3.521 \(\int x^{5/2} (a+b x)^{3/2} \, dx\)

Optimal. Leaf size=143 \[ -\frac{a^3 x^{3/2} \sqrt{a+b x}}{64 b^2}+\frac{3 a^4 \sqrt{x} \sqrt{a+b x}}{128 b^3}-\frac{3 a^5 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{128 b^{7/2}}+\frac{a^2 x^{5/2} \sqrt{a+b x}}{80 b}+\frac{3}{40} a x^{7/2} \sqrt{a+b x}+\frac{1}{5} x^{7/2} (a+b x)^{3/2} \]

[Out]

(3*a^4*Sqrt[x]*Sqrt[a + b*x])/(128*b^3) - (a^3*x^(3/2)*Sqrt[a + b*x])/(64*b^2) + (a^2*x^(5/2)*Sqrt[a + b*x])/(
80*b) + (3*a*x^(7/2)*Sqrt[a + b*x])/40 + (x^(7/2)*(a + b*x)^(3/2))/5 - (3*a^5*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[a
+ b*x]])/(128*b^(7/2))

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Rubi [A]  time = 0.051225, antiderivative size = 143, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {50, 63, 217, 206} \[ -\frac{a^3 x^{3/2} \sqrt{a+b x}}{64 b^2}+\frac{3 a^4 \sqrt{x} \sqrt{a+b x}}{128 b^3}-\frac{3 a^5 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{128 b^{7/2}}+\frac{a^2 x^{5/2} \sqrt{a+b x}}{80 b}+\frac{3}{40} a x^{7/2} \sqrt{a+b x}+\frac{1}{5} x^{7/2} (a+b x)^{3/2} \]

Antiderivative was successfully verified.

[In]

Int[x^(5/2)*(a + b*x)^(3/2),x]

[Out]

(3*a^4*Sqrt[x]*Sqrt[a + b*x])/(128*b^3) - (a^3*x^(3/2)*Sqrt[a + b*x])/(64*b^2) + (a^2*x^(5/2)*Sqrt[a + b*x])/(
80*b) + (3*a*x^(7/2)*Sqrt[a + b*x])/40 + (x^(7/2)*(a + b*x)^(3/2))/5 - (3*a^5*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[a
+ b*x]])/(128*b^(7/2))

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 206

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

Rubi steps

\begin{align*} \int x^{5/2} (a+b x)^{3/2} \, dx &=\frac{1}{5} x^{7/2} (a+b x)^{3/2}+\frac{1}{10} (3 a) \int x^{5/2} \sqrt{a+b x} \, dx\\ &=\frac{3}{40} a x^{7/2} \sqrt{a+b x}+\frac{1}{5} x^{7/2} (a+b x)^{3/2}+\frac{1}{80} \left (3 a^2\right ) \int \frac{x^{5/2}}{\sqrt{a+b x}} \, dx\\ &=\frac{a^2 x^{5/2} \sqrt{a+b x}}{80 b}+\frac{3}{40} a x^{7/2} \sqrt{a+b x}+\frac{1}{5} x^{7/2} (a+b x)^{3/2}-\frac{a^3 \int \frac{x^{3/2}}{\sqrt{a+b x}} \, dx}{32 b}\\ &=-\frac{a^3 x^{3/2} \sqrt{a+b x}}{64 b^2}+\frac{a^2 x^{5/2} \sqrt{a+b x}}{80 b}+\frac{3}{40} a x^{7/2} \sqrt{a+b x}+\frac{1}{5} x^{7/2} (a+b x)^{3/2}+\frac{\left (3 a^4\right ) \int \frac{\sqrt{x}}{\sqrt{a+b x}} \, dx}{128 b^2}\\ &=\frac{3 a^4 \sqrt{x} \sqrt{a+b x}}{128 b^3}-\frac{a^3 x^{3/2} \sqrt{a+b x}}{64 b^2}+\frac{a^2 x^{5/2} \sqrt{a+b x}}{80 b}+\frac{3}{40} a x^{7/2} \sqrt{a+b x}+\frac{1}{5} x^{7/2} (a+b x)^{3/2}-\frac{\left (3 a^5\right ) \int \frac{1}{\sqrt{x} \sqrt{a+b x}} \, dx}{256 b^3}\\ &=\frac{3 a^4 \sqrt{x} \sqrt{a+b x}}{128 b^3}-\frac{a^3 x^{3/2} \sqrt{a+b x}}{64 b^2}+\frac{a^2 x^{5/2} \sqrt{a+b x}}{80 b}+\frac{3}{40} a x^{7/2} \sqrt{a+b x}+\frac{1}{5} x^{7/2} (a+b x)^{3/2}-\frac{\left (3 a^5\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x^2}} \, dx,x,\sqrt{x}\right )}{128 b^3}\\ &=\frac{3 a^4 \sqrt{x} \sqrt{a+b x}}{128 b^3}-\frac{a^3 x^{3/2} \sqrt{a+b x}}{64 b^2}+\frac{a^2 x^{5/2} \sqrt{a+b x}}{80 b}+\frac{3}{40} a x^{7/2} \sqrt{a+b x}+\frac{1}{5} x^{7/2} (a+b x)^{3/2}-\frac{\left (3 a^5\right ) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{\sqrt{x}}{\sqrt{a+b x}}\right )}{128 b^3}\\ &=\frac{3 a^4 \sqrt{x} \sqrt{a+b x}}{128 b^3}-\frac{a^3 x^{3/2} \sqrt{a+b x}}{64 b^2}+\frac{a^2 x^{5/2} \sqrt{a+b x}}{80 b}+\frac{3}{40} a x^{7/2} \sqrt{a+b x}+\frac{1}{5} x^{7/2} (a+b x)^{3/2}-\frac{3 a^5 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{128 b^{7/2}}\\ \end{align*}

Mathematica [A]  time = 0.224587, size = 107, normalized size = 0.75 \[ \frac{\sqrt{a+b x} \left (\sqrt{b} \sqrt{x} \left (8 a^2 b^2 x^2-10 a^3 b x+15 a^4+176 a b^3 x^3+128 b^4 x^4\right )-\frac{15 a^{9/2} \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{\sqrt{\frac{b x}{a}+1}}\right )}{640 b^{7/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(5/2)*(a + b*x)^(3/2),x]

[Out]

(Sqrt[a + b*x]*(Sqrt[b]*Sqrt[x]*(15*a^4 - 10*a^3*b*x + 8*a^2*b^2*x^2 + 176*a*b^3*x^3 + 128*b^4*x^4) - (15*a^(9
/2)*ArcSinh[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/Sqrt[1 + (b*x)/a]))/(640*b^(7/2))

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Maple [A]  time = 0.005, size = 138, normalized size = 1. \begin{align*}{\frac{1}{5\,b}{x}^{{\frac{5}{2}}} \left ( bx+a \right ) ^{{\frac{5}{2}}}}-{\frac{a}{8\,{b}^{2}}{x}^{{\frac{3}{2}}} \left ( bx+a \right ) ^{{\frac{5}{2}}}}+{\frac{{a}^{2}}{16\,{b}^{3}} \left ( bx+a \right ) ^{{\frac{5}{2}}}\sqrt{x}}-{\frac{{a}^{3}}{64\,{b}^{3}} \left ( bx+a \right ) ^{{\frac{3}{2}}}\sqrt{x}}-{\frac{3\,{a}^{4}}{128\,{b}^{3}}\sqrt{x}\sqrt{bx+a}}-{\frac{3\,{a}^{5}}{256}\sqrt{x \left ( bx+a \right ) }\ln \left ({ \left ({\frac{a}{2}}+bx \right ){\frac{1}{\sqrt{b}}}}+\sqrt{b{x}^{2}+ax} \right ){b}^{-{\frac{7}{2}}}{\frac{1}{\sqrt{bx+a}}}{\frac{1}{\sqrt{x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(5/2)*(b*x+a)^(3/2),x)

[Out]

1/5/b*x^(5/2)*(b*x+a)^(5/2)-1/8/b^2*a*x^(3/2)*(b*x+a)^(5/2)+1/16/b^3*a^2*x^(1/2)*(b*x+a)^(5/2)-1/64/b^3*a^3*(b
*x+a)^(3/2)*x^(1/2)-3/128*a^4*x^(1/2)*(b*x+a)^(1/2)/b^3-3/256/b^(7/2)*a^5*(x*(b*x+a))^(1/2)/(b*x+a)^(1/2)/x^(1
/2)*ln((1/2*a+b*x)/b^(1/2)+(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(x^(5/2)*(b*x+a)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.59036, size = 470, normalized size = 3.29 \begin{align*} \left [\frac{15 \, a^{5} \sqrt{b} \log \left (2 \, b x - 2 \, \sqrt{b x + a} \sqrt{b} \sqrt{x} + a\right ) + 2 \,{\left (128 \, b^{5} x^{4} + 176 \, a b^{4} x^{3} + 8 \, a^{2} b^{3} x^{2} - 10 \, a^{3} b^{2} x + 15 \, a^{4} b\right )} \sqrt{b x + a} \sqrt{x}}{1280 \, b^{4}}, \frac{15 \, a^{5} \sqrt{-b} \arctan \left (\frac{\sqrt{b x + a} \sqrt{-b}}{b \sqrt{x}}\right ) +{\left (128 \, b^{5} x^{4} + 176 \, a b^{4} x^{3} + 8 \, a^{2} b^{3} x^{2} - 10 \, a^{3} b^{2} x + 15 \, a^{4} b\right )} \sqrt{b x + a} \sqrt{x}}{640 \, b^{4}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/2)*(b*x+a)^(3/2),x, algorithm="fricas")

[Out]

[1/1280*(15*a^5*sqrt(b)*log(2*b*x - 2*sqrt(b*x + a)*sqrt(b)*sqrt(x) + a) + 2*(128*b^5*x^4 + 176*a*b^4*x^3 + 8*
a^2*b^3*x^2 - 10*a^3*b^2*x + 15*a^4*b)*sqrt(b*x + a)*sqrt(x))/b^4, 1/640*(15*a^5*sqrt(-b)*arctan(sqrt(b*x + a)
*sqrt(-b)/(b*sqrt(x))) + (128*b^5*x^4 + 176*a*b^4*x^3 + 8*a^2*b^3*x^2 - 10*a^3*b^2*x + 15*a^4*b)*sqrt(b*x + a)
*sqrt(x))/b^4]

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Sympy [A]  time = 52.0442, size = 178, normalized size = 1.24 \begin{align*} \frac{3 a^{\frac{9}{2}} \sqrt{x}}{128 b^{3} \sqrt{1 + \frac{b x}{a}}} + \frac{a^{\frac{7}{2}} x^{\frac{3}{2}}}{128 b^{2} \sqrt{1 + \frac{b x}{a}}} - \frac{a^{\frac{5}{2}} x^{\frac{5}{2}}}{320 b \sqrt{1 + \frac{b x}{a}}} + \frac{23 a^{\frac{3}{2}} x^{\frac{7}{2}}}{80 \sqrt{1 + \frac{b x}{a}}} + \frac{19 \sqrt{a} b x^{\frac{9}{2}}}{40 \sqrt{1 + \frac{b x}{a}}} - \frac{3 a^{5} \operatorname{asinh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{128 b^{\frac{7}{2}}} + \frac{b^{2} x^{\frac{11}{2}}}{5 \sqrt{a} \sqrt{1 + \frac{b x}{a}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(5/2)*(b*x+a)**(3/2),x)

[Out]

3*a**(9/2)*sqrt(x)/(128*b**3*sqrt(1 + b*x/a)) + a**(7/2)*x**(3/2)/(128*b**2*sqrt(1 + b*x/a)) - a**(5/2)*x**(5/
2)/(320*b*sqrt(1 + b*x/a)) + 23*a**(3/2)*x**(7/2)/(80*sqrt(1 + b*x/a)) + 19*sqrt(a)*b*x**(9/2)/(40*sqrt(1 + b*
x/a)) - 3*a**5*asinh(sqrt(b)*sqrt(x)/sqrt(a))/(128*b**(7/2)) + b**2*x**(11/2)/(5*sqrt(a)*sqrt(1 + b*x/a))

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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(x^(5/2)*(b*x+a)^(3/2),x, algorithm="giac")

[Out]

Timed out