3.6.2 \(\int x^{3/2} (a+b x)^{5/2} (A+B x) \, dx\) [502]

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

[Out]

1/96*a*(12*A*b-5*B*a)*x^(5/2)*(b*x+a)^(3/2)/b+1/60*(12*A*b-5*B*a)*x^(5/2)*(b*x+a)^(5/2)/b+1/6*B*x^(5/2)*(b*x+a
)^(7/2)/b+1/512*a^5*(12*A*b-5*B*a)*arctanh(b^(1/2)*x^(1/2)/(b*x+a)^(1/2))/b^(7/2)+1/768*a^3*(12*A*b-5*B*a)*x^(
3/2)*(b*x+a)^(1/2)/b^2+1/192*a^2*(12*A*b-5*B*a)*x^(5/2)*(b*x+a)^(1/2)/b-1/512*a^4*(12*A*b-5*B*a)*x^(1/2)*(b*x+
a)^(1/2)/b^3

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Rubi [A]
time = 0.07, 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 = {81, 52, 65, 223, 212} \begin {gather*} \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} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/512*(a^4*(12*A*b - 5*a*B)*Sqrt[x]*Sqrt[a + b*x])/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 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 81

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 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 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 ) \text {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 ) \text {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*}

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Mathematica [A]
time = 0.28, size = 156, normalized size = 0.69 \begin {gather*} \frac {\sqrt {b} \sqrt {x} \sqrt {a+b x} \left (75 a^5 B+40 a^3 b^2 x (3 A+B x)+256 b^5 x^4 (6 A+5 B x)-10 a^4 b (18 A+5 B x)+48 a^2 b^3 x^2 (62 A+45 B x)+64 a b^4 x^3 (63 A+50 B x)\right )+15 a^5 (-12 A b+5 a B) \log \left (-\sqrt {b} \sqrt {x}+\sqrt {a+b x}\right )}{7680 b^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[b]*Sqrt[x]*Sqrt[a + b*x]*(75*a^5*B + 40*a^3*b^2*x*(3*A + B*x) + 256*b^5*x^4*(6*A + 5*B*x) - 10*a^4*b*(18
*A + 5*B*x) + 48*a^2*b^3*x^2*(62*A + 45*B*x) + 64*a*b^4*x^3*(63*A + 50*B*x)) + 15*a^5*(-12*A*b + 5*a*B)*Log[-(
Sqrt[b]*Sqrt[x]) + Sqrt[a + b*x]])/(7680*b^(7/2))

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Maple [A]
time = 0.07, size = 302, normalized size = 1.34

method	result	size
risch	\(-\frac {\left (-1280 b^{5} B \,x^{5}-1536 A \,b^{5} x^{4}-3200 B a \,b^{4} x^{4}-4032 A a \,b^{4} x^{3}-2160 B \,a^{2} b^{3} x^{3}-2976 A \,a^{2} b^{3} x^{2}-40 B \,a^{3} b^{2} x^{2}-120 a^{3} b^{2} A x +50 a^{4} b B x +180 a^{4} b A -75 a^{5} B \right ) \sqrt {b x +a}\, \sqrt {x}}{7680 b^{3}}+\frac {\left (\frac {3 a^{5} \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right ) A}{256 b^{\frac {5}{2}}}-\frac {5 a^{6} \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right ) B}{1024 b^{\frac {7}{2}}}\right ) \sqrt {\left (b x +a \right ) x}}{\sqrt {b x +a}\, \sqrt {x}}\)	\(210\)
default	\(\frac {\sqrt {x}\, \sqrt {b x +a}\, \left (2560 B \,b^{\frac {11}{2}} x^{5} \sqrt {\left (b x +a \right ) x}+3072 A \,b^{\frac {11}{2}} x^{4} \sqrt {\left (b x +a \right ) x}+6400 B a \,b^{\frac {9}{2}} x^{4} \sqrt {\left (b x +a \right ) x}+8064 A a \,b^{\frac {9}{2}} x^{3} \sqrt {\left (b x +a \right ) x}+4320 B \,a^{2} b^{\frac {7}{2}} x^{3} \sqrt {\left (b x +a \right ) x}+5952 A \,a^{2} b^{\frac {7}{2}} x^{2} \sqrt {\left (b x +a \right ) x}+80 B \,a^{3} b^{\frac {5}{2}} x^{2} \sqrt {\left (b x +a \right ) x}+240 A \,b^{\frac {5}{2}} \sqrt {\left (b x +a \right ) x}\, a^{3} x -100 B \,b^{\frac {3}{2}} \sqrt {\left (b x +a \right ) x}\, a^{4} x +180 A \ln \left (\frac {2 \sqrt {\left (b x +a \right ) x}\, \sqrt {b}+2 b x +a}{2 \sqrt {b}}\right ) a^{5} b -360 A \,b^{\frac {3}{2}} \sqrt {\left (b x +a \right ) x}\, a^{4}-75 B \ln \left (\frac {2 \sqrt {\left (b x +a \right ) x}\, \sqrt {b}+2 b x +a}{2 \sqrt {b}}\right ) a^{6}+150 B \sqrt {b}\, \sqrt {\left (b x +a \right ) x}\, a^{5}\right )}{15360 b^{\frac {7}{2}} \sqrt {\left (b x +a \right ) x}}\)	\(302\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 422 vs. \(2 (179) = 358\).
time = 0.29, 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

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Fricas [A]
time = 1.08, 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]

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Sympy [F(-1)] Timed out
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)

[Out]

Timed out

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Giac [F(-1)] Timed out
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

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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)

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