3.9.76 \(\int \frac {(c x^2)^{5/2}}{x^7 (a+b x)} \, dx\) [876]

Optimal. Leaf size=70 \[ -\frac {c^2 \sqrt {c x^2}}{a x^2}-\frac {b c^2 \sqrt {c x^2} \log (x)}{a^2 x}+\frac {b c^2 \sqrt {c x^2} \log (a+b x)}{a^2 x} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 70, 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 c^2 \sqrt {c x^2} \log (x)}{a^2 x}+\frac {b c^2 \sqrt {c x^2} \log (a+b x)}{a^2 x}-\frac {c^2 \sqrt {c x^2}}{a x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c*x^2)^(5/2)/(x^7*(a + b*x)),x]

[Out]

-((c^2*Sqrt[c*x^2])/(a*x^2)) - (b*c^2*Sqrt[c*x^2]*Log[x])/(a^2*x) + (b*c^2*Sqrt[c*x^2]*Log[a + b*x])/(a^2*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 )^{5/2}}{x^7 (a+b x)} \, dx &=\frac {\left (c^2 \sqrt {c x^2}\right ) \int \frac {1}{x^2 (a+b x)} \, dx}{x}\\ &=\frac {\left (c^2 \sqrt {c x^2}\right ) \int \left (\frac {1}{a x^2}-\frac {b}{a^2 x}+\frac {b^2}{a^2 (a+b x)}\right ) \, dx}{x}\\ &=-\frac {c^2 \sqrt {c x^2}}{a x^2}-\frac {b c^2 \sqrt {c x^2} \log (x)}{a^2 x}+\frac {b c^2 \sqrt {c x^2} \log (a+b x)}{a^2 x}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 34, normalized size = 0.49 \begin {gather*} -\frac {c^3 (a+b x \log (x)-b x \log (a+b x))}{a^2 \sqrt {c x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c*x^2)^(5/2)/(x^7*(a + b*x)),x]

[Out]

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

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Maple [A]
time = 0.13, size = 33, normalized size = 0.47

method	result	size
default	\(-\frac {\left (c \,x^{2}\right )^{\frac {5}{2}} \left (b x \ln \left (x \right )-b \ln \left (b x +a \right ) x +a \right )}{a^{2} x^{6}}\)	\(33\)
risch	\(-\frac {c^{2} \sqrt {c \,x^{2}}}{a \,x^{2}}+\frac {c^{2} \sqrt {c \,x^{2}}\, b \ln \left (-b x -a \right )}{x \,a^{2}}-\frac {b \,c^{2} \ln \left (x \right ) \sqrt {c \,x^{2}}}{a^{2} x}\)	\(68\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2)^(5/2)/x^7/(b*x+a),x,method=_RETURNVERBOSE)

[Out]

-(c*x^2)^(5/2)*(b*x*ln(x)-b*ln(b*x+a)*x+a)/a^2/x^6

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Maxima [A]
time = 0.30, size = 37, normalized size = 0.53 \begin {gather*} \frac {b c^{\frac {5}{2}} \log \left (b x + a\right )}{a^{2}} - \frac {b c^{\frac {5}{2}} \log \left (x\right )}{a^{2}} - \frac {c^{\frac {5}{2}}}{a x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2)^(5/2)/x^7/(b*x+a),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.90, size = 37, normalized size = 0.53 \begin {gather*} \frac {{\left (b c^{2} x \log \left (\frac {b x + a}{x}\right ) - a c^{2}\right )} \sqrt {c x^{2}}}{a^{2} x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2)^(5/2)/x^7/(b*x+a),x, algorithm="fricas")

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (c x^{2}\right )^{\frac {5}{2}}}{x^{7} \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**7/(b*x+a),x)

[Out]

Integral((c*x**2)**(5/2)/(x**7*(a + b*x)), x)

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2)^(5/2)/x^7/(b*x+a),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Warning, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):Ch
eck [abs(sa

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (c\,x^2\right )}^{5/2}}{x^7\,\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^7*(a + b*x)),x)

[Out]

int((c*x^2)^(5/2)/(x^7*(a + b*x)), x)

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