3.7.1 \(\int \frac {1}{x^{5/2} (a-b x)^{3/2}} \, dx\) [601]

Optimal. Leaf size=66 \[ \frac {2}{a x^{3/2} \sqrt {a-b x}}-\frac {8 \sqrt {a-b x}}{3 a^2 x^{3/2}}-\frac {16 b \sqrt {a-b x}}{3 a^3 \sqrt {x}} \]

[Out]

2/a/x^(3/2)/(-b*x+a)^(1/2)-8/3*(-b*x+a)^(1/2)/a^2/x^(3/2)-16/3*b*(-b*x+a)^(1/2)/a^3/x^(1/2)

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Rubi [A]
time = 0.01, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {47, 37} \begin {gather*} -\frac {16 b \sqrt {a-b x}}{3 a^3 \sqrt {x}}-\frac {8 \sqrt {a-b x}}{3 a^2 x^{3/2}}+\frac {2}{a x^{3/2} \sqrt {a-b x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^(5/2)*(a - b*x)^(3/2)),x]

[Out]

2/(a*x^(3/2)*Sqrt[a - b*x]) - (8*Sqrt[a - b*x])/(3*a^2*x^(3/2)) - (16*b*Sqrt[a - b*x])/(3*a^3*Sqrt[x])

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n +
1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rule 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n + 1
)/((b*c - a*d)*(m + 1))), x] - Dist[d*(Simplify[m + n + 2]/((b*c - a*d)*(m + 1))), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
 NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

Rubi steps

\begin {align*} \int \frac {1}{x^{5/2} (a-b x)^{3/2}} \, dx &=\frac {2}{a x^{3/2} \sqrt {a-b x}}+\frac {4 \int \frac {1}{x^{5/2} \sqrt {a-b x}} \, dx}{a}\\ &=\frac {2}{a x^{3/2} \sqrt {a-b x}}-\frac {8 \sqrt {a-b x}}{3 a^2 x^{3/2}}+\frac {(8 b) \int \frac {1}{x^{3/2} \sqrt {a-b x}} \, dx}{3 a^2}\\ &=\frac {2}{a x^{3/2} \sqrt {a-b x}}-\frac {8 \sqrt {a-b x}}{3 a^2 x^{3/2}}-\frac {16 b \sqrt {a-b x}}{3 a^3 \sqrt {x}}\\ \end {align*}

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Mathematica [A]
time = 0.09, size = 39, normalized size = 0.59 \begin {gather*} -\frac {2 \left (a^2+4 a b x-8 b^2 x^2\right )}{3 a^3 x^{3/2} \sqrt {a-b x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^(5/2)*(a - b*x)^(3/2)),x]

[Out]

(-2*(a^2 + 4*a*b*x - 8*b^2*x^2))/(3*a^3*x^(3/2)*Sqrt[a - b*x])

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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 2 in optimal.
time = 8.56, size = 311, normalized size = 4.71 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {2 \sqrt {b} \left (-a^3-3 a^2 b x+12 a b^2 x^2-8 b^3 x^3\right ) \sqrt {\frac {a-b x}{b x}}}{3 a^3 x \left (a^2-2 a b x+b^2 x^2\right )},\text {Abs}\left [\frac {a}{b x}\right ]>1\right \}\right \},\frac {-2 I a^3 b^{\frac {9}{2}} \sqrt {1-\frac {a}{b x}}}{3 a^5 b^4 x-6 a^4 b^5 x^2+3 a^3 b^6 x^3}-\frac {6 I a^2 b^{\frac {11}{2}} x \sqrt {1-\frac {a}{b x}}}{3 a^5 b^4 x-6 a^4 b^5 x^2+3 a^3 b^6 x^3}+\frac {I 24 a b^{\frac {13}{2}} x^2 \sqrt {1-\frac {a}{b x}}}{3 a^5 b^4 x-6 a^4 b^5 x^2+3 a^3 b^6 x^3}-\frac {16 I b^{\frac {15}{2}} x^3 \sqrt {1-\frac {a}{b x}}}{3 a^5 b^4 x-6 a^4 b^5 x^2+3 a^3 b^6 x^3}\right ] \end {gather*}

Warning: Unable to verify antiderivative.

[In]

mathics('Integrate[1/(x^(5/2)*(a - b*x)^(3/2)),x]')

[Out]

Piecewise[{{2 Sqrt[b] (-a ^ 3 - 3 a ^ 2 b x + 12 a b ^ 2 x ^ 2 - 8 b ^ 3 x ^ 3) Sqrt[(a - b x) / (b x)] / (3 a
 ^ 3 x (a ^ 2 - 2 a b x + b ^ 2 x ^ 2)), Abs[a / (b x)] > 1}}, -2 I a ^ 3 b ^ (9 / 2) Sqrt[1 - a / (b x)] / (3
 a ^ 5 b ^ 4 x - 6 a ^ 4 b ^ 5 x ^ 2 + 3 a ^ 3 b ^ 6 x ^ 3) - 6 I a ^ 2 b ^ (11 / 2) x Sqrt[1 - a / (b x)] / (
3 a ^ 5 b ^ 4 x - 6 a ^ 4 b ^ 5 x ^ 2 + 3 a ^ 3 b ^ 6 x ^ 3) + I 24 a b ^ (13 / 2) x ^ 2 Sqrt[1 - a / (b x)] /
 (3 a ^ 5 b ^ 4 x - 6 a ^ 4 b ^ 5 x ^ 2 + 3 a ^ 3 b ^ 6 x ^ 3) - 16 I b ^ (15 / 2) x ^ 3 Sqrt[1 - a / (b x)] /
 (3 a ^ 5 b ^ 4 x - 6 a ^ 4 b ^ 5 x ^ 2 + 3 a ^ 3 b ^ 6 x ^ 3)]

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Maple [A]
time = 0.14, size = 58, normalized size = 0.88

method result size
gosper \(-\frac {2 \left (-8 x^{2} b^{2}+4 a b x +a^{2}\right )}{3 x^{\frac {3}{2}} \sqrt {-b x +a}\, a^{3}}\) \(34\)
risch \(-\frac {2 \sqrt {-b x +a}\, \left (5 b x +a \right )}{3 a^{3} x^{\frac {3}{2}}}+\frac {2 b^{2} \sqrt {x}}{a^{3} \sqrt {-b x +a}}\) \(43\)
default \(-\frac {2}{3 a \,x^{\frac {3}{2}} \sqrt {-b x +a}}+\frac {4 b \left (-\frac {2}{a \sqrt {x}\, \sqrt {-b x +a}}+\frac {4 b \sqrt {x}}{a^{2} \sqrt {-b x +a}}\right )}{3 a}\) \(58\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^(5/2)/(-b*x+a)^(3/2),x,method=_RETURNVERBOSE)

[Out]

-2/3/a/x^(3/2)/(-b*x+a)^(1/2)+4/3*b/a*(-2/a/x^(1/2)/(-b*x+a)^(1/2)+4*b/a^2*x^(1/2)/(-b*x+a)^(1/2))

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Maxima [A]
time = 0.27, size = 52, normalized size = 0.79 \begin {gather*} \frac {2 \, b^{2} \sqrt {x}}{\sqrt {-b x + a} a^{3}} - \frac {2 \, {\left (\frac {6 \, \sqrt {-b x + a} b}{\sqrt {x}} + \frac {{\left (-b x + a\right )}^{\frac {3}{2}}}{x^{\frac {3}{2}}}\right )}}{3 \, a^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(5/2)/(-b*x+a)^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.30, size = 51, normalized size = 0.77 \begin {gather*} -\frac {2 \, {\left (8 \, b^{2} x^{2} - 4 \, a b x - a^{2}\right )} \sqrt {-b x + a} \sqrt {x}}{3 \, {\left (a^{3} b x^{3} - a^{4} x^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(5/2)/(-b*x+a)^(3/2),x, algorithm="fricas")

[Out]

-2/3*(8*b^2*x^2 - 4*a*b*x - a^2)*sqrt(-b*x + a)*sqrt(x)/(a^3*b*x^3 - a^4*x^2)

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Sympy [C] Result contains complex when optimal does not.
time = 2.67, size = 452, normalized size = 6.85 \begin {gather*} \begin {cases} - \frac {2 a^{3} b^{\frac {9}{2}} \sqrt {\frac {a}{b x} - 1}}{3 a^{5} b^{4} x - 6 a^{4} b^{5} x^{2} + 3 a^{3} b^{6} x^{3}} - \frac {6 a^{2} b^{\frac {11}{2}} x \sqrt {\frac {a}{b x} - 1}}{3 a^{5} b^{4} x - 6 a^{4} b^{5} x^{2} + 3 a^{3} b^{6} x^{3}} + \frac {24 a b^{\frac {13}{2}} x^{2} \sqrt {\frac {a}{b x} - 1}}{3 a^{5} b^{4} x - 6 a^{4} b^{5} x^{2} + 3 a^{3} b^{6} x^{3}} - \frac {16 b^{\frac {15}{2}} x^{3} \sqrt {\frac {a}{b x} - 1}}{3 a^{5} b^{4} x - 6 a^{4} b^{5} x^{2} + 3 a^{3} b^{6} x^{3}} & \text {for}\: \left |{\frac {a}{b x}}\right | > 1 \\- \frac {2 i a^{3} b^{\frac {9}{2}} \sqrt {- \frac {a}{b x} + 1}}{3 a^{5} b^{4} x - 6 a^{4} b^{5} x^{2} + 3 a^{3} b^{6} x^{3}} - \frac {6 i a^{2} b^{\frac {11}{2}} x \sqrt {- \frac {a}{b x} + 1}}{3 a^{5} b^{4} x - 6 a^{4} b^{5} x^{2} + 3 a^{3} b^{6} x^{3}} + \frac {24 i a b^{\frac {13}{2}} x^{2} \sqrt {- \frac {a}{b x} + 1}}{3 a^{5} b^{4} x - 6 a^{4} b^{5} x^{2} + 3 a^{3} b^{6} x^{3}} - \frac {16 i b^{\frac {15}{2}} x^{3} \sqrt {- \frac {a}{b x} + 1}}{3 a^{5} b^{4} x - 6 a^{4} b^{5} x^{2} + 3 a^{3} b^{6} x^{3}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**(5/2)/(-b*x+a)**(3/2),x)

[Out]

Piecewise((-2*a**3*b**(9/2)*sqrt(a/(b*x) - 1)/(3*a**5*b**4*x - 6*a**4*b**5*x**2 + 3*a**3*b**6*x**3) - 6*a**2*b
**(11/2)*x*sqrt(a/(b*x) - 1)/(3*a**5*b**4*x - 6*a**4*b**5*x**2 + 3*a**3*b**6*x**3) + 24*a*b**(13/2)*x**2*sqrt(
a/(b*x) - 1)/(3*a**5*b**4*x - 6*a**4*b**5*x**2 + 3*a**3*b**6*x**3) - 16*b**(15/2)*x**3*sqrt(a/(b*x) - 1)/(3*a*
*5*b**4*x - 6*a**4*b**5*x**2 + 3*a**3*b**6*x**3), Abs(a/(b*x)) > 1), (-2*I*a**3*b**(9/2)*sqrt(-a/(b*x) + 1)/(3
*a**5*b**4*x - 6*a**4*b**5*x**2 + 3*a**3*b**6*x**3) - 6*I*a**2*b**(11/2)*x*sqrt(-a/(b*x) + 1)/(3*a**5*b**4*x -
 6*a**4*b**5*x**2 + 3*a**3*b**6*x**3) + 24*I*a*b**(13/2)*x**2*sqrt(-a/(b*x) + 1)/(3*a**5*b**4*x - 6*a**4*b**5*
x**2 + 3*a**3*b**6*x**3) - 16*I*b**(15/2)*x**3*sqrt(-a/(b*x) + 1)/(3*a**5*b**4*x - 6*a**4*b**5*x**2 + 3*a**3*b
**6*x**3), True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 135 vs. \(2 (50) = 100\).
time = 0.01, size = 162, normalized size = 2.45 \begin {gather*} 2 \left (\frac {\frac {1}{2}\cdot 2 b^{2} \sqrt {x} \sqrt {a-b x}}{a^{3} \left (a-b x\right )}+\frac {2 \left (3 b \sqrt {-b} \left (\sqrt {a-b x}-\sqrt {-b} \sqrt {x}\right )^{4}-12 b \sqrt {-b} \left (\sqrt {a-b x}-\sqrt {-b} \sqrt {x}\right )^{2} a+5 b \sqrt {-b} a^{2}\right )}{3 a^{2} \left (\left (\sqrt {a-b x}-\sqrt {-b} \sqrt {x}\right )^{2}-a\right )^{3}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(5/2)/(-b*x+a)^(3/2),x)

[Out]

-2*sqrt(-b*x + a)*b^2*sqrt(x)/((b*x - a)*a^3) + 4/3*(3*sqrt(-b)*b*(sqrt(-b)*sqrt(x) - sqrt(-b*x + a))^4 - 12*a
*sqrt(-b)*b*(sqrt(-b)*sqrt(x) - sqrt(-b*x + a))^2 + 5*a^2*sqrt(-b)*b)/(((sqrt(-b)*sqrt(x) - sqrt(-b*x + a))^2
- a)^3*a^2)

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Mupad [B]
time = 0.43, size = 48, normalized size = 0.73 \begin {gather*} \frac {\sqrt {a-b\,x}\,\left (\frac {8\,x}{3\,a^2}+\frac {2}{3\,a\,b}-\frac {16\,b\,x^2}{3\,a^3}\right )}{x^{5/2}-\frac {a\,x^{3/2}}{b}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^(5/2)*(a - b*x)^(3/2)),x)

[Out]

((a - b*x)^(1/2)*((8*x)/(3*a^2) + 2/(3*a*b) - (16*b*x^2)/(3*a^3)))/(x^(5/2) - (a*x^(3/2))/b)

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