3.9.2 \(\int \frac {a+b x}{x^3 (c x^2)^{5/2}} \, dx\) [802]

Optimal. Leaf size=41 \[ -\frac {a}{7 c^2 x^6 \sqrt {c x^2}}-\frac {b}{6 c^2 x^5 \sqrt {c x^2}} \]

[Out]

-1/7*a/c^2/x^6/(c*x^2)^(1/2)-1/6*b/c^2/x^5/(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}{7 c^2 x^6 \sqrt {c x^2}}-\frac {b}{6 c^2 x^5 \sqrt {c x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)/(x^3*(c*x^2)^(5/2)),x]

[Out]

-1/7*a/(c^2*x^6*Sqrt[c*x^2]) - b/(6*c^2*x^5*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^3 \left (c x^2\right )^{5/2}} \, dx &=\frac {x \int \frac {a+b x}{x^8} \, dx}{c^2 \sqrt {c x^2}}\\ &=\frac {x \int \left (\frac {a}{x^8}+\frac {b}{x^7}\right ) \, dx}{c^2 \sqrt {c x^2}}\\ &=-\frac {a}{7 c^2 x^6 \sqrt {c x^2}}-\frac {b}{6 c^2 x^5 \sqrt {c x^2}}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 22, normalized size = 0.54 \begin {gather*} \frac {c (-6 a-7 b x)}{42 \left (c x^2\right )^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)/(x^3*(c*x^2)^(5/2)),x]

[Out]

(c*(-6*a - 7*b*x))/(42*(c*x^2)^(7/2))

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Mathics [A]
time = 1.94, size = 22, normalized size = 0.54 \begin {gather*} \frac {\left (-\frac {a}{7}-\frac {b x}{6}\right ) {\left (c x^2\right )}^{\frac {5}{2}}}{c^5 x^{12}} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[(a + b*x)/(x^3*(c*x^2)^(5/2)),x]')

[Out]

(-a / 7 - b x / 6) (c x ^ 2) ^ (5 / 2) / (c ^ 5 x ^ 12)

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

method	result	size
gosper	\(-\frac {7 b x +6 a}{42 x^{2} \left (c \,x^{2}\right )^{\frac {5}{2}}}\)	\(21\)
default	\(-\frac {7 b x +6 a}{42 x^{2} \left (c \,x^{2}\right )^{\frac {5}{2}}}\)	\(21\)
risch	\(\frac {-\frac {b x}{6}-\frac {a}{7}}{c^{2} x^{6} \sqrt {c \,x^{2}}}\)	\(23\)
trager	\(\frac {\left (-1+x \right ) \left (6 a \,x^{6}+7 b \,x^{6}+6 a \,x^{5}+7 b \,x^{5}+6 a \,x^{4}+7 b \,x^{4}+6 a \,x^{3}+7 b \,x^{3}+6 a \,x^{2}+7 x^{2} b +6 a x +7 b x +6 a \right ) \sqrt {c \,x^{2}}}{42 c^{3} x^{8}}\)	\(91\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)/x^3/(c*x^2)^(5/2),x,method=_RETURNVERBOSE)

[Out]

-1/42*(7*b*x+6*a)/x^2/(c*x^2)^(5/2)

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Maxima [A]
time = 0.27, size = 19, normalized size = 0.46 \begin {gather*} -\frac {b}{6 \, c^{\frac {5}{2}} x^{6}} - \frac {a}{7 \, c^{\frac {5}{2}} x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)/x^3/(c*x^2)^(5/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.29, size = 23, normalized size = 0.56 \begin {gather*} -\frac {\sqrt {c x^{2}} {\left (7 \, b x + 6 \, a\right )}}{42 \, c^{3} x^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)/x^3/(c*x^2)^(5/2),x, algorithm="fricas")

[Out]

-1/42*sqrt(c*x^2)*(7*b*x + 6*a)/(c^3*x^8)

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Sympy [A]
time = 0.43, size = 29, normalized size = 0.71 \begin {gather*} - \frac {a}{7 x^{2} \left (c x^{2}\right )^{\frac {5}{2}}} - \frac {b}{6 x \left (c x^{2}\right )^{\frac {5}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)/x**3/(c*x**2)**(5/2),x)

[Out]

-a/(7*x**2*(c*x**2)**(5/2)) - b/(6*x*(c*x**2)**(5/2))

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Giac [A]
time = 0.00, size = 29, normalized size = 0.71 \begin {gather*} \frac {-7 b x-6 a}{\sqrt {c}\cdot 42 \left (c^{2} x^{7} \mathrm {sign}\left (x\right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)/x^3/(c*x^2)^(5/2),x)

[Out]

-1/42*(7*b*x + 6*a)/(c^(5/2)*x^7*sgn(x))

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Mupad [B]
time = 0.16, size = 26, normalized size = 0.63 \begin {gather*} -\frac {6\,a\,\sqrt {x^2}+7\,b\,x\,\sqrt {x^2}}{42\,c^{5/2}\,x^8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)/(x^3*(c*x^2)^(5/2)),x)

[Out]

-(6*a*(x^2)^(1/2) + 7*b*x*(x^2)^(1/2))/(42*c^(5/2)*x^8)

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