3.553 \(\int \sqrt{x} (a-b x)^{5/2} \, dx\)

Optimal. Leaf size=121 \[ \frac{5 a^4 \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a-b x}}\right )}{64 b^{3/2}}+\frac{5}{32} a^2 x^{3/2} \sqrt{a-b x}-\frac{5 a^3 \sqrt{x} \sqrt{a-b x}}{64 b}+\frac{5}{24} a x^{3/2} (a-b x)^{3/2}+\frac{1}{4} x^{3/2} (a-b x)^{5/2} \]

[Out]

(-5*a^3*Sqrt[x]*Sqrt[a - b*x])/(64*b) + (5*a^2*x^(3/2)*Sqrt[a - b*x])/32 + (5*a*x^(3/2)*(a - b*x)^(3/2))/24 +
(x^(3/2)*(a - b*x)^(5/2))/4 + (5*a^4*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a - b*x]])/(64*b^(3/2))

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Rubi [A]  time = 0.0377049, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {50, 63, 217, 203} \[ \frac{5 a^4 \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a-b x}}\right )}{64 b^{3/2}}+\frac{5}{32} a^2 x^{3/2} \sqrt{a-b x}-\frac{5 a^3 \sqrt{x} \sqrt{a-b x}}{64 b}+\frac{5}{24} a x^{3/2} (a-b x)^{3/2}+\frac{1}{4} x^{3/2} (a-b x)^{5/2} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[x]*(a - b*x)^(5/2),x]

[Out]

(-5*a^3*Sqrt[x]*Sqrt[a - b*x])/(64*b) + (5*a^2*x^(3/2)*Sqrt[a - b*x])/32 + (5*a*x^(3/2)*(a - b*x)^(3/2))/24 +
(x^(3/2)*(a - b*x)^(5/2))/4 + (5*a^4*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a - b*x]])/(64*b^(3/2))

Rule 50

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 63

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 217

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]

Rule 203

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

Rubi steps

\begin{align*} \int \sqrt{x} (a-b x)^{5/2} \, dx &=\frac{1}{4} x^{3/2} (a-b x)^{5/2}+\frac{1}{8} (5 a) \int \sqrt{x} (a-b x)^{3/2} \, dx\\ &=\frac{5}{24} a x^{3/2} (a-b x)^{3/2}+\frac{1}{4} x^{3/2} (a-b x)^{5/2}+\frac{1}{16} \left (5 a^2\right ) \int \sqrt{x} \sqrt{a-b x} \, dx\\ &=\frac{5}{32} a^2 x^{3/2} \sqrt{a-b x}+\frac{5}{24} a x^{3/2} (a-b x)^{3/2}+\frac{1}{4} x^{3/2} (a-b x)^{5/2}+\frac{1}{64} \left (5 a^3\right ) \int \frac{\sqrt{x}}{\sqrt{a-b x}} \, dx\\ &=-\frac{5 a^3 \sqrt{x} \sqrt{a-b x}}{64 b}+\frac{5}{32} a^2 x^{3/2} \sqrt{a-b x}+\frac{5}{24} a x^{3/2} (a-b x)^{3/2}+\frac{1}{4} x^{3/2} (a-b x)^{5/2}+\frac{\left (5 a^4\right ) \int \frac{1}{\sqrt{x} \sqrt{a-b x}} \, dx}{128 b}\\ &=-\frac{5 a^3 \sqrt{x} \sqrt{a-b x}}{64 b}+\frac{5}{32} a^2 x^{3/2} \sqrt{a-b x}+\frac{5}{24} a x^{3/2} (a-b x)^{3/2}+\frac{1}{4} x^{3/2} (a-b x)^{5/2}+\frac{\left (5 a^4\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a-b x^2}} \, dx,x,\sqrt{x}\right )}{64 b}\\ &=-\frac{5 a^3 \sqrt{x} \sqrt{a-b x}}{64 b}+\frac{5}{32} a^2 x^{3/2} \sqrt{a-b x}+\frac{5}{24} a x^{3/2} (a-b x)^{3/2}+\frac{1}{4} x^{3/2} (a-b x)^{5/2}+\frac{\left (5 a^4\right ) \operatorname{Subst}\left (\int \frac{1}{1+b x^2} \, dx,x,\frac{\sqrt{x}}{\sqrt{a-b x}}\right )}{64 b}\\ &=-\frac{5 a^3 \sqrt{x} \sqrt{a-b x}}{64 b}+\frac{5}{32} a^2 x^{3/2} \sqrt{a-b x}+\frac{5}{24} a x^{3/2} (a-b x)^{3/2}+\frac{1}{4} x^{3/2} (a-b x)^{5/2}+\frac{5 a^4 \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a-b x}}\right )}{64 b^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.120693, size = 98, normalized size = 0.81 \[ \frac{\sqrt{a-b x} \left (\sqrt{b} \sqrt{x} \left (118 a^2 b x-15 a^3-136 a b^2 x^2+48 b^3 x^3\right )+\frac{15 a^{7/2} \sin ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{\sqrt{1-\frac{b x}{a}}}\right )}{192 b^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[x]*(a - b*x)^(5/2),x]

[Out]

(Sqrt[a - b*x]*(Sqrt[b]*Sqrt[x]*(-15*a^3 + 118*a^2*b*x - 136*a*b^2*x^2 + 48*b^3*x^3) + (15*a^(7/2)*ArcSin[(Sqr
t[b]*Sqrt[x])/Sqrt[a]])/Sqrt[1 - (b*x)/a]))/(192*b^(3/2))

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Maple [A]  time = 0.005, size = 118, normalized size = 1. \begin{align*}{\frac{1}{4}{x}^{{\frac{3}{2}}} \left ( -bx+a \right ) ^{{\frac{5}{2}}}}+{\frac{5\,a}{24}{x}^{{\frac{3}{2}}} \left ( -bx+a \right ) ^{{\frac{3}{2}}}}+{\frac{5\,{a}^{2}}{32}{x}^{{\frac{3}{2}}}\sqrt{-bx+a}}-{\frac{5\,{a}^{3}}{64\,b}\sqrt{x}\sqrt{-bx+a}}+{\frac{5\,{a}^{4}}{128}\sqrt{x \left ( -bx+a \right ) }\arctan \left ({\sqrt{b} \left ( x-{\frac{a}{2\,b}} \right ){\frac{1}{\sqrt{-b{x}^{2}+ax}}}} \right ){b}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{-bx+a}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*x^(3/2)*(-b*x+a)^(5/2)+5/24*a*x^(3/2)*(-b*x+a)^(3/2)+5/32*a^2*x^(3/2)*(-b*x+a)^(1/2)-5/64*a^3*x^(1/2)*(-b*
x+a)^(1/2)/b+5/128*a^4/b^(3/2)*(x*(-b*x+a))^(1/2)/x^(1/2)/(-b*x+a)^(1/2)*arctan(b^(1/2)*(x-1/2/b*a)/(-b*x^2+a*
x)^(1/2))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.78927, size = 432, normalized size = 3.57 \begin{align*} \left [-\frac{15 \, a^{4} \sqrt{-b} \log \left (-2 \, b x + 2 \, \sqrt{-b x + a} \sqrt{-b} \sqrt{x} + a\right ) - 2 \,{\left (48 \, b^{4} x^{3} - 136 \, a b^{3} x^{2} + 118 \, a^{2} b^{2} x - 15 \, a^{3} b\right )} \sqrt{-b x + a} \sqrt{x}}{384 \, b^{2}}, -\frac{15 \, a^{4} \sqrt{b} \arctan \left (\frac{\sqrt{-b x + a}}{\sqrt{b} \sqrt{x}}\right ) -{\left (48 \, b^{4} x^{3} - 136 \, a b^{3} x^{2} + 118 \, a^{2} b^{2} x - 15 \, a^{3} b\right )} \sqrt{-b x + a} \sqrt{x}}{192 \, b^{2}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/384*(15*a^4*sqrt(-b)*log(-2*b*x + 2*sqrt(-b*x + a)*sqrt(-b)*sqrt(x) + a) - 2*(48*b^4*x^3 - 136*a*b^3*x^2 +
118*a^2*b^2*x - 15*a^3*b)*sqrt(-b*x + a)*sqrt(x))/b^2, -1/192*(15*a^4*sqrt(b)*arctan(sqrt(-b*x + a)/(sqrt(b)*
sqrt(x))) - (48*b^4*x^3 - 136*a*b^3*x^2 + 118*a^2*b^2*x - 15*a^3*b)*sqrt(-b*x + a)*sqrt(x))/b^2]

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Sympy [A]  time = 19.0609, size = 328, normalized size = 2.71 \begin{align*} \begin{cases} \frac{5 i a^{\frac{7}{2}} \sqrt{x}}{64 b \sqrt{-1 + \frac{b x}{a}}} - \frac{133 i a^{\frac{5}{2}} x^{\frac{3}{2}}}{192 \sqrt{-1 + \frac{b x}{a}}} + \frac{127 i a^{\frac{3}{2}} b x^{\frac{5}{2}}}{96 \sqrt{-1 + \frac{b x}{a}}} - \frac{23 i \sqrt{a} b^{2} x^{\frac{7}{2}}}{24 \sqrt{-1 + \frac{b x}{a}}} - \frac{5 i a^{4} \operatorname{acosh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{64 b^{\frac{3}{2}}} + \frac{i b^{3} x^{\frac{9}{2}}}{4 \sqrt{a} \sqrt{-1 + \frac{b x}{a}}} & \text{for}\: \frac{\left |{b x}\right |}{\left |{a}\right |} > 1 \\- \frac{5 a^{\frac{7}{2}} \sqrt{x}}{64 b \sqrt{1 - \frac{b x}{a}}} + \frac{133 a^{\frac{5}{2}} x^{\frac{3}{2}}}{192 \sqrt{1 - \frac{b x}{a}}} - \frac{127 a^{\frac{3}{2}} b x^{\frac{5}{2}}}{96 \sqrt{1 - \frac{b x}{a}}} + \frac{23 \sqrt{a} b^{2} x^{\frac{7}{2}}}{24 \sqrt{1 - \frac{b x}{a}}} + \frac{5 a^{4} \operatorname{asin}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{64 b^{\frac{3}{2}}} - \frac{b^{3} x^{\frac{9}{2}}}{4 \sqrt{a} \sqrt{1 - \frac{b x}{a}}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((5*I*a**(7/2)*sqrt(x)/(64*b*sqrt(-1 + b*x/a)) - 133*I*a**(5/2)*x**(3/2)/(192*sqrt(-1 + b*x/a)) + 127
*I*a**(3/2)*b*x**(5/2)/(96*sqrt(-1 + b*x/a)) - 23*I*sqrt(a)*b**2*x**(7/2)/(24*sqrt(-1 + b*x/a)) - 5*I*a**4*aco
sh(sqrt(b)*sqrt(x)/sqrt(a))/(64*b**(3/2)) + I*b**3*x**(9/2)/(4*sqrt(a)*sqrt(-1 + b*x/a)), Abs(b*x)/Abs(a) > 1)
, (-5*a**(7/2)*sqrt(x)/(64*b*sqrt(1 - b*x/a)) + 133*a**(5/2)*x**(3/2)/(192*sqrt(1 - b*x/a)) - 127*a**(3/2)*b*x
**(5/2)/(96*sqrt(1 - b*x/a)) + 23*sqrt(a)*b**2*x**(7/2)/(24*sqrt(1 - b*x/a)) + 5*a**4*asin(sqrt(b)*sqrt(x)/sqr
t(a))/(64*b**(3/2)) - b**3*x**(9/2)/(4*sqrt(a)*sqrt(1 - b*x/a)), True))

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Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out