3.15.38 \(\int \frac {a+b x}{(c+d x)^{5/2}} \, dx\) [1438]

Optimal. Leaf size=40 \[ \frac {2 (b c-a d)}{3 d^2 (c+d x)^{3/2}}-\frac {2 b}{d^2 \sqrt {c+d x}} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {45} \begin {gather*} \frac {2 (b c-a d)}{3 d^2 (c+d x)^{3/2}}-\frac {2 b}{d^2 \sqrt {c+d x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)/(c + d*x)^(5/2),x]

[Out]

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

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

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Mathematica [A]
time = 0.03, size = 29, normalized size = 0.72 \begin {gather*} -\frac {2 (2 b c+a d+3 b d x)}{3 d^2 (c+d x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 2 in optimal.
time = 2.76, size = 47, normalized size = 1.18 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {2 \left (-2 b c+d \left (-a-3 b x\right )\right )}{3 d^2 \left (c+d x\right )^{\frac {3}{2}}},d\text {!=}0\right \}\right \},\frac {a x+\frac {b x^2}{2}}{c^{\frac {5}{2}}}\right ] \end {gather*}

Warning: Unable to verify antiderivative.

[In]

mathics('Integrate[(a + b*x)^1/(c + d*x)^(5/2),x]')

[Out]

Piecewise[{{2 (-2 b c + d (-a - 3 b x)) / (3 d ^ 2 (c + d x) ^ (3 / 2)), d != 0}}, (a x + b x ^ 2 / 2) / c ^ (
5 / 2)]

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Maple [A]
time = 0.14, size = 34, normalized size = 0.85

method	result	size
gosper	\(-\frac {2 \left (3 b d x +a d +2 b c \right )}{3 \left (d x +c \right )^{\frac {3}{2}} d^{2}}\)	\(26\)
trager	\(-\frac {2 \left (3 b d x +a d +2 b c \right )}{3 \left (d x +c \right )^{\frac {3}{2}} d^{2}}\)	\(26\)
derivativedivides	\(\frac {-\frac {2 \left (a d -b c \right )}{3 \left (d x +c \right )^{\frac {3}{2}}}-\frac {2 b}{\sqrt {d x +c}}}{d^{2}}\)	\(34\)
default	\(\frac {-\frac {2 \left (a d -b c \right )}{3 \left (d x +c \right )^{\frac {3}{2}}}-\frac {2 b}{\sqrt {d x +c}}}{d^{2}}\)	\(34\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.26, size = 28, normalized size = 0.70 \begin {gather*} -\frac {2 \, {\left (3 \, {\left (d x + c\right )} b - b c + a d\right )}}{3 \, {\left (d x + c\right )}^{\frac {3}{2}} d^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.29, size = 46, normalized size = 1.15 \begin {gather*} -\frac {2 \, {\left (3 \, b d x + 2 \, b c + a d\right )} \sqrt {d x + c}}{3 \, {\left (d^{4} x^{2} + 2 \, c d^{3} x + c^{2} d^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*(3*b*d*x + 2*b*c + a*d)*sqrt(d*x + c)/(d^4*x^2 + 2*c*d^3*x + c^2*d^2)

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Sympy [A]
time = 0.50, size = 124, normalized size = 3.10 \begin {gather*} \begin {cases} - \frac {2 a d}{3 c d^{2} \sqrt {c + d x} + 3 d^{3} x \sqrt {c + d x}} - \frac {4 b c}{3 c d^{2} \sqrt {c + d x} + 3 d^{3} x \sqrt {c + d x}} - \frac {6 b d x}{3 c d^{2} \sqrt {c + d x} + 3 d^{3} x \sqrt {c + d x}} & \text {for}\: d \neq 0 \\\frac {a x + \frac {b x^{2}}{2}}{c^{\frac {5}{2}}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-2*a*d/(3*c*d**2*sqrt(c + d*x) + 3*d**3*x*sqrt(c + d*x)) - 4*b*c/(3*c*d**2*sqrt(c + d*x) + 3*d**3*x
*sqrt(c + d*x)) - 6*b*d*x/(3*c*d**2*sqrt(c + d*x) + 3*d**3*x*sqrt(c + d*x)), Ne(d, 0)), ((a*x + b*x**2/2)/c**(
5/2), True))

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Giac [A]
time = 0.00, size = 42, normalized size = 1.05 \begin {gather*} \frac {-6 \left (c+d x\right ) b+2 b c-2 d a}{3 d^{2} \sqrt {c+d x} \left (c+d x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.25, size = 29, normalized size = 0.72 \begin {gather*} -\frac {2\,a\,d-2\,b\,c+6\,b\,\left (c+d\,x\right )}{3\,d^2\,{\left (c+d\,x\right )}^{3/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)/(c + d*x)^(5/2),x)

[Out]

-(2*a*d - 2*b*c + 6*b*(c + d*x))/(3*d^2*(c + d*x)^(3/2))

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