(5/2)/(x^6*(a + b*x)),x]

[Out]

(c^2*Sqrt[c*x^2]*Log[x])/(a*x) - (c^2*Sqrt[c*x^2]*Log[a + b*x])/(a*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 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {\left (c^2 \sqrt {c x^2}\right ) \int \frac {1}{x (a+b x)} \, dx}{x} \\ & = \frac {\left (c^2 \sqrt {c x^2}\right ) \int \frac {1}{x} \, dx}{a x}-\frac {\left (b c^2 \sqrt {c x^2}\right ) \int \frac {1}{a+b x} \, dx}{a x} \\ & = \frac {c^2 \sqrt {c x^2} \log (x)}{a x}-\frac {c^2 \sqrt {c x^2} \log (a+b x)}{a x} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.62 \[ \int \frac {\left (c x^2\right )^{5/2}}{x^6 (a+b x)} \, dx=\frac {c^3 x (\log (x)-\log (a (a+b x)))}{a \sqrt {c x^2}} \]

[In]

Integrate[(c*x^2)^(5/2)/(x^6*(a + b*x)),x]

[Out]

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

Maple [A] (verified)

Time = 0.34 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.54

method	result	size
default	\(\frac {\left (c \,x^{2}\right )^{\frac {5}{2}} \left (\ln \left (x \right )-\ln \left (b x +a \right )\right )}{a \,x^{5}}\)	\(26\)
risch	\(\frac {c^{2} \sqrt {c \,x^{2}}\, \ln \left (-x \right )}{x a}-\frac {c^{2} \ln \left (b x +a \right ) \sqrt {c \,x^{2}}}{a x}\)	\(47\)

[In]

int((c*x^2)^(5/2)/x^6/(b*x+a),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.46 \[ \int \frac {\left (c x^2\right )^{5/2}}{x^6 (a+b x)} \, dx=\left [\frac {\sqrt {c x^{2}} c^{2} \log \left (\frac {x}{b x + a}\right )}{a x}, \frac {2 \, \sqrt {-c} c^{2} \arctan \left (\frac {\sqrt {c x^{2}} {\left (2 \, b x + a\right )} \sqrt {-c}}{a c x}\right )}{a}\right ] \]

[In]

integrate((c*x^2)^(5/2)/x^6/(b*x+a),x, algorithm="fricas")

[Out]

[sqrt(c*x^2)*c^2*log(x/(b*x + a))/(a*x), 2*sqrt(-c)*c^2*arctan(sqrt(c*x^2)*(2*b*x + a)*sqrt(-c)/(a*c*x))/a]

Sympy [F]

\[ \int \frac {\left (c x^2\right )^{5/2}}{x^6 (a+b x)} \, dx=\int \frac {\left (c x^{2}\right )^{\frac {5}{2}}}{x^{6} \left (a + b x\right )}\, dx \]

[In]

integrate((c*x**2)**(5/2)/x**6/(b*x+a),x)

[Out]

Integral((c*x**2)**(5/2)/(x**6*(a + b*x)), x)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.50 \[ \int \frac {\left (c x^2\right )^{5/2}}{x^6 (a+b x)} \, dx=-\frac {c^{\frac {5}{2}} \log \left (b x + a\right )}{a} + \frac {c^{\frac {5}{2}} \log \left (x\right )}{a} \]

[In]

integrate((c*x^2)^(5/2)/x^6/(b*x+a),x, algorithm="maxima")

[Out]

-c^(5/2)*log(b*x + a)/a + c^(5/2)*log(x)/a

Giac [F(-2)]

Exception generated. \[ \int \frac {\left (c x^2\right )^{5/2}}{x^6 (a+b x)} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((c*x^2)^(5/2)/x^6/(b*x+a),x, algorithm="giac")

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Limit: Max order reached or unable to make series expansion Error: Bad Argument Value

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (c x^2\right )^{5/2}}{x^6 (a+b x)} \, dx=\int \frac {{\left (c\,x^2\right )}^{5/2}}{x^6\,\left (a+b\,x\right )} \,d x \]

[In]

int((c*x^2)^(5/2)/(x^6*(a + b*x)),x)

[Out]

int((c*x^2)^(5/2)/(x^6*(a + b*x)), x)