3.5.78 \(\int \sqrt {x} (a+b x)^{5/2} (A+B x) \, dx\)

Optimal. Leaf size=192 \[ -\frac {a^4 (10 A b-3 a B) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{128 b^{5/2}}+\frac {a^3 \sqrt {x} \sqrt {a+b x} (10 A b-3 a B)}{128 b^2}+\frac {a^2 x^{3/2} \sqrt {a+b x} (10 A b-3 a B)}{64 b}+\frac {a x^{3/2} (a+b x)^{3/2} (10 A b-3 a B)}{48 b}+\frac {x^{3/2} (a+b x)^{5/2} (10 A b-3 a B)}{40 b}+\frac {B x^{3/2} (a+b x)^{7/2}}{5 b} \]

________________________________________________________________________________________

Rubi [A]  time = 0.08, antiderivative size = 192, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {80, 50, 63, 217, 206} \begin {gather*} \frac {a^3 \sqrt {x} \sqrt {a+b x} (10 A b-3 a B)}{128 b^2}-\frac {a^4 (10 A b-3 a B) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{128 b^{5/2}}+\frac {a^2 x^{3/2} \sqrt {a+b x} (10 A b-3 a B)}{64 b}+\frac {a x^{3/2} (a+b x)^{3/2} (10 A b-3 a B)}{48 b}+\frac {x^{3/2} (a+b x)^{5/2} (10 A b-3 a B)}{40 b}+\frac {B x^{3/2} (a+b x)^{7/2}}{5 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[x]*(a + b*x)^(5/2)*(A + B*x),x]

[Out]

(a^3*(10*A*b - 3*a*B)*Sqrt[x]*Sqrt[a + b*x])/(128*b^2) + (a^2*(10*A*b - 3*a*B)*x^(3/2)*Sqrt[a + b*x])/(64*b) +
(a*(10*A*b - 3*a*B)*x^(3/2)*(a + b*x)^(3/2))/(48*b) + ((10*A*b - 3*a*B)*x^(3/2)*(a + b*x)^(5/2))/(40*b) + (B*
x^(3/2)*(a + b*x)^(7/2))/(5*b) - (a^4*(10*A*b - 3*a*B)*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[a + b*x]])/(128*b^(5/2))

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 80

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(c + d*x)
^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 2)), x] + Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(
d*f*(n + p + 2)), Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2,
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])

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]

Rubi steps

\begin {align*} \int \sqrt {x} (a+b x)^{5/2} (A+B x) \, dx &=\frac {B x^{3/2} (a+b x)^{7/2}}{5 b}+\frac {\left (5 A b-\frac {3 a B}{2}\right ) \int \sqrt {x} (a+b x)^{5/2} \, dx}{5 b}\\ &=\frac {(10 A b-3 a B) x^{3/2} (a+b x)^{5/2}}{40 b}+\frac {B x^{3/2} (a+b x)^{7/2}}{5 b}+\frac {(a (10 A b-3 a B)) \int \sqrt {x} (a+b x)^{3/2} \, dx}{16 b}\\ &=\frac {a (10 A b-3 a B) x^{3/2} (a+b x)^{3/2}}{48 b}+\frac {(10 A b-3 a B) x^{3/2} (a+b x)^{5/2}}{40 b}+\frac {B x^{3/2} (a+b x)^{7/2}}{5 b}+\frac {\left (a^2 (10 A b-3 a B)\right ) \int \sqrt {x} \sqrt {a+b x} \, dx}{32 b}\\ &=\frac {a^2 (10 A b-3 a B) x^{3/2} \sqrt {a+b x}}{64 b}+\frac {a (10 A b-3 a B) x^{3/2} (a+b x)^{3/2}}{48 b}+\frac {(10 A b-3 a B) x^{3/2} (a+b x)^{5/2}}{40 b}+\frac {B x^{3/2} (a+b x)^{7/2}}{5 b}+\frac {\left (a^3 (10 A b-3 a B)\right ) \int \frac {\sqrt {x}}{\sqrt {a+b x}} \, dx}{128 b}\\ &=\frac {a^3 (10 A b-3 a B) \sqrt {x} \sqrt {a+b x}}{128 b^2}+\frac {a^2 (10 A b-3 a B) x^{3/2} \sqrt {a+b x}}{64 b}+\frac {a (10 A b-3 a B) x^{3/2} (a+b x)^{3/2}}{48 b}+\frac {(10 A b-3 a B) x^{3/2} (a+b x)^{5/2}}{40 b}+\frac {B x^{3/2} (a+b x)^{7/2}}{5 b}-\frac {\left (a^4 (10 A b-3 a B)\right ) \int \frac {1}{\sqrt {x} \sqrt {a+b x}} \, dx}{256 b^2}\\ &=\frac {a^3 (10 A b-3 a B) \sqrt {x} \sqrt {a+b x}}{128 b^2}+\frac {a^2 (10 A b-3 a B) x^{3/2} \sqrt {a+b x}}{64 b}+\frac {a (10 A b-3 a B) x^{3/2} (a+b x)^{3/2}}{48 b}+\frac {(10 A b-3 a B) x^{3/2} (a+b x)^{5/2}}{40 b}+\frac {B x^{3/2} (a+b x)^{7/2}}{5 b}-\frac {\left (a^4 (10 A b-3 a B)\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\sqrt {x}\right )}{128 b^2}\\ &=\frac {a^3 (10 A b-3 a B) \sqrt {x} \sqrt {a+b x}}{128 b^2}+\frac {a^2 (10 A b-3 a B) x^{3/2} \sqrt {a+b x}}{64 b}+\frac {a (10 A b-3 a B) x^{3/2} (a+b x)^{3/2}}{48 b}+\frac {(10 A b-3 a B) x^{3/2} (a+b x)^{5/2}}{40 b}+\frac {B x^{3/2} (a+b x)^{7/2}}{5 b}-\frac {\left (a^4 (10 A b-3 a B)\right ) \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {a+b x}}\right )}{128 b^2}\\ &=\frac {a^3 (10 A b-3 a B) \sqrt {x} \sqrt {a+b x}}{128 b^2}+\frac {a^2 (10 A b-3 a B) x^{3/2} \sqrt {a+b x}}{64 b}+\frac {a (10 A b-3 a B) x^{3/2} (a+b x)^{3/2}}{48 b}+\frac {(10 A b-3 a B) x^{3/2} (a+b x)^{5/2}}{40 b}+\frac {B x^{3/2} (a+b x)^{7/2}}{5 b}-\frac {a^4 (10 A b-3 a B) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{128 b^{5/2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.33, size = 145, normalized size = 0.76 \begin {gather*} \frac {\sqrt {a+b x} \left (\frac {15 a^{7/2} (3 a B-10 A b) \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{\sqrt {\frac {b x}{a}+1}}+\sqrt {b} \sqrt {x} \left (-45 a^4 B+30 a^3 b (5 A+B x)+4 a^2 b^2 x (295 A+186 B x)+16 a b^3 x^2 (85 A+63 B x)+96 b^4 x^3 (5 A+4 B x)\right )\right )}{1920 b^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[x]*(a + b*x)^(5/2)*(A + B*x),x]

[Out]

(Sqrt[a + b*x]*(Sqrt[b]*Sqrt[x]*(-45*a^4*B + 30*a^3*b*(5*A + B*x) + 96*b^4*x^3*(5*A + 4*B*x) + 16*a*b^3*x^2*(8
5*A + 63*B*x) + 4*a^2*b^2*x*(295*A + 186*B*x)) + (15*a^(7/2)*(-10*A*b + 3*a*B)*ArcSinh[(Sqrt[b]*Sqrt[x])/Sqrt[
a]])/Sqrt[1 + (b*x)/a]))/(1920*b^(5/2))

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.33, size = 173, normalized size = 0.90 \begin {gather*} \frac {\left (10 a^4 A b-3 a^5 B\right ) \log \left (\sqrt {a+b x}-\sqrt {b} \sqrt {x}\right )}{128 b^{5/2}}+\frac {\sqrt {a+b x} \left (-45 a^4 B \sqrt {x}+150 a^3 A b \sqrt {x}+30 a^3 b B x^{3/2}+1180 a^2 A b^2 x^{3/2}+744 a^2 b^2 B x^{5/2}+1360 a A b^3 x^{5/2}+1008 a b^3 B x^{7/2}+480 A b^4 x^{7/2}+384 b^4 B x^{9/2}\right )}{1920 b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[Sqrt[x]*(a + b*x)^(5/2)*(A + B*x),x]

[Out]

(Sqrt[a + b*x]*(150*a^3*A*b*Sqrt[x] - 45*a^4*B*Sqrt[x] + 1180*a^2*A*b^2*x^(3/2) + 30*a^3*b*B*x^(3/2) + 1360*a*
A*b^3*x^(5/2) + 744*a^2*b^2*B*x^(5/2) + 480*A*b^4*x^(7/2) + 1008*a*b^3*B*x^(7/2) + 384*b^4*B*x^(9/2)))/(1920*b
^2) + ((10*a^4*A*b - 3*a^5*B)*Log[-(Sqrt[b]*Sqrt[x]) + Sqrt[a + b*x]])/(128*b^(5/2))

________________________________________________________________________________________

fricas [A]  time = 1.31, size = 297, normalized size = 1.55 \begin {gather*} \left [-\frac {15 \, {\left (3 \, B a^{5} - 10 \, A a^{4} b\right )} \sqrt {b} \log \left (2 \, b x - 2 \, \sqrt {b x + a} \sqrt {b} \sqrt {x} + a\right ) - 2 \, {\left (384 \, B b^{5} x^{4} - 45 \, B a^{4} b + 150 \, A a^{3} b^{2} + 48 \, {\left (21 \, B a b^{4} + 10 \, A b^{5}\right )} x^{3} + 8 \, {\left (93 \, B a^{2} b^{3} + 170 \, A a b^{4}\right )} x^{2} + 10 \, {\left (3 \, B a^{3} b^{2} + 118 \, A a^{2} b^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {x}}{3840 \, b^{3}}, -\frac {15 \, {\left (3 \, B a^{5} - 10 \, A a^{4} b\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-b}}{b \sqrt {x}}\right ) - {\left (384 \, B b^{5} x^{4} - 45 \, B a^{4} b + 150 \, A a^{3} b^{2} + 48 \, {\left (21 \, B a b^{4} + 10 \, A b^{5}\right )} x^{3} + 8 \, {\left (93 \, B a^{2} b^{3} + 170 \, A a b^{4}\right )} x^{2} + 10 \, {\left (3 \, B a^{3} b^{2} + 118 \, A a^{2} b^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {x}}{1920 \, b^{3}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/3840*(15*(3*B*a^5 - 10*A*a^4*b)*sqrt(b)*log(2*b*x - 2*sqrt(b*x + a)*sqrt(b)*sqrt(x) + a) - 2*(384*B*b^5*x^
4 - 45*B*a^4*b + 150*A*a^3*b^2 + 48*(21*B*a*b^4 + 10*A*b^5)*x^3 + 8*(93*B*a^2*b^3 + 170*A*a*b^4)*x^2 + 10*(3*B
*a^3*b^2 + 118*A*a^2*b^3)*x)*sqrt(b*x + a)*sqrt(x))/b^3, -1/1920*(15*(3*B*a^5 - 10*A*a^4*b)*sqrt(-b)*arctan(sq
rt(b*x + a)*sqrt(-b)/(b*sqrt(x))) - (384*B*b^5*x^4 - 45*B*a^4*b + 150*A*a^3*b^2 + 48*(21*B*a*b^4 + 10*A*b^5)*x
^3 + 8*(93*B*a^2*b^3 + 170*A*a*b^4)*x^2 + 10*(3*B*a^3*b^2 + 118*A*a^2*b^3)*x)*sqrt(b*x + a)*sqrt(x))/b^3]

________________________________________________________________________________________

giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

maple [A]  time = 0.01, size = 260, normalized size = 1.35 \begin {gather*} -\frac {\sqrt {b x +a}\, \left (-768 \sqrt {\left (b x +a \right ) x}\, B \,b^{\frac {9}{2}} x^{4}-960 \sqrt {\left (b x +a \right ) x}\, A \,b^{\frac {9}{2}} x^{3}-2016 \sqrt {\left (b x +a \right ) x}\, B a \,b^{\frac {7}{2}} x^{3}-2720 \sqrt {\left (b x +a \right ) x}\, A a \,b^{\frac {7}{2}} x^{2}-1488 \sqrt {\left (b x +a \right ) x}\, B \,a^{2} b^{\frac {5}{2}} x^{2}+150 A \,a^{4} b \ln \left (\frac {2 b x +a +2 \sqrt {\left (b x +a \right ) x}\, \sqrt {b}}{2 \sqrt {b}}\right )-45 B \,a^{5} \ln \left (\frac {2 b x +a +2 \sqrt {\left (b x +a \right ) x}\, \sqrt {b}}{2 \sqrt {b}}\right )-2360 \sqrt {\left (b x +a \right ) x}\, A \,a^{2} b^{\frac {5}{2}} x -60 \sqrt {\left (b x +a \right ) x}\, B \,a^{3} b^{\frac {3}{2}} x -300 \sqrt {\left (b x +a \right ) x}\, A \,a^{3} b^{\frac {3}{2}}+90 \sqrt {\left (b x +a \right ) x}\, B \,a^{4} \sqrt {b}\right ) \sqrt {x}}{3840 \sqrt {\left (b x +a \right ) x}\, b^{\frac {5}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3840*(b*x+a)^(1/2)*x^(1/2)/b^(5/2)*(-768*((b*x+a)*x)^(1/2)*B*b^(9/2)*x^4-960*((b*x+a)*x)^(1/2)*A*b^(9/2)*x^
3-2016*((b*x+a)*x)^(1/2)*B*a*b^(7/2)*x^3-2720*((b*x+a)*x)^(1/2)*A*a*b^(7/2)*x^2-1488*((b*x+a)*x)^(1/2)*B*a^2*b
^(5/2)*x^2-2360*((b*x+a)*x)^(1/2)*A*a^2*b^(5/2)*x-60*((b*x+a)*x)^(1/2)*B*a^3*b^(3/2)*x+150*A*a^4*b*ln(1/2*(2*b
*x+a+2*((b*x+a)*x)^(1/2)*b^(1/2))/b^(1/2))-300*((b*x+a)*x)^(1/2)*A*a^3*b^(3/2)-45*B*a^5*ln(1/2*(2*b*x+a+2*((b*
x+a)*x)^(1/2)*b^(1/2))/b^(1/2))+90*((b*x+a)*x)^(1/2)*B*a^4*b^(1/2))/((b*x+a)*x)^(1/2)

________________________________________________________________________________________

maxima [B]  time = 0.96, size = 485, normalized size = 2.53 \begin {gather*} \frac {1}{5} \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} B b x^{2} - \frac {7}{40} \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} B a x + \frac {1}{2} \, \sqrt {b x^{2} + a x} A a^{2} x - \frac {7 \, \sqrt {b x^{2} + a x} B a^{3} x}{64 \, b} + \frac {7 \, B a^{5} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right )}{256 \, b^{\frac {5}{2}}} - \frac {A a^{4} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right )}{8 \, b^{\frac {3}{2}}} - \frac {7 \, \sqrt {b x^{2} + a x} B a^{4}}{128 \, b^{2}} + \frac {7 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} B a^{2}}{48 \, b} + \frac {\sqrt {b x^{2} + a x} A a^{3}}{4 \, b} + \frac {5 \, {\left (2 \, B a b + A b^{2}\right )} \sqrt {b x^{2} + a x} a^{2} x}{32 \, b^{2}} + \frac {{\left (2 \, B a b + A b^{2}\right )} {\left (b x^{2} + a x\right )}^{\frac {3}{2}} x}{4 \, b} - \frac {{\left (B a^{2} + 2 \, A a b\right )} \sqrt {b x^{2} + a x} a x}{4 \, b} - \frac {5 \, {\left (2 \, B a b + A b^{2}\right )} a^{4} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right )}{128 \, b^{\frac {7}{2}}} + \frac {{\left (B a^{2} + 2 \, A a b\right )} a^{3} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right )}{16 \, b^{\frac {5}{2}}} + \frac {5 \, {\left (2 \, B a b + A b^{2}\right )} \sqrt {b x^{2} + a x} a^{3}}{64 \, b^{3}} - \frac {5 \, {\left (2 \, B a b + A b^{2}\right )} {\left (b x^{2} + a x\right )}^{\frac {3}{2}} a}{24 \, b^{2}} - \frac {{\left (B a^{2} + 2 \, A a b\right )} \sqrt {b x^{2} + a x} a^{2}}{8 \, b^{2}} + \frac {{\left (B a^{2} + 2 \, A a b\right )} {\left (b x^{2} + a x\right )}^{\frac {3}{2}}}{3 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*(b*x^2 + a*x)^(3/2)*B*b*x^2 - 7/40*(b*x^2 + a*x)^(3/2)*B*a*x + 1/2*sqrt(b*x^2 + a*x)*A*a^2*x - 7/64*sqrt(b
*x^2 + a*x)*B*a^3*x/b + 7/256*B*a^5*log(2*b*x + a + 2*sqrt(b*x^2 + a*x)*sqrt(b))/b^(5/2) - 1/8*A*a^4*log(2*b*x
+ a + 2*sqrt(b*x^2 + a*x)*sqrt(b))/b^(3/2) - 7/128*sqrt(b*x^2 + a*x)*B*a^4/b^2 + 7/48*(b*x^2 + a*x)^(3/2)*B*a
^2/b + 1/4*sqrt(b*x^2 + a*x)*A*a^3/b + 5/32*(2*B*a*b + A*b^2)*sqrt(b*x^2 + a*x)*a^2*x/b^2 + 1/4*(2*B*a*b + A*b
^2)*(b*x^2 + a*x)^(3/2)*x/b - 1/4*(B*a^2 + 2*A*a*b)*sqrt(b*x^2 + a*x)*a*x/b - 5/128*(2*B*a*b + A*b^2)*a^4*log(
2*b*x + a + 2*sqrt(b*x^2 + a*x)*sqrt(b))/b^(7/2) + 1/16*(B*a^2 + 2*A*a*b)*a^3*log(2*b*x + a + 2*sqrt(b*x^2 + a
*x)*sqrt(b))/b^(5/2) + 5/64*(2*B*a*b + A*b^2)*sqrt(b*x^2 + a*x)*a^3/b^3 - 5/24*(2*B*a*b + A*b^2)*(b*x^2 + a*x)
^(3/2)*a/b^2 - 1/8*(B*a^2 + 2*A*a*b)*sqrt(b*x^2 + a*x)*a^2/b^2 + 1/3*(B*a^2 + 2*A*a*b)*(b*x^2 + a*x)^(3/2)/b

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \sqrt {x}\,\left (A+B\,x\right )\,{\left (a+b\,x\right )}^{5/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [B]  time = 51.62, size = 359, normalized size = 1.87 \begin {gather*} \frac {5 A a^{\frac {7}{2}} \sqrt {x}}{64 b \sqrt {1 + \frac {b x}{a}}} + \frac {133 A a^{\frac {5}{2}} x^{\frac {3}{2}}}{192 \sqrt {1 + \frac {b x}{a}}} + \frac {127 A a^{\frac {3}{2}} b x^{\frac {5}{2}}}{96 \sqrt {1 + \frac {b x}{a}}} + \frac {23 A \sqrt {a} b^{2} x^{\frac {7}{2}}}{24 \sqrt {1 + \frac {b x}{a}}} - \frac {5 A a^{4} \operatorname {asinh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{64 b^{\frac {3}{2}}} + \frac {A b^{3} x^{\frac {9}{2}}}{4 \sqrt {a} \sqrt {1 + \frac {b x}{a}}} - \frac {3 B a^{\frac {9}{2}} \sqrt {x}}{128 b^{2} \sqrt {1 + \frac {b x}{a}}} - \frac {B a^{\frac {7}{2}} x^{\frac {3}{2}}}{128 b \sqrt {1 + \frac {b x}{a}}} + \frac {129 B a^{\frac {5}{2}} x^{\frac {5}{2}}}{320 \sqrt {1 + \frac {b x}{a}}} + \frac {73 B a^{\frac {3}{2}} b x^{\frac {7}{2}}}{80 \sqrt {1 + \frac {b x}{a}}} + \frac {29 B \sqrt {a} b^{2} x^{\frac {9}{2}}}{40 \sqrt {1 + \frac {b x}{a}}} + \frac {3 B a^{5} \operatorname {asinh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{128 b^{\frac {5}{2}}} + \frac {B b^{3} x^{\frac {11}{2}}}{5 \sqrt {a} \sqrt {1 + \frac {b x}{a}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5*A*a**(7/2)*sqrt(x)/(64*b*sqrt(1 + b*x/a)) + 133*A*a**(5/2)*x**(3/2)/(192*sqrt(1 + b*x/a)) + 127*A*a**(3/2)*b
*x**(5/2)/(96*sqrt(1 + b*x/a)) + 23*A*sqrt(a)*b**2*x**(7/2)/(24*sqrt(1 + b*x/a)) - 5*A*a**4*asinh(sqrt(b)*sqrt
(x)/sqrt(a))/(64*b**(3/2)) + A*b**3*x**(9/2)/(4*sqrt(a)*sqrt(1 + b*x/a)) - 3*B*a**(9/2)*sqrt(x)/(128*b**2*sqrt
(1 + b*x/a)) - B*a**(7/2)*x**(3/2)/(128*b*sqrt(1 + b*x/a)) + 129*B*a**(5/2)*x**(5/2)/(320*sqrt(1 + b*x/a)) + 7
3*B*a**(3/2)*b*x**(7/2)/(80*sqrt(1 + b*x/a)) + 29*B*sqrt(a)*b**2*x**(9/2)/(40*sqrt(1 + b*x/a)) + 3*B*a**5*asin
h(sqrt(b)*sqrt(x)/sqrt(a))/(128*b**(5/2)) + B*b**3*x**(11/2)/(5*sqrt(a)*sqrt(1 + b*x/a))

________________________________________________________________________________________