3.6.83 \(\int \frac {1}{x^{7/2} (a+b x)^{3/2}} \, dx\) [583]

Optimal. Leaf size=87 \[ \frac {2}{a x^{5/2} \sqrt {a+b x}}-\frac {12 \sqrt {a+b x}}{5 a^2 x^{5/2}}+\frac {16 b \sqrt {a+b x}}{5 a^3 x^{3/2}}-\frac {32 b^2 \sqrt {a+b x}}{5 a^4 \sqrt {x}} \]

[Out]

2/a/x^(5/2)/(b*x+a)^(1/2)-12/5*(b*x+a)^(1/2)/a^2/x^(5/2)+16/5*b*(b*x+a)^(1/2)/a^3/x^(3/2)-32/5*b^2*(b*x+a)^(1/
2)/a^4/x^(1/2)

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Rubi [A]
time = 0.01, antiderivative size = 87, 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^2 \sqrt {a+b x}}{5 a^4 \sqrt {x}}+\frac {16 b \sqrt {a+b x}}{5 a^3 x^{3/2}}-\frac {12 \sqrt {a+b x}}{5 a^2 x^{5/2}}+\frac {2}{a x^{5/2} \sqrt {a+b x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^(7/2)*(a + b*x)^(3/2)),x]

[Out]

2/(a*x^(5/2)*Sqrt[a + b*x]) - (12*Sqrt[a + b*x])/(5*a^2*x^(5/2)) + (16*b*Sqrt[a + b*x])/(5*a^3*x^(3/2)) - (32*
b^2*Sqrt[a + b*x])/(5*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^{7/2} (a+b x)^{3/2}} \, dx &=\frac {2}{a x^{5/2} \sqrt {a+b x}}+\frac {6 \int \frac {1}{x^{7/2} \sqrt {a+b x}} \, dx}{a}\\ &=\frac {2}{a x^{5/2} \sqrt {a+b x}}-\frac {12 \sqrt {a+b x}}{5 a^2 x^{5/2}}-\frac {(24 b) \int \frac {1}{x^{5/2} \sqrt {a+b x}} \, dx}{5 a^2}\\ &=\frac {2}{a x^{5/2} \sqrt {a+b x}}-\frac {12 \sqrt {a+b x}}{5 a^2 x^{5/2}}+\frac {16 b \sqrt {a+b x}}{5 a^3 x^{3/2}}+\frac {\left (16 b^2\right ) \int \frac {1}{x^{3/2} \sqrt {a+b x}} \, dx}{5 a^3}\\ &=\frac {2}{a x^{5/2} \sqrt {a+b x}}-\frac {12 \sqrt {a+b x}}{5 a^2 x^{5/2}}+\frac {16 b \sqrt {a+b x}}{5 a^3 x^{3/2}}-\frac {32 b^2 \sqrt {a+b x}}{5 a^4 \sqrt {x}}\\ \end {align*}

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Mathematica [A]
time = 0.09, size = 49, normalized size = 0.56 \begin {gather*} -\frac {2 \left (a^3-2 a^2 b x+8 a b^2 x^2+16 b^3 x^3\right )}{5 a^4 x^{5/2} \sqrt {a+b x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^(7/2)*(a + b*x)^(3/2)),x]

[Out]

(-2*(a^3 - 2*a^2*b*x + 8*a*b^2*x^2 + 16*b^3*x^3))/(5*a^4*x^(5/2)*Sqrt[a + b*x])

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Mathics [A]
time = 10.44, size = 99, normalized size = 1.14 \begin {gather*} \frac {2 \sqrt {b} \left (-a^5-5 a^3 b^2 x^2-30 a^2 b^3 x^3-40 a b^4 x^4-16 b^5 x^5\right ) \sqrt {\frac {a+b x}{b x}}}{5 a^4 x^2 \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^(7/2)*(a + b*x)^(3/2)),x]')

[Out]

2 Sqrt[b] (-a ^ 5 - 5 a ^ 3 b ^ 2 x ^ 2 - 30 a ^ 2 b ^ 3 x ^ 3 - 40 a b ^ 4 x ^ 4 - 16 b ^ 5 x ^ 5) Sqrt[(a +
b x) / (b x)] / (5 a ^ 4 x ^ 2 (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 = 77, normalized size = 0.89

method	result	size
gosper	\(-\frac {2 \left (16 b^{3} x^{3}+8 a \,b^{2} x^{2}-2 a^{2} b x +a^{3}\right )}{5 x^{\frac {5}{2}} \sqrt {b x +a}\, a^{4}}\)	\(44\)
risch	\(-\frac {2 \sqrt {b x +a}\, \left (11 x^{2} b^{2}-3 a b x +a^{2}\right )}{5 a^{4} x^{\frac {5}{2}}}-\frac {2 b^{3} \sqrt {x}}{a^{4} \sqrt {b x +a}}\)	\(52\)
default	\(-\frac {2}{5 a \,x^{\frac {5}{2}} \sqrt {b x +a}}-\frac {6 b \left (-\frac {2}{3 a \,x^{\frac {3}{2}} \sqrt {b x +a}}-\frac {4 b \left (-\frac {2}{a \sqrt {x}\, \sqrt {b x +a}}-\frac {4 b \sqrt {x}}{a^{2} \sqrt {b x +a}}\right )}{3 a}\right )}{5 a}\)	\(77\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^(7/2)/(b*x+a)^(3/2),x,method=_RETURNVERBOSE)

[Out]

-2/5/a/x^(5/2)/(b*x+a)^(1/2)-6/5*b/a*(-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 = 0.26, size = 64, normalized size = 0.74 \begin {gather*} -\frac {2 \, b^{3} \sqrt {x}}{\sqrt {b x + a} a^{4}} - \frac {2 \, {\left (\frac {15 \, \sqrt {b x + a} b^{2}}{\sqrt {x}} - \frac {5 \, {\left (b x + a\right )}^{\frac {3}{2}} b}{x^{\frac {3}{2}}} + \frac {{\left (b x + a\right )}^{\frac {5}{2}}}{x^{\frac {5}{2}}}\right )}}{5 \, a^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(7/2)/(b*x+a)^(3/2),x, algorithm="maxima")

[Out]

-2*b^3*sqrt(x)/(sqrt(b*x + a)*a^4) - 2/5*(15*sqrt(b*x + a)*b^2/sqrt(x) - 5*(b*x + a)^(3/2)*b/x^(3/2) + (b*x +
a)^(5/2)/x^(5/2))/a^4

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Fricas [A]
time = 0.31, size = 58, normalized size = 0.67 \begin {gather*} -\frac {2 \, {\left (16 \, b^{3} x^{3} + 8 \, a b^{2} x^{2} - 2 \, a^{2} b x + a^{3}\right )} \sqrt {b x + a} \sqrt {x}}{5 \, {\left (a^{4} b x^{4} + a^{5} x^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(7/2)/(b*x+a)^(3/2),x, algorithm="fricas")

[Out]

-2/5*(16*b^3*x^3 + 8*a*b^2*x^2 - 2*a^2*b*x + a^3)*sqrt(b*x + a)*sqrt(x)/(a^4*b*x^4 + a^5*x^3)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 348 vs. \(2 (82) = 164\).
time = 8.27, size = 348, normalized size = 4.00 \begin {gather*} - \frac {2 a^{5} b^{\frac {19}{2}} \sqrt {\frac {a}{b x} + 1}}{5 a^{7} b^{9} x^{2} + 15 a^{6} b^{10} x^{3} + 15 a^{5} b^{11} x^{4} + 5 a^{4} b^{12} x^{5}} - \frac {10 a^{3} b^{\frac {23}{2}} x^{2} \sqrt {\frac {a}{b x} + 1}}{5 a^{7} b^{9} x^{2} + 15 a^{6} b^{10} x^{3} + 15 a^{5} b^{11} x^{4} + 5 a^{4} b^{12} x^{5}} - \frac {60 a^{2} b^{\frac {25}{2}} x^{3} \sqrt {\frac {a}{b x} + 1}}{5 a^{7} b^{9} x^{2} + 15 a^{6} b^{10} x^{3} + 15 a^{5} b^{11} x^{4} + 5 a^{4} b^{12} x^{5}} - \frac {80 a b^{\frac {27}{2}} x^{4} \sqrt {\frac {a}{b x} + 1}}{5 a^{7} b^{9} x^{2} + 15 a^{6} b^{10} x^{3} + 15 a^{5} b^{11} x^{4} + 5 a^{4} b^{12} x^{5}} - \frac {32 b^{\frac {29}{2}} x^{5} \sqrt {\frac {a}{b x} + 1}}{5 a^{7} b^{9} x^{2} + 15 a^{6} b^{10} x^{3} + 15 a^{5} b^{11} x^{4} + 5 a^{4} b^{12} x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**(7/2)/(b*x+a)**(3/2),x)

[Out]

-2*a**5*b**(19/2)*sqrt(a/(b*x) + 1)/(5*a**7*b**9*x**2 + 15*a**6*b**10*x**3 + 15*a**5*b**11*x**4 + 5*a**4*b**12
*x**5) - 10*a**3*b**(23/2)*x**2*sqrt(a/(b*x) + 1)/(5*a**7*b**9*x**2 + 15*a**6*b**10*x**3 + 15*a**5*b**11*x**4
+ 5*a**4*b**12*x**5) - 60*a**2*b**(25/2)*x**3*sqrt(a/(b*x) + 1)/(5*a**7*b**9*x**2 + 15*a**6*b**10*x**3 + 15*a*
*5*b**11*x**4 + 5*a**4*b**12*x**5) - 80*a*b**(27/2)*x**4*sqrt(a/(b*x) + 1)/(5*a**7*b**9*x**2 + 15*a**6*b**10*x
**3 + 15*a**5*b**11*x**4 + 5*a**4*b**12*x**5) - 32*b**(29/2)*x**5*sqrt(a/(b*x) + 1)/(5*a**7*b**9*x**2 + 15*a**
6*b**10*x**3 + 15*a**5*b**11*x**4 + 5*a**4*b**12*x**5)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 161 vs. \(2 (65) = 130\).
time = 0.02, size = 233, normalized size = 2.68 \begin {gather*} 2 \left (-\frac {\frac {1}{2}\cdot 2 b^{3} \sqrt {x} \sqrt {a+b x}}{a^{4} \left (a+b x\right )}+\frac {2 \left (5 b^{2} \sqrt {b} \left (\sqrt {a+b x}-\sqrt {b} \sqrt {x}\right )^{8}-30 b^{2} \sqrt {b} \left (\sqrt {a+b x}-\sqrt {b} \sqrt {x}\right )^{6} a+80 b^{2} \sqrt {b} \left (\sqrt {a+b x}-\sqrt {b} \sqrt {x}\right )^{4} a^{2}-50 b^{2} \sqrt {b} \left (\sqrt {a+b x}-\sqrt {b} \sqrt {x}\right )^{2} a^{3}+11 b^{2} \sqrt {b} a^{4}\right )}{5 a^{3} \left (\left (\sqrt {a+b x}-\sqrt {b} \sqrt {x}\right )^{2}-a\right )^{5}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(7/2)/(b*x+a)^(3/2),x)

[Out]

-2*b^3*sqrt(x)/(sqrt(b*x + a)*a^4) + 4/5*(5*b^(5/2)*(sqrt(b)*sqrt(x) - sqrt(b*x + a))^8 - 30*a*b^(5/2)*(sqrt(b
)*sqrt(x) - sqrt(b*x + a))^6 + 80*a^2*b^(5/2)*(sqrt(b)*sqrt(x) - sqrt(b*x + a))^4 - 50*a^3*b^(5/2)*(sqrt(b)*sq
rt(x) - sqrt(b*x + a))^2 + 11*a^4*b^(5/2))/(((sqrt(b)*sqrt(x) - sqrt(b*x + a))^2 - a)^5*a^3)

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Mupad [B]
time = 0.43, size = 58, normalized size = 0.67 \begin {gather*} -\frac {\sqrt {a+b\,x}\,\left (\frac {2}{5\,a\,b}-\frac {4\,x}{5\,a^2}+\frac {16\,b\,x^2}{5\,a^3}+\frac {32\,b^2\,x^3}{5\,a^4}\right )}{x^{7/2}+\frac {a\,x^{5/2}}{b}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^(7/2)*(a + b*x)^(3/2)),x)

[Out]

-((a + b*x)^(1/2)*(2/(5*a*b) - (4*x)/(5*a^2) + (16*b*x^2)/(5*a^3) + (32*b^2*x^3)/(5*a^4)))/(x^(7/2) + (a*x^(5/
2))/b)

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