3.6.89 \(\int \frac {1}{x^{5/2} (a+b x)^{5/2}} \, dx\) [589]

Optimal. Leaf size=84 \[ \frac {2}{3 a x^{3/2} (a+b x)^{3/2}}+\frac {4}{a^2 x^{3/2} \sqrt {a+b x}}-\frac {16 \sqrt {a+b x}}{3 a^3 x^{3/2}}+\frac {32 b \sqrt {a+b x}}{3 a^4 \sqrt {x}} \]

[Out]

2/3/a/x^(3/2)/(b*x+a)^(3/2)+4/a^2/x^(3/2)/(b*x+a)^(1/2)-16/3*(b*x+a)^(1/2)/a^3/x^(3/2)+32/3*b*(b*x+a)^(1/2)/a^
4/x^(1/2)

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Rubi [A]
time = 0.01, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {47, 37} \begin {gather*} \frac {32 b \sqrt {a+b x}}{3 a^4 \sqrt {x}}-\frac {16 \sqrt {a+b x}}{3 a^3 x^{3/2}}+\frac {4}{a^2 x^{3/2} \sqrt {a+b x}}+\frac {2}{3 a x^{3/2} (a+b x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^(5/2)*(a + b*x)^(5/2)),x]

[Out]

2/(3*a*x^(3/2)*(a + b*x)^(3/2)) + 4/(a^2*x^(3/2)*Sqrt[a + b*x]) - (16*Sqrt[a + b*x])/(3*a^3*x^(3/2)) + (32*b*S
qrt[a + b*x])/(3*a^4*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)^{5/2}} \, dx &=\frac {2}{3 a x^{3/2} (a+b x)^{3/2}}+\frac {2 \int \frac {1}{x^{5/2} (a+b x)^{3/2}} \, dx}{a}\\ &=\frac {2}{3 a x^{3/2} (a+b x)^{3/2}}+\frac {4}{a^2 x^{3/2} \sqrt {a+b x}}+\frac {8 \int \frac {1}{x^{5/2} \sqrt {a+b x}} \, dx}{a^2}\\ &=\frac {2}{3 a x^{3/2} (a+b x)^{3/2}}+\frac {4}{a^2 x^{3/2} \sqrt {a+b x}}-\frac {16 \sqrt {a+b x}}{3 a^3 x^{3/2}}-\frac {(16 b) \int \frac {1}{x^{3/2} \sqrt {a+b x}} \, dx}{3 a^3}\\ &=\frac {2}{3 a x^{3/2} (a+b x)^{3/2}}+\frac {4}{a^2 x^{3/2} \sqrt {a+b x}}-\frac {16 \sqrt {a+b x}}{3 a^3 x^{3/2}}+\frac {32 b \sqrt {a+b x}}{3 a^4 \sqrt {x}}\\ \end {align*}

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Mathematica [A]
time = 0.08, size = 49, normalized size = 0.58 \begin {gather*} -\frac {2 \left (a^3-6 a^2 b x-24 a b^2 x^2-16 b^3 x^3\right )}{3 a^4 x^{3/2} (a+b x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(a^3 - 6*a^2*b*x - 24*a*b^2*x^2 - 16*b^3*x^3))/(3*a^4*x^(3/2)*(a + b*x)^(3/2))

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Mathics [A]
time = 7.83, size = 95, normalized size = 1.13 \begin {gather*} \frac {2 \sqrt {b} \left (-a^4+5 a^3 b x+30 a^2 b^2 x^2+40 a b^3 x^3+16 b^4 x^4\right ) \sqrt {\frac {a+b x}{b x}}}{3 a^4 x \left (a^3+3 a^2 b x+3 a b^2 x^2+b^3 x^3\right )} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

2 Sqrt[b] (-a ^ 4 + 5 a ^ 3 b x + 30 a ^ 2 b ^ 2 x ^ 2 + 40 a b ^ 3 x ^ 3 + 16 b ^ 4 x ^ 4) Sqrt[(a + b x) / (
b x)] / (3 a ^ 4 x (a ^ 3 + 3 a ^ 2 b x + 3 a b ^ 2 x ^ 2 + b ^ 3 x ^ 3))

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Maple [A]
time = 0.13, size = 76, normalized size = 0.90

method	result	size
gosper	\(-\frac {2 \left (-16 b^{3} x^{3}-24 a \,b^{2} x^{2}-6 a^{2} b x +a^{3}\right )}{3 x^{\frac {3}{2}} \left (b x +a \right )^{\frac {3}{2}} a^{4}}\)	\(44\)
risch	\(-\frac {2 \sqrt {b x +a}\, \left (-8 b x +a \right )}{3 a^{4} x^{\frac {3}{2}}}+\frac {2 b^{2} \left (8 b x +9 a \right ) \sqrt {x}}{3 \left (b x +a \right )^{\frac {3}{2}} a^{4}}\)	\(49\)
default	\(-\frac {2}{3 a \,x^{\frac {3}{2}} \left (b x +a \right )^{\frac {3}{2}}}-\frac {2 b \left (-\frac {2}{a \left (b x +a \right )^{\frac {3}{2}} \sqrt {x}}-\frac {4 b \left (\frac {2 \sqrt {x}}{3 a \left (b x +a \right )^{\frac {3}{2}}}+\frac {4 \sqrt {x}}{3 a^{2} \sqrt {b x +a}}\right )}{a}\right )}{a}\)	\(76\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.29, size = 64, normalized size = 0.76 \begin {gather*} \frac {2 \, {\left (\frac {9 \, \sqrt {b x + a} b}{\sqrt {x}} - \frac {{\left (b x + a\right )}^{\frac {3}{2}}}{x^{\frac {3}{2}}}\right )}}{3 \, a^{4}} - \frac {2 \, {\left (b^{3} - \frac {9 \, {\left (b x + a\right )} b^{2}}{x}\right )} x^{\frac {3}{2}}}{3 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(9*sqrt(b*x + a)*b/sqrt(x) - (b*x + a)^(3/2)/x^(3/2))/a^4 - 2/3*(b^3 - 9*(b*x + a)*b^2/x)*x^(3/2)/((b*x +
a)^(3/2)*a^4)

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Fricas [A]
time = 0.32, size = 71, normalized size = 0.85 \begin {gather*} \frac {2 \, {\left (16 \, b^{3} x^{3} + 24 \, a b^{2} x^{2} + 6 \, a^{2} b x - a^{3}\right )} \sqrt {b x + a} \sqrt {x}}{3 \, {\left (a^{4} b^{2} x^{4} + 2 \, a^{5} b x^{3} + a^{6} x^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(16*b^3*x^3 + 24*a*b^2*x^2 + 6*a^2*b*x - a^3)*sqrt(b*x + a)*sqrt(x)/(a^4*b^2*x^4 + 2*a^5*b*x^3 + a^6*x^2)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 337 vs. \(2 (78) = 156\).
time = 4.66, size = 337, normalized size = 4.01 \begin {gather*} - \frac {2 a^{4} b^{\frac {19}{2}} \sqrt {\frac {a}{b x} + 1}}{3 a^{7} b^{9} x + 9 a^{6} b^{10} x^{2} + 9 a^{5} b^{11} x^{3} + 3 a^{4} b^{12} x^{4}} + \frac {10 a^{3} b^{\frac {21}{2}} x \sqrt {\frac {a}{b x} + 1}}{3 a^{7} b^{9} x + 9 a^{6} b^{10} x^{2} + 9 a^{5} b^{11} x^{3} + 3 a^{4} b^{12} x^{4}} + \frac {60 a^{2} b^{\frac {23}{2}} x^{2} \sqrt {\frac {a}{b x} + 1}}{3 a^{7} b^{9} x + 9 a^{6} b^{10} x^{2} + 9 a^{5} b^{11} x^{3} + 3 a^{4} b^{12} x^{4}} + \frac {80 a b^{\frac {25}{2}} x^{3} \sqrt {\frac {a}{b x} + 1}}{3 a^{7} b^{9} x + 9 a^{6} b^{10} x^{2} + 9 a^{5} b^{11} x^{3} + 3 a^{4} b^{12} x^{4}} + \frac {32 b^{\frac {27}{2}} x^{4} \sqrt {\frac {a}{b x} + 1}}{3 a^{7} b^{9} x + 9 a^{6} b^{10} x^{2} + 9 a^{5} b^{11} x^{3} + 3 a^{4} b^{12} x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*a**4*b**(19/2)*sqrt(a/(b*x) + 1)/(3*a**7*b**9*x + 9*a**6*b**10*x**2 + 9*a**5*b**11*x**3 + 3*a**4*b**12*x**4
) + 10*a**3*b**(21/2)*x*sqrt(a/(b*x) + 1)/(3*a**7*b**9*x + 9*a**6*b**10*x**2 + 9*a**5*b**11*x**3 + 3*a**4*b**1
2*x**4) + 60*a**2*b**(23/2)*x**2*sqrt(a/(b*x) + 1)/(3*a**7*b**9*x + 9*a**6*b**10*x**2 + 9*a**5*b**11*x**3 + 3*
a**4*b**12*x**4) + 80*a*b**(25/2)*x**3*sqrt(a/(b*x) + 1)/(3*a**7*b**9*x + 9*a**6*b**10*x**2 + 9*a**5*b**11*x**
3 + 3*a**4*b**12*x**4) + 32*b**(27/2)*x**4*sqrt(a/(b*x) + 1)/(3*a**7*b**9*x + 9*a**6*b**10*x**2 + 9*a**5*b**11
*x**3 + 3*a**4*b**12*x**4)

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Giac [A]
time = 0.01, size = 189, normalized size = 2.25 \begin {gather*} 2 \left (\frac {2 \left (\frac {\frac {1}{18}\cdot 24 b^{4} a^{3} \sqrt {x} \sqrt {x}}{b a^{7}}+\frac {\frac {1}{18}\cdot 27 b^{3} a^{4}}{b a^{7}}\right ) \sqrt {x} \sqrt {a+b x}}{\left (a+b x\right )^{2}}+\frac {2 \left (-6 b \sqrt {b} \left (\sqrt {a+b x}-\sqrt {b} \sqrt {x}\right )^{4}+18 b \sqrt {b} \left (\sqrt {a+b x}-\sqrt {b} \sqrt {x}\right )^{2} a-8 b \sqrt {b} a^{2}\right )}{3 a^{3} \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)^(5/2),x)

[Out]

2/3*sqrt(x)*(8*b^3*x/a^4 + 9*b^2/a^3)/(b*x + a)^(3/2) - 8/3*(3*b^(3/2)*(sqrt(b)*sqrt(x) - sqrt(b*x + a))^4 - 9
*a*b^(3/2)*(sqrt(b)*sqrt(x) - sqrt(b*x + a))^2 + 4*a^2*b^(3/2))/(((sqrt(b)*sqrt(x) - sqrt(b*x + a))^2 - a)^3*a
^3)

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Mupad [B]
time = 0.47, size = 88, normalized size = 1.05 \begin {gather*} \frac {32\,b^3\,x^3\,\sqrt {a+b\,x}-2\,a^3\,\sqrt {a+b\,x}+12\,a^2\,b\,x\,\sqrt {a+b\,x}+48\,a\,b^2\,x^2\,\sqrt {a+b\,x}}{x^{3/2}\,\left (x\,\left (6\,a^5\,b+3\,x\,a^4\,b^2\right )+3\,a^6\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^(5/2)*(a + b*x)^(5/2)),x)

[Out]

(32*b^3*x^3*(a + b*x)^(1/2) - 2*a^3*(a + b*x)^(1/2) + 12*a^2*b*x*(a + b*x)^(1/2) + 48*a*b^2*x^2*(a + b*x)^(1/2
))/(x^(3/2)*(x*(6*a^5*b + 3*a^4*b^2*x) + 3*a^6))

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