3.6.75 \(\int \frac {1}{x^{7/2} \sqrt {a+b x}} \, dx\) [575]

Optimal. Leaf size=68 \[ -\frac {2 \sqrt {a+b x}}{5 a x^{5/2}}+\frac {8 b \sqrt {a+b x}}{15 a^2 x^{3/2}}-\frac {16 b^2 \sqrt {a+b x}}{15 a^3 \sqrt {x}} \]

[Out]

-2/5*(b*x+a)^(1/2)/a/x^(5/2)+8/15*b*(b*x+a)^(1/2)/a^2/x^(3/2)-16/15*b^2*(b*x+a)^(1/2)/a^3/x^(1/2)

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Rubi [A]
time = 0.01, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 3, 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 {16 b^2 \sqrt {a+b x}}{15 a^3 \sqrt {x}}+\frac {8 b \sqrt {a+b x}}{15 a^2 x^{3/2}}-\frac {2 \sqrt {a+b x}}{5 a x^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^(7/2)*Sqrt[a + b*x]),x]

[Out]

(-2*Sqrt[a + b*x])/(5*a*x^(5/2)) + (8*b*Sqrt[a + b*x])/(15*a^2*x^(3/2)) - (16*b^2*Sqrt[a + b*x])/(15*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^{7/2} \sqrt {a+b x}} \, dx &=-\frac {2 \sqrt {a+b x}}{5 a x^{5/2}}-\frac {(4 b) \int \frac {1}{x^{5/2} \sqrt {a+b x}} \, dx}{5 a}\\ &=-\frac {2 \sqrt {a+b x}}{5 a x^{5/2}}+\frac {8 b \sqrt {a+b x}}{15 a^2 x^{3/2}}+\frac {\left (8 b^2\right ) \int \frac {1}{x^{3/2} \sqrt {a+b x}} \, dx}{15 a^2}\\ &=-\frac {2 \sqrt {a+b x}}{5 a x^{5/2}}+\frac {8 b \sqrt {a+b x}}{15 a^2 x^{3/2}}-\frac {16 b^2 \sqrt {a+b x}}{15 a^3 \sqrt {x}}\\ \end {align*}

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Mathematica [A]
time = 0.07, size = 40, normalized size = 0.59 \begin {gather*} -\frac {2 \sqrt {a+b x} \left (3 a^2-4 a b x+8 b^2 x^2\right )}{15 a^3 x^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[a + b*x]*(3*a^2 - 4*a*b*x + 8*b^2*x^2))/(15*a^3*x^(5/2))

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Maple [A]
time = 0.12, size = 55, normalized size = 0.81

method	result	size
gosper	\(-\frac {2 \sqrt {b x +a}\, \left (8 x^{2} b^{2}-4 a b x +3 a^{2}\right )}{15 x^{\frac {5}{2}} a^{3}}\)	\(35\)
risch	\(-\frac {2 \sqrt {b x +a}\, \left (8 x^{2} b^{2}-4 a b x +3 a^{2}\right )}{15 x^{\frac {5}{2}} a^{3}}\)	\(35\)
default	\(-\frac {2 \sqrt {b x +a}}{5 a \,x^{\frac {5}{2}}}-\frac {4 b \left (-\frac {2 \sqrt {b x +a}}{3 a \,x^{\frac {3}{2}}}+\frac {4 b \sqrt {b x +a}}{3 a^{2} \sqrt {x}}\right )}{5 a}\)	\(55\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.27, size = 46, normalized size = 0.68 \begin {gather*} -\frac {2 \, {\left (\frac {15 \, \sqrt {b x + a} b^{2}}{\sqrt {x}} - \frac {10 \, {\left (b x + a\right )}^{\frac {3}{2}} b}{x^{\frac {3}{2}}} + \frac {3 \, {\left (b x + a\right )}^{\frac {5}{2}}}{x^{\frac {5}{2}}}\right )}}{15 \, a^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/15*(15*sqrt(b*x + a)*b^2/sqrt(x) - 10*(b*x + a)^(3/2)*b/x^(3/2) + 3*(b*x + a)^(5/2)/x^(5/2))/a^3

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Fricas [A]
time = 0.69, size = 34, normalized size = 0.50 \begin {gather*} -\frac {2 \, {\left (8 \, b^{2} x^{2} - 4 \, a b x + 3 \, a^{2}\right )} \sqrt {b x + a}}{15 \, a^{3} x^{\frac {5}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/15*(8*b^2*x^2 - 4*a*b*x + 3*a^2)*sqrt(b*x + a)/(a^3*x^(5/2))

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 287 vs. \(2 (63) = 126\).
time = 3.50, size = 287, normalized size = 4.22 \begin {gather*} - \frac {6 a^{4} b^{\frac {9}{2}} \sqrt {\frac {a}{b x} + 1}}{15 a^{5} b^{4} x^{2} + 30 a^{4} b^{5} x^{3} + 15 a^{3} b^{6} x^{4}} - \frac {4 a^{3} b^{\frac {11}{2}} x \sqrt {\frac {a}{b x} + 1}}{15 a^{5} b^{4} x^{2} + 30 a^{4} b^{5} x^{3} + 15 a^{3} b^{6} x^{4}} - \frac {6 a^{2} b^{\frac {13}{2}} x^{2} \sqrt {\frac {a}{b x} + 1}}{15 a^{5} b^{4} x^{2} + 30 a^{4} b^{5} x^{3} + 15 a^{3} b^{6} x^{4}} - \frac {24 a b^{\frac {15}{2}} x^{3} \sqrt {\frac {a}{b x} + 1}}{15 a^{5} b^{4} x^{2} + 30 a^{4} b^{5} x^{3} + 15 a^{3} b^{6} x^{4}} - \frac {16 b^{\frac {17}{2}} x^{4} \sqrt {\frac {a}{b x} + 1}}{15 a^{5} b^{4} x^{2} + 30 a^{4} b^{5} x^{3} + 15 a^{3} b^{6} x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-6*a**4*b**(9/2)*sqrt(a/(b*x) + 1)/(15*a**5*b**4*x**2 + 30*a**4*b**5*x**3 + 15*a**3*b**6*x**4) - 4*a**3*b**(11
/2)*x*sqrt(a/(b*x) + 1)/(15*a**5*b**4*x**2 + 30*a**4*b**5*x**3 + 15*a**3*b**6*x**4) - 6*a**2*b**(13/2)*x**2*sq
rt(a/(b*x) + 1)/(15*a**5*b**4*x**2 + 30*a**4*b**5*x**3 + 15*a**3*b**6*x**4) - 24*a*b**(15/2)*x**3*sqrt(a/(b*x)
+ 1)/(15*a**5*b**4*x**2 + 30*a**4*b**5*x**3 + 15*a**3*b**6*x**4) - 16*b**(17/2)*x**4*sqrt(a/(b*x) + 1)/(15*a*
*5*b**4*x**2 + 30*a**4*b**5*x**3 + 15*a**3*b**6*x**4)

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Giac [A]
time = 1.20, size = 66, normalized size = 0.97 \begin {gather*} -\frac {2 \, {\left (\frac {15 \, b^{5}}{a} + 4 \, {\left (\frac {2 \, {\left (b x + a\right )} b^{5}}{a^{3}} - \frac {5 \, b^{5}}{a^{2}}\right )} {\left (b x + a\right )}\right )} \sqrt {b x + a} b}{15 \, {\left ({\left (b x + a\right )} b - a b\right )}^{\frac {5}{2}} {\left | b \right |}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.35, size = 36, normalized size = 0.53 \begin {gather*} -\frac {\sqrt {a+b\,x}\,\left (\frac {2}{5\,a}+\frac {16\,b^2\,x^2}{15\,a^3}-\frac {8\,b\,x}{15\,a^2}\right )}{x^{5/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^(7/2)*(a + b*x)^(1/2)),x)

[Out]

-((a + b*x)^(1/2)*(2/(5*a) + (16*b^2*x^2)/(15*a^3) - (8*b*x)/(15*a^2)))/x^(5/2)

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