3.9.67 \(\int \frac {(c x^2)^{3/2}}{x^6 (a+b x)} \, dx\) [867]

Optimal. Leaf size=88 \[ -\frac {c \sqrt {c x^2}}{2 a x^3}+\frac {b c \sqrt {c x^2}}{a^2 x^2}+\frac {b^2 c \sqrt {c x^2} \log (x)}{a^3 x}-\frac {b^2 c \sqrt {c x^2} \log (a+b x)}{a^3 x} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.01, antiderivative size = 88, 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, 46} \begin {gather*} \frac {b^2 c \sqrt {c x^2} \log (x)}{a^3 x}-\frac {b^2 c \sqrt {c x^2} \log (a+b x)}{a^3 x}+\frac {b c \sqrt {c x^2}}{a^2 x^2}-\frac {c \sqrt {c x^2}}{2 a x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c*x^2)^(3/2)/(x^6*(a + b*x)),x]

[Out]

-1/2*(c*Sqrt[c*x^2])/(a*x^3) + (b*c*Sqrt[c*x^2])/(a^2*x^2) + (b^2*c*Sqrt[c*x^2]*Log[x])/(a^3*x) - (b^2*c*Sqrt[
c*x^2]*Log[a + b*x])/(a^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 46

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}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && Lt
Q[m + n + 2, 0])

Rubi steps

\begin {align*} \int \frac {\left (c x^2\right )^{3/2}}{x^6 (a+b x)} \, dx &=\frac {\left (c \sqrt {c x^2}\right ) \int \frac {1}{x^3 (a+b x)} \, dx}{x}\\ &=\frac {\left (c \sqrt {c x^2}\right ) \int \left (\frac {1}{a x^3}-\frac {b}{a^2 x^2}+\frac {b^2}{a^3 x}-\frac {b^3}{a^3 (a+b x)}\right ) \, dx}{x}\\ &=-\frac {c \sqrt {c x^2}}{2 a x^3}+\frac {b c \sqrt {c x^2}}{a^2 x^2}+\frac {b^2 c \sqrt {c x^2} \log (x)}{a^3 x}-\frac {b^2 c \sqrt {c x^2} \log (a+b x)}{a^3 x}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.01, size = 53, normalized size = 0.60 \begin {gather*} \frac {\left (c x^2\right )^{3/2} \left (-a (a-2 b x)+2 b^2 x^2 \log (x)-2 b^2 x^2 \log (a+b x)\right )}{2 a^3 x^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c*x^2)^(3/2)/(x^6*(a + b*x)),x]

[Out]

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

________________________________________________________________________________________

Mathics [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {cought exception: maximum recursion depth exceeded while calling a Python object} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

mathics('Integrate[(c*x^2)^(3/2)/(x^6*(a + b*x)),x]')

[Out]

cought exception: maximum recursion depth exceeded while calling a Python object

________________________________________________________________________________________

Maple [A]
time = 0.14, size = 51, normalized size = 0.58

method	result	size
default	\(\frac {\left (c \,x^{2}\right )^{\frac {3}{2}} \left (2 b^{2} \ln \left (x \right ) x^{2}-2 b^{2} \ln \left (b x +a \right ) x^{2}+2 a b x -a^{2}\right )}{2 a^{3} x^{5}}\)	\(51\)
risch	\(\frac {c \sqrt {c \,x^{2}}\, \left (\frac {b x}{a^{2}}-\frac {1}{2 a}\right )}{x^{3}}+\frac {c \sqrt {c \,x^{2}}\, b^{2} \ln \left (-x \right )}{x \,a^{3}}-\frac {b^{2} c \ln \left (b x +a \right ) \sqrt {c \,x^{2}}}{a^{3} x}\)	\(73\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2)^(3/2)/x^6/(b*x+a),x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.29, size = 52, normalized size = 0.59 \begin {gather*} -\frac {b^{2} c^{\frac {3}{2}} \log \left (b x + a\right )}{a^{3}} + \frac {b^{2} c^{\frac {3}{2}} \log \left (x\right )}{a^{3}} + \frac {2 \, b c^{\frac {3}{2}} x - a c^{\frac {3}{2}}}{2 \, a^{2} x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2)^(3/2)/x^6/(b*x+a),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]
time = 0.30, size = 47, normalized size = 0.53 \begin {gather*} \frac {{\left (2 \, b^{2} c x^{2} \log \left (\frac {x}{b x + a}\right ) + 2 \, a b c x - a^{2} c\right )} \sqrt {c x^{2}}}{2 \, a^{3} x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2)^(3/2)/x^6/(b*x+a),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (c x^{2}\right )^{\frac {3}{2}}}{x^{6} \left (a + b x\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2)**(3/2)/x**6/(b*x+a),x)

[Out]

Integral((c*x**2)**(3/2)/(x**6*(a + b*x)), x)

________________________________________________________________________________________

Giac [F] N/A
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Limit: Max order reached or unable to make series expansion Error: Bad Argument Value} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2)^(3/2)/x^6/(b*x+a),x)

[Out]

Limit: Max order reached or unable to make series expansion Error: Bad Argument Value

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (c\,x^2\right )}^{3/2}}{x^6\,\left (a+b\,x\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2)^(3/2)/(x^6*(a + b*x)),x)

[Out]

int((c*x^2)^(3/2)/(x^6*(a + b*x)), x)

________________________________________________________________________________________