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

Optimal. Leaf size=225 \[ \frac {a^5 (12 A b-5 a B) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{512 b^{7/2}}-\frac {a^4 \sqrt {x} \sqrt {a+b x} (12 A b-5 a B)}{512 b^3}+\frac {a^3 x^{3/2} \sqrt {a+b x} (12 A b-5 a B)}{768 b^2}+\frac {a^2 x^{5/2} \sqrt {a+b x} (12 A b-5 a B)}{192 b}+\frac {a x^{5/2} (a+b x)^{3/2} (12 A b-5 a B)}{96 b}+\frac {x^{5/2} (a+b x)^{5/2} (12 A b-5 a B)}{60 b}+\frac {B x^{5/2} (a+b x)^{7/2}}{6 b} \]

________________________________________________________________________________________

Rubi [A]  time = 0.11, antiderivative size = 225, normalized size of antiderivative = 1.00, number of steps used = 9, 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 x^{3/2} \sqrt {a+b x} (12 A b-5 a B)}{768 b^2}-\frac {a^4 \sqrt {x} \sqrt {a+b x} (12 A b-5 a B)}{512 b^3}+\frac {a^5 (12 A b-5 a B) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{512 b^{7/2}}+\frac {a^2 x^{5/2} \sqrt {a+b x} (12 A b-5 a B)}{192 b}+\frac {a x^{5/2} (a+b x)^{3/2} (12 A b-5 a B)}{96 b}+\frac {x^{5/2} (a+b x)^{5/2} (12 A b-5 a B)}{60 b}+\frac {B x^{5/2} (a+b x)^{7/2}}{6 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^(3/2)*(a + b*x)^(5/2)*(A + B*x),x]

[Out]

-(a^4*(12*A*b - 5*a*B)*Sqrt[x]*Sqrt[a + b*x])/(512*b^3) + (a^3*(12*A*b - 5*a*B)*x^(3/2)*Sqrt[a + b*x])/(768*b^
2) + (a^2*(12*A*b - 5*a*B)*x^(5/2)*Sqrt[a + b*x])/(192*b) + (a*(12*A*b - 5*a*B)*x^(5/2)*(a + b*x)^(3/2))/(96*b
) + ((12*A*b - 5*a*B)*x^(5/2)*(a + b*x)^(5/2))/(60*b) + (B*x^(5/2)*(a + b*x)^(7/2))/(6*b) + (a^5*(12*A*b - 5*a
*B)*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[a + b*x]])/(512*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 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 x^{3/2} (a+b x)^{5/2} (A+B x) \, dx &=\frac {B x^{5/2} (a+b x)^{7/2}}{6 b}+\frac {\left (6 A b-\frac {5 a B}{2}\right ) \int x^{3/2} (a+b x)^{5/2} \, dx}{6 b}\\ &=\frac {(12 A b-5 a B) x^{5/2} (a+b x)^{5/2}}{60 b}+\frac {B x^{5/2} (a+b x)^{7/2}}{6 b}+\frac {(a (12 A b-5 a B)) \int x^{3/2} (a+b x)^{3/2} \, dx}{24 b}\\ &=\frac {a (12 A b-5 a B) x^{5/2} (a+b x)^{3/2}}{96 b}+\frac {(12 A b-5 a B) x^{5/2} (a+b x)^{5/2}}{60 b}+\frac {B x^{5/2} (a+b x)^{7/2}}{6 b}+\frac {\left (a^2 (12 A b-5 a B)\right ) \int x^{3/2} \sqrt {a+b x} \, dx}{64 b}\\ &=\frac {a^2 (12 A b-5 a B) x^{5/2} \sqrt {a+b x}}{192 b}+\frac {a (12 A b-5 a B) x^{5/2} (a+b x)^{3/2}}{96 b}+\frac {(12 A b-5 a B) x^{5/2} (a+b x)^{5/2}}{60 b}+\frac {B x^{5/2} (a+b x)^{7/2}}{6 b}+\frac {\left (a^3 (12 A b-5 a B)\right ) \int \frac {x^{3/2}}{\sqrt {a+b x}} \, dx}{384 b}\\ &=\frac {a^3 (12 A b-5 a B) x^{3/2} \sqrt {a+b x}}{768 b^2}+\frac {a^2 (12 A b-5 a B) x^{5/2} \sqrt {a+b x}}{192 b}+\frac {a (12 A b-5 a B) x^{5/2} (a+b x)^{3/2}}{96 b}+\frac {(12 A b-5 a B) x^{5/2} (a+b x)^{5/2}}{60 b}+\frac {B x^{5/2} (a+b x)^{7/2}}{6 b}-\frac {\left (a^4 (12 A b-5 a B)\right ) \int \frac {\sqrt {x}}{\sqrt {a+b x}} \, dx}{512 b^2}\\ &=-\frac {a^4 (12 A b-5 a B) \sqrt {x} \sqrt {a+b x}}{512 b^3}+\frac {a^3 (12 A b-5 a B) x^{3/2} \sqrt {a+b x}}{768 b^2}+\frac {a^2 (12 A b-5 a B) x^{5/2} \sqrt {a+b x}}{192 b}+\frac {a (12 A b-5 a B) x^{5/2} (a+b x)^{3/2}}{96 b}+\frac {(12 A b-5 a B) x^{5/2} (a+b x)^{5/2}}{60 b}+\frac {B x^{5/2} (a+b x)^{7/2}}{6 b}+\frac {\left (a^5 (12 A b-5 a B)\right ) \int \frac {1}{\sqrt {x} \sqrt {a+b x}} \, dx}{1024 b^3}\\ &=-\frac {a^4 (12 A b-5 a B) \sqrt {x} \sqrt {a+b x}}{512 b^3}+\frac {a^3 (12 A b-5 a B) x^{3/2} \sqrt {a+b x}}{768 b^2}+\frac {a^2 (12 A b-5 a B) x^{5/2} \sqrt {a+b x}}{192 b}+\frac {a (12 A b-5 a B) x^{5/2} (a+b x)^{3/2}}{96 b}+\frac {(12 A b-5 a B) x^{5/2} (a+b x)^{5/2}}{60 b}+\frac {B x^{5/2} (a+b x)^{7/2}}{6 b}+\frac {\left (a^5 (12 A b-5 a B)\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\sqrt {x}\right )}{512 b^3}\\ &=-\frac {a^4 (12 A b-5 a B) \sqrt {x} \sqrt {a+b x}}{512 b^3}+\frac {a^3 (12 A b-5 a B) x^{3/2} \sqrt {a+b x}}{768 b^2}+\frac {a^2 (12 A b-5 a B) x^{5/2} \sqrt {a+b x}}{192 b}+\frac {a (12 A b-5 a B) x^{5/2} (a+b x)^{3/2}}{96 b}+\frac {(12 A b-5 a B) x^{5/2} (a+b x)^{5/2}}{60 b}+\frac {B x^{5/2} (a+b x)^{7/2}}{6 b}+\frac {\left (a^5 (12 A b-5 a B)\right ) \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {a+b x}}\right )}{512 b^3}\\ &=-\frac {a^4 (12 A b-5 a B) \sqrt {x} \sqrt {a+b x}}{512 b^3}+\frac {a^3 (12 A b-5 a B) x^{3/2} \sqrt {a+b x}}{768 b^2}+\frac {a^2 (12 A b-5 a B) x^{5/2} \sqrt {a+b x}}{192 b}+\frac {a (12 A b-5 a B) x^{5/2} (a+b x)^{3/2}}{96 b}+\frac {(12 A b-5 a B) x^{5/2} (a+b x)^{5/2}}{60 b}+\frac {B x^{5/2} (a+b x)^{7/2}}{6 b}+\frac {a^5 (12 A b-5 a B) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{512 b^{7/2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.41, size = 151, normalized size = 0.67 \begin {gather*} \frac {\sqrt {a+b x} (12 A b-5 a B) \left (15 a^{9/2} \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )+\sqrt {b} \sqrt {x} \sqrt {\frac {b x}{a}+1} \left (-15 a^4+10 a^3 b x+248 a^2 b^2 x^2+336 a b^3 x^3+128 b^4 x^4\right )\right )}{7680 b^{7/2} \sqrt {\frac {b x}{a}+1}}+\frac {B x^{5/2} (a+b x)^{7/2}}{6 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^(3/2)*(a + b*x)^(5/2)*(A + B*x),x]

[Out]

(B*x^(5/2)*(a + b*x)^(7/2))/(6*b) + ((12*A*b - 5*a*B)*Sqrt[a + b*x]*(Sqrt[b]*Sqrt[x]*Sqrt[1 + (b*x)/a]*(-15*a^
4 + 10*a^3*b*x + 248*a^2*b^2*x^2 + 336*a*b^3*x^3 + 128*b^4*x^4) + 15*a^(9/2)*ArcSinh[(Sqrt[b]*Sqrt[x])/Sqrt[a]
]))/(7680*b^(7/2)*Sqrt[1 + (b*x)/a])

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.34, size = 201, normalized size = 0.89 \begin {gather*} \frac {\left (5 a^6 B-12 a^5 A b\right ) \log \left (\sqrt {a+b x}-\sqrt {b} \sqrt {x}\right )}{512 b^{7/2}}+\frac {\sqrt {a+b x} \left (75 a^5 B \sqrt {x}-180 a^4 A b \sqrt {x}-50 a^4 b B x^{3/2}+120 a^3 A b^2 x^{3/2}+40 a^3 b^2 B x^{5/2}+2976 a^2 A b^3 x^{5/2}+2160 a^2 b^3 B x^{7/2}+4032 a A b^4 x^{7/2}+3200 a b^4 B x^{9/2}+1536 A b^5 x^{9/2}+1280 b^5 B x^{11/2}\right )}{7680 b^3} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[x^(3/2)*(a + b*x)^(5/2)*(A + B*x),x]

[Out]

(Sqrt[a + b*x]*(-180*a^4*A*b*Sqrt[x] + 75*a^5*B*Sqrt[x] + 120*a^3*A*b^2*x^(3/2) - 50*a^4*b*B*x^(3/2) + 2976*a^
2*A*b^3*x^(5/2) + 40*a^3*b^2*B*x^(5/2) + 4032*a*A*b^4*x^(7/2) + 2160*a^2*b^3*B*x^(7/2) + 1536*A*b^5*x^(9/2) +
3200*a*b^4*B*x^(9/2) + 1280*b^5*B*x^(11/2)))/(7680*b^3) + ((-12*a^5*A*b + 5*a^6*B)*Log[-(Sqrt[b]*Sqrt[x]) + Sq
rt[a + b*x]])/(512*b^(7/2))

________________________________________________________________________________________

fricas [A]  time = 1.81, size = 344, normalized size = 1.53 \begin {gather*} \left [-\frac {15 \, {\left (5 \, B a^{6} - 12 \, A a^{5} b\right )} \sqrt {b} \log \left (2 \, b x + 2 \, \sqrt {b x + a} \sqrt {b} \sqrt {x} + a\right ) - 2 \, {\left (1280 \, B b^{6} x^{5} + 75 \, B a^{5} b - 180 \, A a^{4} b^{2} + 128 \, {\left (25 \, B a b^{5} + 12 \, A b^{6}\right )} x^{4} + 144 \, {\left (15 \, B a^{2} b^{4} + 28 \, A a b^{5}\right )} x^{3} + 8 \, {\left (5 \, B a^{3} b^{3} + 372 \, A a^{2} b^{4}\right )} x^{2} - 10 \, {\left (5 \, B a^{4} b^{2} - 12 \, A a^{3} b^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {x}}{15360 \, b^{4}}, \frac {15 \, {\left (5 \, B a^{6} - 12 \, A a^{5} b\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-b}}{b \sqrt {x}}\right ) + {\left (1280 \, B b^{6} x^{5} + 75 \, B a^{5} b - 180 \, A a^{4} b^{2} + 128 \, {\left (25 \, B a b^{5} + 12 \, A b^{6}\right )} x^{4} + 144 \, {\left (15 \, B a^{2} b^{4} + 28 \, A a b^{5}\right )} x^{3} + 8 \, {\left (5 \, B a^{3} b^{3} + 372 \, A a^{2} b^{4}\right )} x^{2} - 10 \, {\left (5 \, B a^{4} b^{2} - 12 \, A a^{3} b^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {x}}{7680 \, b^{4}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/15360*(15*(5*B*a^6 - 12*A*a^5*b)*sqrt(b)*log(2*b*x + 2*sqrt(b*x + a)*sqrt(b)*sqrt(x) + a) - 2*(1280*B*b^6*
x^5 + 75*B*a^5*b - 180*A*a^4*b^2 + 128*(25*B*a*b^5 + 12*A*b^6)*x^4 + 144*(15*B*a^2*b^4 + 28*A*a*b^5)*x^3 + 8*(
5*B*a^3*b^3 + 372*A*a^2*b^4)*x^2 - 10*(5*B*a^4*b^2 - 12*A*a^3*b^3)*x)*sqrt(b*x + a)*sqrt(x))/b^4, 1/7680*(15*(
5*B*a^6 - 12*A*a^5*b)*sqrt(-b)*arctan(sqrt(b*x + a)*sqrt(-b)/(b*sqrt(x))) + (1280*B*b^6*x^5 + 75*B*a^5*b - 180
*A*a^4*b^2 + 128*(25*B*a*b^5 + 12*A*b^6)*x^4 + 144*(15*B*a^2*b^4 + 28*A*a*b^5)*x^3 + 8*(5*B*a^3*b^3 + 372*A*a^
2*b^4)*x^2 - 10*(5*B*a^4*b^2 - 12*A*a^3*b^3)*x)*sqrt(b*x + a)*sqrt(x))/b^4]

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

maple [A]  time = 0.01, size = 302, normalized size = 1.34 \begin {gather*} \frac {\sqrt {b x +a}\, \left (2560 \sqrt {\left (b x +a \right ) x}\, B \,b^{\frac {11}{2}} x^{5}+3072 \sqrt {\left (b x +a \right ) x}\, A \,b^{\frac {11}{2}} x^{4}+6400 \sqrt {\left (b x +a \right ) x}\, B a \,b^{\frac {9}{2}} x^{4}+8064 \sqrt {\left (b x +a \right ) x}\, A a \,b^{\frac {9}{2}} x^{3}+4320 \sqrt {\left (b x +a \right ) x}\, B \,a^{2} b^{\frac {7}{2}} x^{3}+5952 \sqrt {\left (b x +a \right ) x}\, A \,a^{2} b^{\frac {7}{2}} x^{2}+80 \sqrt {\left (b x +a \right ) x}\, B \,a^{3} b^{\frac {5}{2}} x^{2}+180 A \,a^{5} b \ln \left (\frac {2 b x +a +2 \sqrt {\left (b x +a \right ) x}\, \sqrt {b}}{2 \sqrt {b}}\right )-75 B \,a^{6} \ln \left (\frac {2 b x +a +2 \sqrt {\left (b x +a \right ) x}\, \sqrt {b}}{2 \sqrt {b}}\right )+240 \sqrt {\left (b x +a \right ) x}\, A \,a^{3} b^{\frac {5}{2}} x -100 \sqrt {\left (b x +a \right ) x}\, B \,a^{4} b^{\frac {3}{2}} x -360 \sqrt {\left (b x +a \right ) x}\, A \,a^{4} b^{\frac {3}{2}}+150 \sqrt {\left (b x +a \right ) x}\, B \,a^{5} \sqrt {b}\right ) \sqrt {x}}{15360 \sqrt {\left (b x +a \right ) x}\, b^{\frac {7}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/15360*x^(1/2)*(b*x+a)^(1/2)/b^(7/2)*(2560*((b*x+a)*x)^(1/2)*B*b^(11/2)*x^5+3072*((b*x+a)*x)^(1/2)*A*b^(11/2)
*x^4+6400*((b*x+a)*x)^(1/2)*B*a*b^(9/2)*x^4+8064*((b*x+a)*x)^(1/2)*A*a*b^(9/2)*x^3+4320*((b*x+a)*x)^(1/2)*B*a^
2*b^(7/2)*x^3+5952*((b*x+a)*x)^(1/2)*A*a^2*b^(7/2)*x^2+80*((b*x+a)*x)^(1/2)*B*a^3*b^(5/2)*x^2+240*((b*x+a)*x)^
(1/2)*A*a^3*b^(5/2)*x-100*((b*x+a)*x)^(1/2)*B*a^4*b^(3/2)*x+180*A*a^5*b*ln(1/2*(2*b*x+a+2*((b*x+a)*x)^(1/2)*b^
(1/2))/b^(1/2))-360*((b*x+a)*x)^(1/2)*A*a^4*b^(3/2)-75*B*a^6*ln(1/2*(2*b*x+a+2*((b*x+a)*x)^(1/2)*b^(1/2))/b^(1
/2))+150*((b*x+a)*x)^(1/2)*B*a^5*b^(1/2))/((b*x+a)*x)^(1/2)

________________________________________________________________________________________

maxima [B]  time = 0.90, size = 422, normalized size = 1.88 \begin {gather*} \frac {1}{6} \, {\left (b x^{2} + a x\right )}^{\frac {5}{2}} B x + \frac {1}{4} \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} A a x - \frac {7 \, \sqrt {b x^{2} + a x} B a^{4} x}{256 \, b^{2}} + \frac {7 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} B a^{2} x}{96 \, b} - \frac {3 \, \sqrt {b x^{2} + a x} A a^{3} x}{32 \, b} + \frac {7 \, B a^{6} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right )}{1024 \, b^{\frac {7}{2}}} + \frac {3 \, A a^{5} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right )}{128 \, b^{\frac {5}{2}}} - \frac {7 \, \sqrt {b x^{2} + a x} B a^{5}}{512 \, b^{3}} + \frac {7 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} B a^{3}}{192 \, b^{2}} - \frac {3 \, \sqrt {b x^{2} + a x} A a^{4}}{64 \, b^{2}} - \frac {7 \, {\left (b x^{2} + a x\right )}^{\frac {5}{2}} B a}{60 \, b} + \frac {{\left (b x^{2} + a x\right )}^{\frac {3}{2}} A a^{2}}{8 \, b} + \frac {3 \, \sqrt {b x^{2} + a x} {\left (B a + A b\right )} a^{3} x}{64 \, b^{2}} - \frac {{\left (b x^{2} + a x\right )}^{\frac {3}{2}} {\left (B a + A b\right )} a x}{8 \, b} - \frac {3 \, {\left (B a + A b\right )} a^{5} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right )}{256 \, b^{\frac {7}{2}}} + \frac {3 \, \sqrt {b x^{2} + a x} {\left (B a + A b\right )} a^{4}}{128 \, b^{3}} - \frac {{\left (b x^{2} + a x\right )}^{\frac {3}{2}} {\left (B a + A b\right )} a^{2}}{16 \, b^{2}} + \frac {{\left (b x^{2} + a x\right )}^{\frac {5}{2}} {\left (B a + A b\right )}}{5 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(b*x^2 + a*x)^(5/2)*B*x + 1/4*(b*x^2 + a*x)^(3/2)*A*a*x - 7/256*sqrt(b*x^2 + a*x)*B*a^4*x/b^2 + 7/96*(b*x^
2 + a*x)^(3/2)*B*a^2*x/b - 3/32*sqrt(b*x^2 + a*x)*A*a^3*x/b + 7/1024*B*a^6*log(2*b*x + a + 2*sqrt(b*x^2 + a*x)
*sqrt(b))/b^(7/2) + 3/128*A*a^5*log(2*b*x + a + 2*sqrt(b*x^2 + a*x)*sqrt(b))/b^(5/2) - 7/512*sqrt(b*x^2 + a*x)
*B*a^5/b^3 + 7/192*(b*x^2 + a*x)^(3/2)*B*a^3/b^2 - 3/64*sqrt(b*x^2 + a*x)*A*a^4/b^2 - 7/60*(b*x^2 + a*x)^(5/2)
*B*a/b + 1/8*(b*x^2 + a*x)^(3/2)*A*a^2/b + 3/64*sqrt(b*x^2 + a*x)*(B*a + A*b)*a^3*x/b^2 - 1/8*(b*x^2 + a*x)^(3
/2)*(B*a + A*b)*a*x/b - 3/256*(B*a + A*b)*a^5*log(2*b*x + a + 2*sqrt(b*x^2 + a*x)*sqrt(b))/b^(7/2) + 3/128*sqr
t(b*x^2 + a*x)*(B*a + A*b)*a^4/b^3 - 1/16*(b*x^2 + a*x)^(3/2)*(B*a + A*b)*a^2/b^2 + 1/5*(b*x^2 + a*x)^(5/2)*(B
*a + A*b)/b

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^{3/2}\,\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^(3/2)*(A + B*x)*(a + b*x)^(5/2),x)

[Out]

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

________________________________________________________________________________________

sympy [C]  time = 160.75, size = 2190, normalized size = 9.73

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*A*a*Piecewise((5*a**(7/2)*sqrt(a + b*x)/(128*sqrt(b)*sqrt(b*x/a)) - 5*a**(5/2)*(a + b*x)**(3/2)/(384*sqrt(b
)*sqrt(b*x/a)) - a**(3/2)*(a + b*x)**(5/2)/(192*sqrt(b)*sqrt(b*x/a)) - 7*sqrt(a)*(a + b*x)**(7/2)/(48*sqrt(b)*
sqrt(b*x/a)) - 5*a**4*acosh(sqrt(a + b*x)/sqrt(a))/(128*sqrt(b)) + (a + b*x)**(9/2)/(8*sqrt(a)*sqrt(b)*sqrt(b*
x/a)), Abs(1 + b*x/a) > 1), (-5*I*a**(7/2)*sqrt(a + b*x)/(128*sqrt(b)*sqrt(-b*x/a)) + 5*I*a**(5/2)*(a + b*x)**
(3/2)/(384*sqrt(b)*sqrt(-b*x/a)) + I*a**(3/2)*(a + b*x)**(5/2)/(192*sqrt(b)*sqrt(-b*x/a)) + 7*I*sqrt(a)*(a + b
*x)**(7/2)/(48*sqrt(b)*sqrt(-b*x/a)) + 5*I*a**4*asin(sqrt(a + b*x)/sqrt(a))/(128*sqrt(b)) - I*(a + b*x)**(9/2)
/(8*sqrt(a)*sqrt(b)*sqrt(-b*x/a)), True))/b**2 + 2*A*Piecewise((7*a**(9/2)*sqrt(a + b*x)/(256*sqrt(b)*sqrt(b*x
/a)) - 7*a**(7/2)*(a + b*x)**(3/2)/(768*sqrt(b)*sqrt(b*x/a)) - 7*a**(5/2)*(a + b*x)**(5/2)/(1920*sqrt(b)*sqrt(
b*x/a)) - a**(3/2)*(a + b*x)**(7/2)/(480*sqrt(b)*sqrt(b*x/a)) - 9*sqrt(a)*(a + b*x)**(9/2)/(80*sqrt(b)*sqrt(b*
x/a)) - 7*a**5*acosh(sqrt(a + b*x)/sqrt(a))/(256*sqrt(b)) + (a + b*x)**(11/2)/(10*sqrt(a)*sqrt(b)*sqrt(b*x/a))
, Abs(1 + b*x/a) > 1), (-7*I*a**(9/2)*sqrt(a + b*x)/(256*sqrt(b)*sqrt(-b*x/a)) + 7*I*a**(7/2)*(a + b*x)**(3/2)
/(768*sqrt(b)*sqrt(-b*x/a)) + 7*I*a**(5/2)*(a + b*x)**(5/2)/(1920*sqrt(b)*sqrt(-b*x/a)) + I*a**(3/2)*(a + b*x)
**(7/2)/(480*sqrt(b)*sqrt(-b*x/a)) + 9*I*sqrt(a)*(a + b*x)**(9/2)/(80*sqrt(b)*sqrt(-b*x/a)) + 7*I*a**5*asin(sq
rt(a + b*x)/sqrt(a))/(256*sqrt(b)) - I*(a + b*x)**(11/2)/(10*sqrt(a)*sqrt(b)*sqrt(-b*x/a)), True))/b**2 + 2*B*
a**2*Piecewise((5*a**(7/2)*sqrt(a + b*x)/(128*sqrt(b)*sqrt(b*x/a)) - 5*a**(5/2)*(a + b*x)**(3/2)/(384*sqrt(b)*
sqrt(b*x/a)) - a**(3/2)*(a + b*x)**(5/2)/(192*sqrt(b)*sqrt(b*x/a)) - 7*sqrt(a)*(a + b*x)**(7/2)/(48*sqrt(b)*sq
rt(b*x/a)) - 5*a**4*acosh(sqrt(a + b*x)/sqrt(a))/(128*sqrt(b)) + (a + b*x)**(9/2)/(8*sqrt(a)*sqrt(b)*sqrt(b*x/
a)), Abs(1 + b*x/a) > 1), (-5*I*a**(7/2)*sqrt(a + b*x)/(128*sqrt(b)*sqrt(-b*x/a)) + 5*I*a**(5/2)*(a + b*x)**(3
/2)/(384*sqrt(b)*sqrt(-b*x/a)) + I*a**(3/2)*(a + b*x)**(5/2)/(192*sqrt(b)*sqrt(-b*x/a)) + 7*I*sqrt(a)*(a + b*x
)**(7/2)/(48*sqrt(b)*sqrt(-b*x/a)) + 5*I*a**4*asin(sqrt(a + b*x)/sqrt(a))/(128*sqrt(b)) - I*(a + b*x)**(9/2)/(
8*sqrt(a)*sqrt(b)*sqrt(-b*x/a)), True))/b**3 - 4*B*a*Piecewise((7*a**(9/2)*sqrt(a + b*x)/(256*sqrt(b)*sqrt(b*x
/a)) - 7*a**(7/2)*(a + b*x)**(3/2)/(768*sqrt(b)*sqrt(b*x/a)) - 7*a**(5/2)*(a + b*x)**(5/2)/(1920*sqrt(b)*sqrt(
b*x/a)) - a**(3/2)*(a + b*x)**(7/2)/(480*sqrt(b)*sqrt(b*x/a)) - 9*sqrt(a)*(a + b*x)**(9/2)/(80*sqrt(b)*sqrt(b*
x/a)) - 7*a**5*acosh(sqrt(a + b*x)/sqrt(a))/(256*sqrt(b)) + (a + b*x)**(11/2)/(10*sqrt(a)*sqrt(b)*sqrt(b*x/a))
, Abs(1 + b*x/a) > 1), (-7*I*a**(9/2)*sqrt(a + b*x)/(256*sqrt(b)*sqrt(-b*x/a)) + 7*I*a**(7/2)*(a + b*x)**(3/2)
/(768*sqrt(b)*sqrt(-b*x/a)) + 7*I*a**(5/2)*(a + b*x)**(5/2)/(1920*sqrt(b)*sqrt(-b*x/a)) + I*a**(3/2)*(a + b*x)
**(7/2)/(480*sqrt(b)*sqrt(-b*x/a)) + 9*I*sqrt(a)*(a + b*x)**(9/2)/(80*sqrt(b)*sqrt(-b*x/a)) + 7*I*a**5*asin(sq
rt(a + b*x)/sqrt(a))/(256*sqrt(b)) - I*(a + b*x)**(11/2)/(10*sqrt(a)*sqrt(b)*sqrt(-b*x/a)), True))/b**3 + 2*B*
Piecewise((21*a**(11/2)*sqrt(a + b*x)/(1024*sqrt(b)*sqrt(b*x/a)) - 7*a**(9/2)*(a + b*x)**(3/2)/(1024*sqrt(b)*s
qrt(b*x/a)) - 7*a**(7/2)*(a + b*x)**(5/2)/(2560*sqrt(b)*sqrt(b*x/a)) - a**(5/2)*(a + b*x)**(7/2)/(640*sqrt(b)*
sqrt(b*x/a)) - a**(3/2)*(a + b*x)**(9/2)/(960*sqrt(b)*sqrt(b*x/a)) - 11*sqrt(a)*(a + b*x)**(11/2)/(120*sqrt(b)
*sqrt(b*x/a)) - 21*a**6*acosh(sqrt(a + b*x)/sqrt(a))/(1024*sqrt(b)) + (a + b*x)**(13/2)/(12*sqrt(a)*sqrt(b)*sq
rt(b*x/a)), Abs(1 + b*x/a) > 1), (-21*I*a**(11/2)*sqrt(a + b*x)/(1024*sqrt(b)*sqrt(-b*x/a)) + 7*I*a**(9/2)*(a
+ b*x)**(3/2)/(1024*sqrt(b)*sqrt(-b*x/a)) + 7*I*a**(7/2)*(a + b*x)**(5/2)/(2560*sqrt(b)*sqrt(-b*x/a)) + I*a**(
5/2)*(a + b*x)**(7/2)/(640*sqrt(b)*sqrt(-b*x/a)) + I*a**(3/2)*(a + b*x)**(9/2)/(960*sqrt(b)*sqrt(-b*x/a)) + 11
*I*sqrt(a)*(a + b*x)**(11/2)/(120*sqrt(b)*sqrt(-b*x/a)) + 21*I*a**6*asin(sqrt(a + b*x)/sqrt(a))/(1024*sqrt(b))
- I*(a + b*x)**(13/2)/(12*sqrt(a)*sqrt(b)*sqrt(-b*x/a)), True))/b**3

________________________________________________________________________________________