3.23.25 \(\int (a+b x)^{5/2} (A+B x) \sqrt {d+e x} \, dx\) [2225]

Optimal. Leaf size=304 \[ -\frac {(b d-a e)^3 (7 b B d-10 A b e+3 a B e) \sqrt {a+b x} \sqrt {d+e x}}{128 b^2 e^4}+\frac {(b d-a e)^2 (7 b B d-10 A b e+3 a B e) (a+b x)^{3/2} \sqrt {d+e x}}{192 b^2 e^3}-\frac {(b d-a e) (7 b B d-10 A b e+3 a B e) (a+b x)^{5/2} \sqrt {d+e x}}{240 b^2 e^2}-\frac {(7 b B d-10 A b e+3 a B e) (a+b x)^{7/2} \sqrt {d+e x}}{40 b^2 e}+\frac {B (a+b x)^{7/2} (d+e x)^{3/2}}{5 b e}+\frac {(b d-a e)^4 (7 b B d-10 A b e+3 a B e) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{128 b^{5/2} e^{9/2}} \]

[Out]

1/5*B*(b*x+a)^(7/2)*(e*x+d)^(3/2)/b/e+1/128*(-a*e+b*d)^4*(-10*A*b*e+3*B*a*e+7*B*b*d)*arctanh(e^(1/2)*(b*x+a)^(
1/2)/b^(1/2)/(e*x+d)^(1/2))/b^(5/2)/e^(9/2)+1/192*(-a*e+b*d)^2*(-10*A*b*e+3*B*a*e+7*B*b*d)*(b*x+a)^(3/2)*(e*x+
d)^(1/2)/b^2/e^3-1/240*(-a*e+b*d)*(-10*A*b*e+3*B*a*e+7*B*b*d)*(b*x+a)^(5/2)*(e*x+d)^(1/2)/b^2/e^2-1/40*(-10*A*
b*e+3*B*a*e+7*B*b*d)*(b*x+a)^(7/2)*(e*x+d)^(1/2)/b^2/e-1/128*(-a*e+b*d)^3*(-10*A*b*e+3*B*a*e+7*B*b*d)*(b*x+a)^
(1/2)*(e*x+d)^(1/2)/b^2/e^4

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Rubi [A]
time = 0.15, antiderivative size = 304, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {81, 52, 65, 223, 212} \begin {gather*} \frac {(b d-a e)^4 (3 a B e-10 A b e+7 b B d) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{128 b^{5/2} e^{9/2}}-\frac {\sqrt {a+b x} \sqrt {d+e x} (b d-a e)^3 (3 a B e-10 A b e+7 b B d)}{128 b^2 e^4}+\frac {(a+b x)^{3/2} \sqrt {d+e x} (b d-a e)^2 (3 a B e-10 A b e+7 b B d)}{192 b^2 e^3}-\frac {(a+b x)^{5/2} \sqrt {d+e x} (b d-a e) (3 a B e-10 A b e+7 b B d)}{240 b^2 e^2}-\frac {(a+b x)^{7/2} \sqrt {d+e x} (3 a B e-10 A b e+7 b B d)}{40 b^2 e}+\frac {B (a+b x)^{7/2} (d+e x)^{3/2}}{5 b e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/128*((b*d - a*e)^3*(7*b*B*d - 10*A*b*e + 3*a*B*e)*Sqrt[a + b*x]*Sqrt[d + e*x])/(b^2*e^4) + ((b*d - a*e)^2*(
7*b*B*d - 10*A*b*e + 3*a*B*e)*(a + b*x)^(3/2)*Sqrt[d + e*x])/(192*b^2*e^3) - ((b*d - a*e)*(7*b*B*d - 10*A*b*e
+ 3*a*B*e)*(a + b*x)^(5/2)*Sqrt[d + e*x])/(240*b^2*e^2) - ((7*b*B*d - 10*A*b*e + 3*a*B*e)*(a + b*x)^(7/2)*Sqrt
[d + e*x])/(40*b^2*e) + (B*(a + b*x)^(7/2)*(d + e*x)^(3/2))/(5*b*e) + ((b*d - a*e)^4*(7*b*B*d - 10*A*b*e + 3*a
*B*e)*ArcTanh[(Sqrt[e]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[d + e*x])])/(128*b^(5/2)*e^(9/2))

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 (a+b x)^{5/2} (A+B x) \sqrt {d+e x} \, dx &=\frac {B (a+b x)^{7/2} (d+e x)^{3/2}}{5 b e}+\frac {\left (5 A b e-B \left (\frac {7 b d}{2}+\frac {3 a e}{2}\right )\right ) \int (a+b x)^{5/2} \sqrt {d+e x} \, dx}{5 b e}\\ &=-\frac {(7 b B d-10 A b e+3 a B e) (a+b x)^{7/2} \sqrt {d+e x}}{40 b^2 e}+\frac {B (a+b x)^{7/2} (d+e x)^{3/2}}{5 b e}-\frac {((b d-a e) (7 b B d-10 A b e+3 a B e)) \int \frac {(a+b x)^{5/2}}{\sqrt {d+e x}} \, dx}{80 b^2 e}\\ &=-\frac {(b d-a e) (7 b B d-10 A b e+3 a B e) (a+b x)^{5/2} \sqrt {d+e x}}{240 b^2 e^2}-\frac {(7 b B d-10 A b e+3 a B e) (a+b x)^{7/2} \sqrt {d+e x}}{40 b^2 e}+\frac {B (a+b x)^{7/2} (d+e x)^{3/2}}{5 b e}+\frac {\left ((b d-a e)^2 (7 b B d-10 A b e+3 a B e)\right ) \int \frac {(a+b x)^{3/2}}{\sqrt {d+e x}} \, dx}{96 b^2 e^2}\\ &=\frac {(b d-a e)^2 (7 b B d-10 A b e+3 a B e) (a+b x)^{3/2} \sqrt {d+e x}}{192 b^2 e^3}-\frac {(b d-a e) (7 b B d-10 A b e+3 a B e) (a+b x)^{5/2} \sqrt {d+e x}}{240 b^2 e^2}-\frac {(7 b B d-10 A b e+3 a B e) (a+b x)^{7/2} \sqrt {d+e x}}{40 b^2 e}+\frac {B (a+b x)^{7/2} (d+e x)^{3/2}}{5 b e}-\frac {\left ((b d-a e)^3 (7 b B d-10 A b e+3 a B e)\right ) \int \frac {\sqrt {a+b x}}{\sqrt {d+e x}} \, dx}{128 b^2 e^3}\\ &=-\frac {(b d-a e)^3 (7 b B d-10 A b e+3 a B e) \sqrt {a+b x} \sqrt {d+e x}}{128 b^2 e^4}+\frac {(b d-a e)^2 (7 b B d-10 A b e+3 a B e) (a+b x)^{3/2} \sqrt {d+e x}}{192 b^2 e^3}-\frac {(b d-a e) (7 b B d-10 A b e+3 a B e) (a+b x)^{5/2} \sqrt {d+e x}}{240 b^2 e^2}-\frac {(7 b B d-10 A b e+3 a B e) (a+b x)^{7/2} \sqrt {d+e x}}{40 b^2 e}+\frac {B (a+b x)^{7/2} (d+e x)^{3/2}}{5 b e}+\frac {\left ((b d-a e)^4 (7 b B d-10 A b e+3 a B e)\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {d+e x}} \, dx}{256 b^2 e^4}\\ &=-\frac {(b d-a e)^3 (7 b B d-10 A b e+3 a B e) \sqrt {a+b x} \sqrt {d+e x}}{128 b^2 e^4}+\frac {(b d-a e)^2 (7 b B d-10 A b e+3 a B e) (a+b x)^{3/2} \sqrt {d+e x}}{192 b^2 e^3}-\frac {(b d-a e) (7 b B d-10 A b e+3 a B e) (a+b x)^{5/2} \sqrt {d+e x}}{240 b^2 e^2}-\frac {(7 b B d-10 A b e+3 a B e) (a+b x)^{7/2} \sqrt {d+e x}}{40 b^2 e}+\frac {B (a+b x)^{7/2} (d+e x)^{3/2}}{5 b e}+\frac {\left ((b d-a e)^4 (7 b B d-10 A b e+3 a B e)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {d-\frac {a e}{b}+\frac {e x^2}{b}}} \, dx,x,\sqrt {a+b x}\right )}{128 b^3 e^4}\\ &=-\frac {(b d-a e)^3 (7 b B d-10 A b e+3 a B e) \sqrt {a+b x} \sqrt {d+e x}}{128 b^2 e^4}+\frac {(b d-a e)^2 (7 b B d-10 A b e+3 a B e) (a+b x)^{3/2} \sqrt {d+e x}}{192 b^2 e^3}-\frac {(b d-a e) (7 b B d-10 A b e+3 a B e) (a+b x)^{5/2} \sqrt {d+e x}}{240 b^2 e^2}-\frac {(7 b B d-10 A b e+3 a B e) (a+b x)^{7/2} \sqrt {d+e x}}{40 b^2 e}+\frac {B (a+b x)^{7/2} (d+e x)^{3/2}}{5 b e}+\frac {\left ((b d-a e)^4 (7 b B d-10 A b e+3 a B e)\right ) \text {Subst}\left (\int \frac {1}{1-\frac {e x^2}{b}} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {d+e x}}\right )}{128 b^3 e^4}\\ &=-\frac {(b d-a e)^3 (7 b B d-10 A b e+3 a B e) \sqrt {a+b x} \sqrt {d+e x}}{128 b^2 e^4}+\frac {(b d-a e)^2 (7 b B d-10 A b e+3 a B e) (a+b x)^{3/2} \sqrt {d+e x}}{192 b^2 e^3}-\frac {(b d-a e) (7 b B d-10 A b e+3 a B e) (a+b x)^{5/2} \sqrt {d+e x}}{240 b^2 e^2}-\frac {(7 b B d-10 A b e+3 a B e) (a+b x)^{7/2} \sqrt {d+e x}}{40 b^2 e}+\frac {B (a+b x)^{7/2} (d+e x)^{3/2}}{5 b e}+\frac {(b d-a e)^4 (7 b B d-10 A b e+3 a B e) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{128 b^{5/2} e^{9/2}}\\ \end {align*}

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Mathematica [A]
time = 0.86, size = 321, normalized size = 1.06 \begin {gather*} \frac {\sqrt {a+b x} \sqrt {d+e x} \left (-45 a^4 B e^4+30 a^3 b e^3 (2 B d+5 A e+B e x)+2 a^2 b^2 e^2 \left (5 A e (73 d+118 e x)+B \left (-173 d^2+109 d e x+372 e^2 x^2\right )\right )+2 a b^3 e \left (5 A e \left (-55 d^2+36 d e x+136 e^2 x^2\right )+B \left (170 d^3-111 d^2 e x+88 d e^2 x^2+504 e^3 x^3\right )\right )+b^4 \left (10 A e \left (15 d^3-10 d^2 e x+8 d e^2 x^2+48 e^3 x^3\right )+B \left (-105 d^4+70 d^3 e x-56 d^2 e^2 x^2+48 d e^3 x^3+384 e^4 x^4\right )\right )\right )}{1920 b^2 e^4}+\frac {(b d-a e)^4 (7 b B d-10 A b e+3 a B e) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {e} \sqrt {a+b x}}\right )}{128 b^{5/2} e^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[a + b*x]*Sqrt[d + e*x]*(-45*a^4*B*e^4 + 30*a^3*b*e^3*(2*B*d + 5*A*e + B*e*x) + 2*a^2*b^2*e^2*(5*A*e*(73*
d + 118*e*x) + B*(-173*d^2 + 109*d*e*x + 372*e^2*x^2)) + 2*a*b^3*e*(5*A*e*(-55*d^2 + 36*d*e*x + 136*e^2*x^2) +
B*(170*d^3 - 111*d^2*e*x + 88*d*e^2*x^2 + 504*e^3*x^3)) + b^4*(10*A*e*(15*d^3 - 10*d^2*e*x + 8*d*e^2*x^2 + 48
*e^3*x^3) + B*(-105*d^4 + 70*d^3*e*x - 56*d^2*e^2*x^2 + 48*d*e^3*x^3 + 384*e^4*x^4))))/(1920*b^2*e^4) + ((b*d
- a*e)^4*(7*b*B*d - 10*A*b*e + 3*a*B*e)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/(Sqrt[e]*Sqrt[a + b*x])])/(128*b^(5/2)
*e^(9/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1371\) vs. \(2(260)=520\).
time = 0.10, size = 1372, normalized size = 4.51

method	result	size
default	\(\text {Expression too large to display}\)	\(1372\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^(5/2)*(B*x+A)*(e*x+d)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/3840*(b*x+a)^(1/2)*(e*x+d)^(1/2)*(-768*B*b^4*e^4*x^4*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-960*A*b^4*e^4*x^3*
((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-45*B*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^
(1/2))*a^5*e^5-105*B*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*b^5*d^5+75*B*
ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a^4*b*d*e^4-600*A*ln(1/2*(2*b*e*x+
2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a^3*b^2*d*e^4+900*A*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*
x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a^2*b^3*d^2*e^3-600*A*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*
(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a*b^4*d^3*e^2+150*B*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*
e+b*d)/(b*e)^(1/2))*a^3*b^2*d^2*e^3-450*B*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)
^(1/2))*a^2*b^3*d^3*e^2+375*B*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a*b^
4*d^4*e-300*A*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)*a^3*b*e^4-300*A*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)*b^4*d^3*
e-1460*A*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)*a^2*b^2*d*e^3-60*B*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)*a^3*b*e^4*
x-140*B*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)*b^4*d^3*e*x+1100*A*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)*a*b^3*d^2*e
^2-120*B*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)*a^3*b*d*e^3+444*B*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)*a*b^3*d^2*e
^2*x-2016*B*a*b^3*e^4*x^3*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-96*B*b^4*d*e^3*x^3*((b*x+a)*(e*x+d))^(1/2)*(b*e)
^(1/2)-720*A*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)*a*b^3*d*e^3*x-436*B*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)*a^2*b
^2*d*e^3*x+692*B*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)*a^2*b^2*d^2*e^2-680*B*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)
*a*b^3*d^3*e-2360*A*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)*a^2*b^2*e^4*x+200*A*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2
)*b^4*d^2*e^2*x-2720*A*a*b^3*e^4*x^2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-160*A*b^4*d*e^3*x^2*((b*x+a)*(e*x+d))
^(1/2)*(b*e)^(1/2)-1488*B*a^2*b^2*e^4*x^2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+112*B*b^4*d^2*e^2*x^2*((b*x+a)*(
e*x+d))^(1/2)*(b*e)^(1/2)+150*A*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a^
4*b*e^5+150*A*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*b^5*d^4*e+90*B*(b*e)
^(1/2)*((b*x+a)*(e*x+d))^(1/2)*a^4*e^4+210*B*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)*b^4*d^4-352*B*a*b^3*d*e^3*x^2
*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2))/b^2/((b*x+a)*(e*x+d))^(1/2)/e^4/(b*e)^(1/2)

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(b*d-%e*a>0)', see `assume?` fo
r more detai

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Fricas [A]
time = 0.65, size = 998, normalized size = 3.28 \begin {gather*} \left [-\frac {{\left (15 \, {\left (7 \, B b^{5} d^{5} - 5 \, {\left (5 \, B a b^{4} + 2 \, A b^{5}\right )} d^{4} e + 10 \, {\left (3 \, B a^{2} b^{3} + 4 \, A a b^{4}\right )} d^{3} e^{2} - 10 \, {\left (B a^{3} b^{2} + 6 \, A a^{2} b^{3}\right )} d^{2} e^{3} - 5 \, {\left (B a^{4} b - 8 \, A a^{3} b^{2}\right )} d e^{4} + {\left (3 \, B a^{5} - 10 \, A a^{4} b\right )} e^{5}\right )} \sqrt {b} e^{\frac {1}{2}} \log \left (b^{2} d^{2} - 4 \, {\left (b d + {\left (2 \, b x + a\right )} e\right )} \sqrt {b x + a} \sqrt {x e + d} \sqrt {b} e^{\frac {1}{2}} + {\left (8 \, b^{2} x^{2} + 8 \, a b x + a^{2}\right )} e^{2} + 2 \, {\left (4 \, b^{2} d x + 3 \, a b d\right )} e\right ) + 4 \, {\left (105 \, B b^{5} d^{4} e - {\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 )} e^{5} - 2 \, {\left (24 \, B b^{5} d x^{3} + 8 \, {\left (11 \, B a b^{4} + 5 \, A b^{5}\right )} d x^{2} + {\left (109 \, B a^{2} b^{3} + 180 \, A a b^{4}\right )} d x + 5 \, {\left (6 \, B a^{3} b^{2} + 73 \, A a^{2} b^{3}\right )} d\right )} e^{4} + 2 \, {\left (28 \, B b^{5} d^{2} x^{2} + {\left (111 \, B a b^{4} + 50 \, A b^{5}\right )} d^{2} x + {\left (173 \, B a^{2} b^{3} + 275 \, A a b^{4}\right )} d^{2}\right )} e^{3} - 10 \, {\left (7 \, B b^{5} d^{3} x + {\left (34 \, B a b^{4} + 15 \, A b^{5}\right )} d^{3}\right )} e^{2}\right )} \sqrt {b x + a} \sqrt {x e + d}\right )} e^{\left (-5\right )}}{7680 \, b^{3}}, -\frac {{\left (15 \, {\left (7 \, B b^{5} d^{5} - 5 \, {\left (5 \, B a b^{4} + 2 \, A b^{5}\right )} d^{4} e + 10 \, {\left (3 \, B a^{2} b^{3} + 4 \, A a b^{4}\right )} d^{3} e^{2} - 10 \, {\left (B a^{3} b^{2} + 6 \, A a^{2} b^{3}\right )} d^{2} e^{3} - 5 \, {\left (B a^{4} b - 8 \, A a^{3} b^{2}\right )} d e^{4} + {\left (3 \, B a^{5} - 10 \, A a^{4} b\right )} e^{5}\right )} \sqrt {-b e} \arctan \left (\frac {{\left (b d + {\left (2 \, b x + a\right )} e\right )} \sqrt {b x + a} \sqrt {-b e} \sqrt {x e + d}}{2 \, {\left ({\left (b^{2} x^{2} + a b x\right )} e^{2} + {\left (b^{2} d x + a b d\right )} e\right )}}\right ) + 2 \, {\left (105 \, B b^{5} d^{4} e - {\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 )} e^{5} - 2 \, {\left (24 \, B b^{5} d x^{3} + 8 \, {\left (11 \, B a b^{4} + 5 \, A b^{5}\right )} d x^{2} + {\left (109 \, B a^{2} b^{3} + 180 \, A a b^{4}\right )} d x + 5 \, {\left (6 \, B a^{3} b^{2} + 73 \, A a^{2} b^{3}\right )} d\right )} e^{4} + 2 \, {\left (28 \, B b^{5} d^{2} x^{2} + {\left (111 \, B a b^{4} + 50 \, A b^{5}\right )} d^{2} x + {\left (173 \, B a^{2} b^{3} + 275 \, A a b^{4}\right )} d^{2}\right )} e^{3} - 10 \, {\left (7 \, B b^{5} d^{3} x + {\left (34 \, B a b^{4} + 15 \, A b^{5}\right )} d^{3}\right )} e^{2}\right )} \sqrt {b x + a} \sqrt {x e + d}\right )} e^{\left (-5\right )}}{3840 \, 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)*(e*x+d)^(1/2),x, algorithm="fricas")

[Out]

[-1/7680*(15*(7*B*b^5*d^5 - 5*(5*B*a*b^4 + 2*A*b^5)*d^4*e + 10*(3*B*a^2*b^3 + 4*A*a*b^4)*d^3*e^2 - 10*(B*a^3*b
^2 + 6*A*a^2*b^3)*d^2*e^3 - 5*(B*a^4*b - 8*A*a^3*b^2)*d*e^4 + (3*B*a^5 - 10*A*a^4*b)*e^5)*sqrt(b)*e^(1/2)*log(
b^2*d^2 - 4*(b*d + (2*b*x + a)*e)*sqrt(b*x + a)*sqrt(x*e + d)*sqrt(b)*e^(1/2) + (8*b^2*x^2 + 8*a*b*x + a^2)*e^
2 + 2*(4*b^2*d*x + 3*a*b*d)*e) + 4*(105*B*b^5*d^4*e - (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)*e^5 - 2*(24*B*
b^5*d*x^3 + 8*(11*B*a*b^4 + 5*A*b^5)*d*x^2 + (109*B*a^2*b^3 + 180*A*a*b^4)*d*x + 5*(6*B*a^3*b^2 + 73*A*a^2*b^3
)*d)*e^4 + 2*(28*B*b^5*d^2*x^2 + (111*B*a*b^4 + 50*A*b^5)*d^2*x + (173*B*a^2*b^3 + 275*A*a*b^4)*d^2)*e^3 - 10*
(7*B*b^5*d^3*x + (34*B*a*b^4 + 15*A*b^5)*d^3)*e^2)*sqrt(b*x + a)*sqrt(x*e + d))*e^(-5)/b^3, -1/3840*(15*(7*B*b
^5*d^5 - 5*(5*B*a*b^4 + 2*A*b^5)*d^4*e + 10*(3*B*a^2*b^3 + 4*A*a*b^4)*d^3*e^2 - 10*(B*a^3*b^2 + 6*A*a^2*b^3)*d
^2*e^3 - 5*(B*a^4*b - 8*A*a^3*b^2)*d*e^4 + (3*B*a^5 - 10*A*a^4*b)*e^5)*sqrt(-b*e)*arctan(1/2*(b*d + (2*b*x + a
)*e)*sqrt(b*x + a)*sqrt(-b*e)*sqrt(x*e + d)/((b^2*x^2 + a*b*x)*e^2 + (b^2*d*x + a*b*d)*e)) + 2*(105*B*b^5*d^4*
e - (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)*e^5 - 2*(24*B*b^5*d*x^3 + 8*(11*B*a*b^4 + 5*A*b^5)*d*x^2 + (109*
B*a^2*b^3 + 180*A*a*b^4)*d*x + 5*(6*B*a^3*b^2 + 73*A*a^2*b^3)*d)*e^4 + 2*(28*B*b^5*d^2*x^2 + (111*B*a*b^4 + 50
*A*b^5)*d^2*x + (173*B*a^2*b^3 + 275*A*a*b^4)*d^2)*e^3 - 10*(7*B*b^5*d^3*x + (34*B*a*b^4 + 15*A*b^5)*d^3)*e^2)
*sqrt(b*x + a)*sqrt(x*e + d))*e^(-5)/b^3]

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

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1670 vs. \(2 (273) = 546\).
time = 1.48, size = 1670, normalized size = 5.49 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1920*(240*(sqrt(b^2*d + (b*x + a)*b*e - a*b*e)*sqrt(b*x + a)*(2*(b*x + a)*(4*(b*x + a)/b^2 + (b^6*d*e^3 - 13
*a*b^5*e^4)*e^(-4)/b^7) - 3*(b^7*d^2*e^2 + 2*a*b^6*d*e^3 - 11*a^2*b^5*e^4)*e^(-4)/b^7) - 3*(b^3*d^3 + a*b^2*d^
2*e + 3*a^2*b*d*e^2 - 5*a^3*e^3)*e^(-5/2)*log(abs(-sqrt(b*x + a)*sqrt(b)*e^(1/2) + sqrt(b^2*d + (b*x + a)*b*e
- a*b*e)))/b^(3/2))*A*a*abs(b) + 30*(sqrt(b^2*d + (b*x + a)*b*e - a*b*e)*(2*(b*x + a)*(4*(b*x + a)*(6*(b*x + a
)/b^3 + (b^12*d*e^5 - 25*a*b^11*e^6)*e^(-6)/b^14) - (5*b^13*d^2*e^4 + 14*a*b^12*d*e^5 - 163*a^2*b^11*e^6)*e^(-
6)/b^14) + 3*(5*b^14*d^3*e^3 + 9*a*b^13*d^2*e^4 + 15*a^2*b^12*d*e^5 - 93*a^3*b^11*e^6)*e^(-6)/b^14)*sqrt(b*x +
a) + 3*(5*b^4*d^4 + 4*a*b^3*d^3*e + 6*a^2*b^2*d^2*e^2 + 20*a^3*b*d*e^3 - 35*a^4*e^4)*e^(-7/2)*log(abs(-sqrt(b
*x + a)*sqrt(b)*e^(1/2) + sqrt(b^2*d + (b*x + a)*b*e - a*b*e)))/b^(5/2))*B*a*abs(b) - 1920*((b^2*d - a*b*e)*e^
(-1/2)*log(abs(-sqrt(b*x + a)*sqrt(b)*e^(1/2) + sqrt(b^2*d + (b*x + a)*b*e - a*b*e)))/sqrt(b) - sqrt(b^2*d + (
b*x + a)*b*e - a*b*e)*sqrt(b*x + a))*A*a^3*abs(b)/b^2 + 240*(sqrt(b^2*d + (b*x + a)*b*e - a*b*e)*sqrt(b*x + a)
*(2*(b*x + a)*(4*(b*x + a)/b^2 + (b^6*d*e^3 - 13*a*b^5*e^4)*e^(-4)/b^7) - 3*(b^7*d^2*e^2 + 2*a*b^6*d*e^3 - 11*
a^2*b^5*e^4)*e^(-4)/b^7) - 3*(b^3*d^3 + a*b^2*d^2*e + 3*a^2*b*d*e^2 - 5*a^3*e^3)*e^(-5/2)*log(abs(-sqrt(b*x +
a)*sqrt(b)*e^(1/2) + sqrt(b^2*d + (b*x + a)*b*e - a*b*e)))/b^(3/2))*B*a^2*abs(b)/b + 10*(sqrt(b^2*d + (b*x + a
)*b*e - a*b*e)*(2*(b*x + a)*(4*(b*x + a)*(6*(b*x + a)/b^3 + (b^12*d*e^5 - 25*a*b^11*e^6)*e^(-6)/b^14) - (5*b^1
3*d^2*e^4 + 14*a*b^12*d*e^5 - 163*a^2*b^11*e^6)*e^(-6)/b^14) + 3*(5*b^14*d^3*e^3 + 9*a*b^13*d^2*e^4 + 15*a^2*b
^12*d*e^5 - 93*a^3*b^11*e^6)*e^(-6)/b^14)*sqrt(b*x + a) + 3*(5*b^4*d^4 + 4*a*b^3*d^3*e + 6*a^2*b^2*d^2*e^2 + 2
0*a^3*b*d*e^3 - 35*a^4*e^4)*e^(-7/2)*log(abs(-sqrt(b*x + a)*sqrt(b)*e^(1/2) + sqrt(b^2*d + (b*x + a)*b*e - a*b
*e)))/b^(5/2))*A*b*abs(b) + (sqrt(b^2*d + (b*x + a)*b*e - a*b*e)*(2*(4*(b*x + a)*(6*(b*x + a)*(8*(b*x + a)/b^4
+ (b^20*d*e^7 - 41*a*b^19*e^8)*e^(-8)/b^23) - (7*b^21*d^2*e^6 + 26*a*b^20*d*e^7 - 513*a^2*b^19*e^8)*e^(-8)/b^
23) + 5*(7*b^22*d^3*e^5 + 19*a*b^21*d^2*e^6 + 37*a^2*b^20*d*e^7 - 447*a^3*b^19*e^8)*e^(-8)/b^23)*(b*x + a) - 1
5*(7*b^23*d^4*e^4 + 12*a*b^22*d^3*e^5 + 18*a^2*b^21*d^2*e^6 + 28*a^3*b^20*d*e^7 - 193*a^4*b^19*e^8)*e^(-8)/b^2
3)*sqrt(b*x + a) - 15*(7*b^5*d^5 + 5*a*b^4*d^4*e + 6*a^2*b^3*d^3*e^2 + 10*a^3*b^2*d^2*e^3 + 35*a^4*b*d*e^4 - 6
3*a^5*e^5)*e^(-9/2)*log(abs(-sqrt(b*x + a)*sqrt(b)*e^(1/2) + sqrt(b^2*d + (b*x + a)*b*e - a*b*e)))/b^(7/2))*B*
b*abs(b) + 480*((b^3*d^2 + 2*a*b^2*d*e - 3*a^2*b*e^2)*e^(-3/2)*log(abs(-sqrt(b*x + a)*sqrt(b)*e^(1/2) + sqrt(b
^2*d + (b*x + a)*b*e - a*b*e)))/sqrt(b) + sqrt(b^2*d + (b*x + a)*b*e - a*b*e)*(2*b*x + (b*d*e - 5*a*e^2)*e^(-2
) + 2*a)*sqrt(b*x + a))*B*a^3*abs(b)/b^3 + 1440*((b^3*d^2 + 2*a*b^2*d*e - 3*a^2*b*e^2)*e^(-3/2)*log(abs(-sqrt(
b*x + a)*sqrt(b)*e^(1/2) + sqrt(b^2*d + (b*x + a)*b*e - a*b*e)))/sqrt(b) + sqrt(b^2*d + (b*x + a)*b*e - a*b*e)
*(2*b*x + (b*d*e - 5*a*e^2)*e^(-2) + 2*a)*sqrt(b*x + a))*A*a^2*abs(b)/b^2)/b

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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