3.6.69 \(\int \frac {x^{5/2}}{\sqrt {a+b x}} \, dx\) [569]

Optimal. Leaf size=101 \[ \frac {5 a^2 \sqrt {x} \sqrt {a+b x}}{8 b^3}-\frac {5 a x^{3/2} \sqrt {a+b x}}{12 b^2}+\frac {x^{5/2} \sqrt {a+b x}}{3 b}-\frac {5 a^3 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{8 b^{7/2}} \]

[Out]

-5/8*a^3*arctanh(b^(1/2)*x^(1/2)/(b*x+a)^(1/2))/b^(7/2)-5/12*a*x^(3/2)*(b*x+a)^(1/2)/b^2+1/3*x^(5/2)*(b*x+a)^(
1/2)/b+5/8*a^2*x^(1/2)*(b*x+a)^(1/2)/b^3

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Rubi [A]
time = 0.02, antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {52, 65, 223, 212} \begin {gather*} -\frac {5 a^3 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{8 b^{7/2}}+\frac {5 a^2 \sqrt {x} \sqrt {a+b x}}{8 b^3}-\frac {5 a x^{3/2} \sqrt {a+b x}}{12 b^2}+\frac {x^{5/2} \sqrt {a+b x}}{3 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^(5/2)/Sqrt[a + b*x],x]

[Out]

(5*a^2*Sqrt[x]*Sqrt[a + b*x])/(8*b^3) - (5*a*x^(3/2)*Sqrt[a + b*x])/(12*b^2) + (x^(5/2)*Sqrt[a + b*x])/(3*b) -
(5*a^3*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[a + b*x]])/(8*b^(7/2))

Rule 52

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

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rubi steps

\begin {align*} \int \frac {x^{5/2}}{\sqrt {a+b x}} \, dx &=\frac {x^{5/2} \sqrt {a+b x}}{3 b}-\frac {(5 a) \int \frac {x^{3/2}}{\sqrt {a+b x}} \, dx}{6 b}\\ &=-\frac {5 a x^{3/2} \sqrt {a+b x}}{12 b^2}+\frac {x^{5/2} \sqrt {a+b x}}{3 b}+\frac {\left (5 a^2\right ) \int \frac {\sqrt {x}}{\sqrt {a+b x}} \, dx}{8 b^2}\\ &=\frac {5 a^2 \sqrt {x} \sqrt {a+b x}}{8 b^3}-\frac {5 a x^{3/2} \sqrt {a+b x}}{12 b^2}+\frac {x^{5/2} \sqrt {a+b x}}{3 b}-\frac {\left (5 a^3\right ) \int \frac {1}{\sqrt {x} \sqrt {a+b x}} \, dx}{16 b^3}\\ &=\frac {5 a^2 \sqrt {x} \sqrt {a+b x}}{8 b^3}-\frac {5 a x^{3/2} \sqrt {a+b x}}{12 b^2}+\frac {x^{5/2} \sqrt {a+b x}}{3 b}-\frac {\left (5 a^3\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\sqrt {x}\right )}{8 b^3}\\ &=\frac {5 a^2 \sqrt {x} \sqrt {a+b x}}{8 b^3}-\frac {5 a x^{3/2} \sqrt {a+b x}}{12 b^2}+\frac {x^{5/2} \sqrt {a+b x}}{3 b}-\frac {\left (5 a^3\right ) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {a+b x}}\right )}{8 b^3}\\ &=\frac {5 a^2 \sqrt {x} \sqrt {a+b x}}{8 b^3}-\frac {5 a x^{3/2} \sqrt {a+b x}}{12 b^2}+\frac {x^{5/2} \sqrt {a+b x}}{3 b}-\frac {5 a^3 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{8 b^{7/2}}\\ \end {align*}

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Mathematica [A]
time = 0.09, size = 76, normalized size = 0.75 \begin {gather*} \frac {\sqrt {x} \sqrt {a+b x} \left (15 a^2-10 a b x+8 b^2 x^2\right )}{24 b^3}+\frac {5 a^3 \log \left (-\sqrt {b} \sqrt {x}+\sqrt {a+b x}\right )}{8 b^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^(5/2)/Sqrt[a + b*x],x]

[Out]

(Sqrt[x]*Sqrt[a + b*x]*(15*a^2 - 10*a*b*x + 8*b^2*x^2))/(24*b^3) + (5*a^3*Log[-(Sqrt[b]*Sqrt[x]) + Sqrt[a + b*
x]])/(8*b^(7/2))

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Maple [A]
time = 0.12, size = 109, normalized size = 1.08

method	result	size
risch	\(\frac {\left (8 x^{2} b^{2}-10 a b x +15 a^{2}\right ) \sqrt {x}\, \sqrt {b x +a}}{24 b^{3}}-\frac {5 a^{3} \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {x^{2} b +a x}\right ) \sqrt {x \left (b x +a \right )}}{16 b^{\frac {7}{2}} \sqrt {x}\, \sqrt {b x +a}}\)	\(87\)
default	\(\frac {x^{\frac {5}{2}} \sqrt {b x +a}}{3 b}-\frac {5 a \left (\frac {x^{\frac {3}{2}} \sqrt {b x +a}}{2 b}-\frac {3 a \left (\frac {\sqrt {x}\, \sqrt {b x +a}}{b}-\frac {a \sqrt {x \left (b x +a \right )}\, \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {x^{2} b +a x}\right )}{2 b^{\frac {3}{2}} \sqrt {x}\, \sqrt {b x +a}}\right )}{4 b}\right )}{6 b}\)	\(109\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.51, size = 146, normalized size = 1.45 \begin {gather*} \frac {5 \, a^{3} \log \left (-\frac {\sqrt {b} - \frac {\sqrt {b x + a}}{\sqrt {x}}}{\sqrt {b} + \frac {\sqrt {b x + a}}{\sqrt {x}}}\right )}{16 \, b^{\frac {7}{2}}} - \frac {\frac {33 \, \sqrt {b x + a} a^{3} b^{2}}{\sqrt {x}} - \frac {40 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{3} b}{x^{\frac {3}{2}}} + \frac {15 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{3}}{x^{\frac {5}{2}}}}{24 \, {\left (b^{6} - \frac {3 \, {\left (b x + a\right )} b^{5}}{x} + \frac {3 \, {\left (b x + a\right )}^{2} b^{4}}{x^{2}} - \frac {{\left (b x + a\right )}^{3} b^{3}}{x^{3}}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5/16*a^3*log(-(sqrt(b) - sqrt(b*x + a)/sqrt(x))/(sqrt(b) + sqrt(b*x + a)/sqrt(x)))/b^(7/2) - 1/24*(33*sqrt(b*x
+ a)*a^3*b^2/sqrt(x) - 40*(b*x + a)^(3/2)*a^3*b/x^(3/2) + 15*(b*x + a)^(5/2)*a^3/x^(5/2))/(b^6 - 3*(b*x + a)*
b^5/x + 3*(b*x + a)^2*b^4/x^2 - (b*x + a)^3*b^3/x^3)

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Fricas [A]
time = 0.47, size = 140, normalized size = 1.39 \begin {gather*} \left [\frac {15 \, a^{3} \sqrt {b} \log \left (2 \, b x - 2 \, \sqrt {b x + a} \sqrt {b} \sqrt {x} + a\right ) + 2 \, {\left (8 \, b^{3} x^{2} - 10 \, a b^{2} x + 15 \, a^{2} b\right )} \sqrt {b x + a} \sqrt {x}}{48 \, b^{4}}, \frac {15 \, a^{3} \sqrt {-b} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-b}}{b \sqrt {x}}\right ) + {\left (8 \, b^{3} x^{2} - 10 \, a b^{2} x + 15 \, a^{2} b\right )} \sqrt {b x + a} \sqrt {x}}{24 \, b^{4}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/48*(15*a^3*sqrt(b)*log(2*b*x - 2*sqrt(b*x + a)*sqrt(b)*sqrt(x) + a) + 2*(8*b^3*x^2 - 10*a*b^2*x + 15*a^2*b)
*sqrt(b*x + a)*sqrt(x))/b^4, 1/24*(15*a^3*sqrt(-b)*arctan(sqrt(b*x + a)*sqrt(-b)/(b*sqrt(x))) + (8*b^3*x^2 - 1
0*a*b^2*x + 15*a^2*b)*sqrt(b*x + a)*sqrt(x))/b^4]

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Sympy [A]
time = 9.21, size = 128, normalized size = 1.27 \begin {gather*} \frac {5 a^{\frac {5}{2}} \sqrt {x}}{8 b^{3} \sqrt {1 + \frac {b x}{a}}} + \frac {5 a^{\frac {3}{2}} x^{\frac {3}{2}}}{24 b^{2} \sqrt {1 + \frac {b x}{a}}} - \frac {\sqrt {a} x^{\frac {5}{2}}}{12 b \sqrt {1 + \frac {b x}{a}}} - \frac {5 a^{3} \operatorname {asinh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{8 b^{\frac {7}{2}}} + \frac {x^{\frac {7}{2}}}{3 \sqrt {a} \sqrt {1 + \frac {b x}{a}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5*a**(5/2)*sqrt(x)/(8*b**3*sqrt(1 + b*x/a)) + 5*a**(3/2)*x**(3/2)/(24*b**2*sqrt(1 + b*x/a)) - sqrt(a)*x**(5/2)
/(12*b*sqrt(1 + b*x/a)) - 5*a**3*asinh(sqrt(b)*sqrt(x)/sqrt(a))/(8*b**(7/2)) + x**(7/2)/(3*sqrt(a)*sqrt(1 + b*
x/a))

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: NotImplementedError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: Warning, choosing root of [1,0,%%%{-4,[1
,0,0]%%%}+%%%{-4,[0,1,1]%%%}+%%%{-4,[0,1,0]%%%}+%%%{-4,[0,0,1]%%%},0,%%%{6,[2,0,0]%%%}+%%%{12,[1,1,1]%%%}+%%%{
4,[1,1,0]%%%}+%%%{4,[

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^{5/2}}{\sqrt {a+b\,x}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(5/2)/(a + b*x)^(1/2),x)

[Out]

int(x^(5/2)/(a + b*x)^(1/2), x)

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