3.8.95 \(\int \frac {a+b x}{x^4 (c x^2)^{3/2}} \, dx\) [795]

Optimal. Leaf size=41 \[ -\frac {a}{6 c x^5 \sqrt {c x^2}}-\frac {b}{5 c x^4 \sqrt {c x^2}} \]

[Out]

-1/6*a/c/x^5/(c*x^2)^(1/2)-1/5*b/c/x^4/(c*x^2)^(1/2)

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Rubi [A]
time = 0.01, antiderivative size = 41, 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}{6 c x^5 \sqrt {c x^2}}-\frac {b}{5 c x^4 \sqrt {c x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)/(x^4*(c*x^2)^(3/2)),x]

[Out]

-1/6*a/(c*x^5*Sqrt[c*x^2]) - b/(5*c*x^4*Sqrt[c*x^2])

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 {a+b x}{x^4 \left (c x^2\right )^{3/2}} \, dx &=\frac {x \int \frac {a+b x}{x^7} \, dx}{c \sqrt {c x^2}}\\ &=\frac {x \int \left (\frac {a}{x^7}+\frac {b}{x^6}\right ) \, dx}{c \sqrt {c x^2}}\\ &=-\frac {a}{6 c x^5 \sqrt {c x^2}}-\frac {b}{5 c x^4 \sqrt {c x^2}}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 24, normalized size = 0.59 \begin {gather*} \frac {-5 a-6 b x}{30 x^3 \left (c x^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-5*a - 6*b*x)/(30*x^3*(c*x^2)^(3/2))

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Mathics [A]
time = 1.85, size = 22, normalized size = 0.54 \begin {gather*} \frac {\left (-\frac {a}{6}-\frac {b x}{5}\right ) {\left (c x^2\right )}^{\frac {3}{2}}}{c^3 x^9} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-a / 6 - b x / 5) (c x ^ 2) ^ (3 / 2) / (c ^ 3 x ^ 9)

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Maple [A]
time = 0.03, size = 21, normalized size = 0.51

method	result	size
gosper	\(-\frac {6 b x +5 a}{30 x^{3} \left (c \,x^{2}\right )^{\frac {3}{2}}}\)	\(21\)
default	\(-\frac {6 b x +5 a}{30 x^{3} \left (c \,x^{2}\right )^{\frac {3}{2}}}\)	\(21\)
risch	\(\frac {-\frac {b x}{5}-\frac {a}{6}}{c \,x^{5} \sqrt {c \,x^{2}}}\)	\(23\)
trager	\(\frac {\left (-1+x \right ) \left (5 a \,x^{5}+6 b \,x^{5}+5 a \,x^{4}+6 b \,x^{4}+5 a \,x^{3}+6 b \,x^{3}+5 a \,x^{2}+6 x^{2} b +5 a x +6 b x +5 a \right ) \sqrt {c \,x^{2}}}{30 c^{2} x^{7}}\)	\(79\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/30*(6*b*x+5*a)/x^3/(c*x^2)^(3/2)

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Maxima [A]
time = 0.27, size = 19, normalized size = 0.46 \begin {gather*} -\frac {b}{5 \, c^{\frac {3}{2}} x^{5}} - \frac {a}{6 \, c^{\frac {3}{2}} x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/5*b/(c^(3/2)*x^5) - 1/6*a/(c^(3/2)*x^6)

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Fricas [A]
time = 0.29, size = 23, normalized size = 0.56 \begin {gather*} -\frac {\sqrt {c x^{2}} {\left (6 \, b x + 5 \, a\right )}}{30 \, c^{2} x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/30*sqrt(c*x^2)*(6*b*x + 5*a)/(c^2*x^7)

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Sympy [A]
time = 0.35, size = 31, normalized size = 0.76 \begin {gather*} - \frac {a}{6 x^{3} \left (c x^{2}\right )^{\frac {3}{2}}} - \frac {b}{5 x^{2} \left (c x^{2}\right )^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a/(6*x**3*(c*x**2)**(3/2)) - b/(5*x**2*(c*x**2)**(3/2))

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Giac [A]
time = 0.00, size = 28, normalized size = 0.68 \begin {gather*} \frac {-6 b x-5 a}{\sqrt {c} c\cdot 30 \left (x^{6} \mathrm {sign}\left (x\right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/30*(6*b*x + 5*a)/(c^(3/2)*x^6*sgn(x))

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Mupad [B]
time = 0.15, size = 26, normalized size = 0.63 \begin {gather*} -\frac {5\,a\,\sqrt {x^2}+6\,b\,x\,\sqrt {x^2}}{30\,c^{3/2}\,x^7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)/(x^4*(c*x^2)^(3/2)),x)

[Out]

-(5*a*(x^2)^(1/2) + 6*b*x*(x^2)^(1/2))/(30*c^(3/2)*x^7)

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