3.8.71 \(\int \frac {(c x^2)^{3/2} (a+b x)}{x^4} \, dx\) [771]

Optimal. Leaf size=30 \[ b c \sqrt {c x^2}+\frac {a c \sqrt {c x^2} \log (x)}{x} \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {15, 45} \begin {gather*} \frac {a c \sqrt {c x^2} \log (x)}{x}+b c \sqrt {c x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((c*x^2)^(3/2)*(a + b*x))/x^4,x]

[Out]

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

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Mathematica [A]
time = 0.00, size = 21, normalized size = 0.70 \begin {gather*} \frac {\left (c x^2\right )^{3/2} (b x+a \log (x))}{x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((c*x^2)^(3/2)*(a + b*x))/x^4,x]

[Out]

((c*x^2)^(3/2)*(b*x + a*Log[x]))/x^3

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

Warning: Unable to verify antiderivative.

[In]

mathics('Integrate[((c*x^2)^(3/2)*(a + b*x))/x^4,x]')

[Out]

cought exception: maximum recursion depth exceeded

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Maple [A]
time = 0.02, size = 20, normalized size = 0.67

method	result	size
default	\(\frac {\left (c \,x^{2}\right )^{\frac {3}{2}} \left (b x +a \ln \left (x \right )\right )}{x^{3}}\)	\(20\)
risch	\(b c \sqrt {c \,x^{2}}+\frac {a c \ln \left (x \right ) \sqrt {c \,x^{2}}}{x}\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2)^(3/2)*(b*x+a)/x^4,x,method=_RETURNVERBOSE)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2)^(3/2)*(b*x+a)/x^4,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: Error executing code in Maxima: expt: undefined: 0 to a negative e
xponent.

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Fricas [A]
time = 0.32, size = 21, normalized size = 0.70 \begin {gather*} \frac {{\left (b c x + a c \log \left (x\right )\right )} \sqrt {c x^{2}}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2)^(3/2)*(b*x+a)/x^4,x, algorithm="fricas")

[Out]

(b*c*x + a*c*log(x))*sqrt(c*x^2)/x

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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 {3}{2}} \left (a + b x\right )}{x^{4}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2)**(3/2)*(b*x+a)/x**4,x)

[Out]

Integral((c*x**2)**(3/2)*(a + b*x)/x**4, x)

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Giac [A]
time = 0.00, size = 20, normalized size = 0.67 \begin {gather*} \sqrt {c} c \left (a \mathrm {sign}\left (x\right ) \ln \left |x\right |+b x \mathrm {sign}\left (x\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2)^(3/2)*(b*x+a)/x^4,x)

[Out]

(b*x*sgn(x) + a*log(abs(x))*sgn(x))*c^(3/2)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {{\left (c\,x^2\right )}^{3/2}\,\left (a+b\,x\right )}{x^4} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((c*x^2)^(3/2)*(a + b*x))/x^4,x)

[Out]

int(((c*x^2)^(3/2)*(a + b*x))/x^4, x)

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