3.22.100 \(\int \sqrt {a+b x} (A+B x) (d+e x)^{5/2} \, dx\) [2200]

Optimal. Leaf size=304 \[ -\frac {(b d-a e)^3 (3 b B d-10 A b e+7 a B e) \sqrt {a+b x} \sqrt {d+e x}}{128 b^4 e^2}-\frac {(b d-a e)^2 (3 b B d-10 A b e+7 a B e) (a+b x)^{3/2} \sqrt {d+e x}}{64 b^4 e}-\frac {(b d-a e) (3 b B d-10 A b e+7 a B e) (a+b x)^{3/2} (d+e x)^{3/2}}{48 b^3 e}-\frac {(3 b B d-10 A b e+7 a B e) (a+b x)^{3/2} (d+e x)^{5/2}}{40 b^2 e}+\frac {B (a+b x)^{3/2} (d+e x)^{7/2}}{5 b e}+\frac {(b d-a e)^4 (3 b B d-10 A b e+7 a B e) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{128 b^{9/2} e^{5/2}} \]

[Out]

-1/48*(-a*e+b*d)*(-10*A*b*e+7*B*a*e+3*B*b*d)*(b*x+a)^(3/2)*(e*x+d)^(3/2)/b^3/e-1/40*(-10*A*b*e+7*B*a*e+3*B*b*d
)*(b*x+a)^(3/2)*(e*x+d)^(5/2)/b^2/e+1/5*B*(b*x+a)^(3/2)*(e*x+d)^(7/2)/b/e+1/128*(-a*e+b*d)^4*(-10*A*b*e+7*B*a*
e+3*B*b*d)*arctanh(e^(1/2)*(b*x+a)^(1/2)/b^(1/2)/(e*x+d)^(1/2))/b^(9/2)/e^(5/2)-1/64*(-a*e+b*d)^2*(-10*A*b*e+7
*B*a*e+3*B*b*d)*(b*x+a)^(3/2)*(e*x+d)^(1/2)/b^4/e-1/128*(-a*e+b*d)^3*(-10*A*b*e+7*B*a*e+3*B*b*d)*(b*x+a)^(1/2)
*(e*x+d)^(1/2)/b^4/e^2

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Rubi [A]
time = 0.17, 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 (7 a B e-10 A b e+3 b B d) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{128 b^{9/2} e^{5/2}}-\frac {\sqrt {a+b x} \sqrt {d+e x} (b d-a e)^3 (7 a B e-10 A b e+3 b B d)}{128 b^4 e^2}-\frac {(a+b x)^{3/2} \sqrt {d+e x} (b d-a e)^2 (7 a B e-10 A b e+3 b B d)}{64 b^4 e}-\frac {(a+b x)^{3/2} (d+e x)^{3/2} (b d-a e) (7 a B e-10 A b e+3 b B d)}{48 b^3 e}-\frac {(a+b x)^{3/2} (d+e x)^{5/2} (7 a B e-10 A b e+3 b B d)}{40 b^2 e}+\frac {B (a+b x)^{3/2} (d+e x)^{7/2}}{5 b e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/128*((b*d - a*e)^3*(3*b*B*d - 10*A*b*e + 7*a*B*e)*Sqrt[a + b*x]*Sqrt[d + e*x])/(b^4*e^2) - ((b*d - a*e)^2*(
3*b*B*d - 10*A*b*e + 7*a*B*e)*(a + b*x)^(3/2)*Sqrt[d + e*x])/(64*b^4*e) - ((b*d - a*e)*(3*b*B*d - 10*A*b*e + 7
*a*B*e)*(a + b*x)^(3/2)*(d + e*x)^(3/2))/(48*b^3*e) - ((3*b*B*d - 10*A*b*e + 7*a*B*e)*(a + b*x)^(3/2)*(d + e*x
)^(5/2))/(40*b^2*e) + (B*(a + b*x)^(3/2)*(d + e*x)^(7/2))/(5*b*e) + ((b*d - a*e)^4*(3*b*B*d - 10*A*b*e + 7*a*B
*e)*ArcTanh[(Sqrt[e]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[d + e*x])])/(128*b^(9/2)*e^(5/2))

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 \sqrt {a+b x} (A+B x) (d+e x)^{5/2} \, dx &=\frac {B (a+b x)^{3/2} (d+e x)^{7/2}}{5 b e}+\frac {\left (5 A b e-B \left (\frac {3 b d}{2}+\frac {7 a e}{2}\right )\right ) \int \sqrt {a+b x} (d+e x)^{5/2} \, dx}{5 b e}\\ &=-\frac {(3 b B d-10 A b e+7 a B e) (a+b x)^{3/2} (d+e x)^{5/2}}{40 b^2 e}+\frac {B (a+b x)^{3/2} (d+e x)^{7/2}}{5 b e}-\frac {((b d-a e) (3 b B d-10 A b e+7 a B e)) \int \sqrt {a+b x} (d+e x)^{3/2} \, dx}{16 b^2 e}\\ &=-\frac {(b d-a e) (3 b B d-10 A b e+7 a B e) (a+b x)^{3/2} (d+e x)^{3/2}}{48 b^3 e}-\frac {(3 b B d-10 A b e+7 a B e) (a+b x)^{3/2} (d+e x)^{5/2}}{40 b^2 e}+\frac {B (a+b x)^{3/2} (d+e x)^{7/2}}{5 b e}-\frac {\left ((b d-a e)^2 (3 b B d-10 A b e+7 a B e)\right ) \int \sqrt {a+b x} \sqrt {d+e x} \, dx}{32 b^3 e}\\ &=-\frac {(b d-a e)^2 (3 b B d-10 A b e+7 a B e) (a+b x)^{3/2} \sqrt {d+e x}}{64 b^4 e}-\frac {(b d-a e) (3 b B d-10 A b e+7 a B e) (a+b x)^{3/2} (d+e x)^{3/2}}{48 b^3 e}-\frac {(3 b B d-10 A b e+7 a B e) (a+b x)^{3/2} (d+e x)^{5/2}}{40 b^2 e}+\frac {B (a+b x)^{3/2} (d+e x)^{7/2}}{5 b e}-\frac {\left ((b d-a e)^3 (3 b B d-10 A b e+7 a B e)\right ) \int \frac {\sqrt {a+b x}}{\sqrt {d+e x}} \, dx}{128 b^4 e}\\ &=-\frac {(b d-a e)^3 (3 b B d-10 A b e+7 a B e) \sqrt {a+b x} \sqrt {d+e x}}{128 b^4 e^2}-\frac {(b d-a e)^2 (3 b B d-10 A b e+7 a B e) (a+b x)^{3/2} \sqrt {d+e x}}{64 b^4 e}-\frac {(b d-a e) (3 b B d-10 A b e+7 a B e) (a+b x)^{3/2} (d+e x)^{3/2}}{48 b^3 e}-\frac {(3 b B d-10 A b e+7 a B e) (a+b x)^{3/2} (d+e x)^{5/2}}{40 b^2 e}+\frac {B (a+b x)^{3/2} (d+e x)^{7/2}}{5 b e}+\frac {\left ((b d-a e)^4 (3 b B d-10 A b e+7 a B e)\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {d+e x}} \, dx}{256 b^4 e^2}\\ &=-\frac {(b d-a e)^3 (3 b B d-10 A b e+7 a B e) \sqrt {a+b x} \sqrt {d+e x}}{128 b^4 e^2}-\frac {(b d-a e)^2 (3 b B d-10 A b e+7 a B e) (a+b x)^{3/2} \sqrt {d+e x}}{64 b^4 e}-\frac {(b d-a e) (3 b B d-10 A b e+7 a B e) (a+b x)^{3/2} (d+e x)^{3/2}}{48 b^3 e}-\frac {(3 b B d-10 A b e+7 a B e) (a+b x)^{3/2} (d+e x)^{5/2}}{40 b^2 e}+\frac {B (a+b x)^{3/2} (d+e x)^{7/2}}{5 b e}+\frac {\left ((b d-a e)^4 (3 b B d-10 A b e+7 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^5 e^2}\\ &=-\frac {(b d-a e)^3 (3 b B d-10 A b e+7 a B e) \sqrt {a+b x} \sqrt {d+e x}}{128 b^4 e^2}-\frac {(b d-a e)^2 (3 b B d-10 A b e+7 a B e) (a+b x)^{3/2} \sqrt {d+e x}}{64 b^4 e}-\frac {(b d-a e) (3 b B d-10 A b e+7 a B e) (a+b x)^{3/2} (d+e x)^{3/2}}{48 b^3 e}-\frac {(3 b B d-10 A b e+7 a B e) (a+b x)^{3/2} (d+e x)^{5/2}}{40 b^2 e}+\frac {B (a+b x)^{3/2} (d+e x)^{7/2}}{5 b e}+\frac {\left ((b d-a e)^4 (3 b B d-10 A b e+7 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^5 e^2}\\ &=-\frac {(b d-a e)^3 (3 b B d-10 A b e+7 a B e) \sqrt {a+b x} \sqrt {d+e x}}{128 b^4 e^2}-\frac {(b d-a e)^2 (3 b B d-10 A b e+7 a B e) (a+b x)^{3/2} \sqrt {d+e x}}{64 b^4 e}-\frac {(b d-a e) (3 b B d-10 A b e+7 a B e) (a+b x)^{3/2} (d+e x)^{3/2}}{48 b^3 e}-\frac {(3 b B d-10 A b e+7 a B e) (a+b x)^{3/2} (d+e x)^{5/2}}{40 b^2 e}+\frac {B (a+b x)^{3/2} (d+e x)^{7/2}}{5 b e}+\frac {(b d-a e)^4 (3 b B d-10 A b e+7 a B e) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{128 b^{9/2} e^{5/2}}\\ \end {align*}

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Mathematica [A]
time = 0.79, size = 322, normalized size = 1.06 \begin {gather*} \frac {\sqrt {a+b x} \sqrt {d+e x} \left (-105 a^4 B e^4+10 a^3 b e^3 (34 B d+15 A e+7 B e x)-2 a^2 b^2 e^2 \left (25 A e (11 d+2 e x)+B \left (173 d^2+111 d e x+28 e^2 x^2\right )\right )+2 a b^3 e \left (5 A e \left (73 d^2+36 d e x+8 e^2 x^2\right )+B \left (30 d^3+109 d^2 e x+88 d e^2 x^2+24 e^3 x^3\right )\right )+b^4 \left (10 A e \left (15 d^3+118 d^2 e x+136 d e^2 x^2+48 e^3 x^3\right )+B \left (-45 d^4+30 d^3 e x+744 d^2 e^2 x^2+1008 d e^3 x^3+384 e^4 x^4\right )\right )\right )}{1920 b^4 e^2}+\frac {(b d-a e)^4 (3 b B d-10 A b e+7 a B e) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{128 b^{9/2} e^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[a + b*x]*Sqrt[d + e*x]*(-105*a^4*B*e^4 + 10*a^3*b*e^3*(34*B*d + 15*A*e + 7*B*e*x) - 2*a^2*b^2*e^2*(25*A*
e*(11*d + 2*e*x) + B*(173*d^2 + 111*d*e*x + 28*e^2*x^2)) + 2*a*b^3*e*(5*A*e*(73*d^2 + 36*d*e*x + 8*e^2*x^2) +
B*(30*d^3 + 109*d^2*e*x + 88*d*e^2*x^2 + 24*e^3*x^3)) + b^4*(10*A*e*(15*d^3 + 118*d^2*e*x + 136*d*e^2*x^2 + 48
*e^3*x^3) + B*(-45*d^4 + 30*d^3*e*x + 744*d^2*e^2*x^2 + 1008*d*e^3*x^3 + 384*e^4*x^4))))/(1920*b^4*e^2) + ((b*
d - a*e)^4*(3*b*B*d - 10*A*b*e + 7*a*B*e)*ArcTanh[(Sqrt[e]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[d + e*x])])/(128*b^(9/
2)*e^(5/2))

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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 = 0.13, 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)*(e*x+d)^(5/2)*(b*x+a)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/3840*(e*x+d)^(1/2)*(b*x+a)^(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)-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))*a^5*e^5-45*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))*b^5*d^5+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^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^2-450*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^3*b^2*d^2*e^3+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^2*b^3*d^3*e^2+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*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+1100*A*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)*a^2*b^2*d*e^3-140*B*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)*a^3*b*e^4
*x-60*B*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)*b^4*d^3*e*x-1460*A*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)*a*b^3*d^2*e
^2-680*B*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)*a^3*b*d*e^3-436*B*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)*a*b^3*d^2*e
^2*x-96*B*a*b^3*e^4*x^3*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-2016*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+444*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^2-120*B*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)
*a*b^3*d^3*e+200*A*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)*a^2*b^2*e^4*x-2360*A*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2
)*b^4*d^2*e^2*x-160*A*a*b^3*e^4*x^2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-2720*A*b^4*d*e^3*x^2*((b*x+a)*(e*x+d))
^(1/2)*(b*e)^(1/2)+112*B*a^2*b^2*e^4*x^2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-1488*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+210*B*(b*e
)^(1/2)*((b*x+a)*(e*x+d))^(1/2)*a^4*e^4+90*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))/e^2/((b*x+a)*(e*x+d))^(1/2)/b^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)*(e*x+d)^(5/2)*(b*x+a)^(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 = 1.54, size = 996, normalized size = 3.28 \begin {gather*} \left [-\frac {{\left (15 \, {\left (3 \, B b^{5} d^{5} - 5 \, {\left (B a b^{4} + 2 \, A b^{5}\right )} d^{4} e - 10 \, {\left (B a^{2} b^{3} - 4 \, A a b^{4}\right )} d^{3} e^{2} + 30 \, {\left (B a^{3} b^{2} - 2 \, A a^{2} b^{3}\right )} d^{2} e^{3} - 5 \, {\left (5 \, B a^{4} b - 8 \, A a^{3} b^{2}\right )} d e^{4} + {\left (7 \, 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 (45 \, B b^{5} d^{4} e - {\left (384 \, B b^{5} x^{4} - 105 \, B a^{4} b + 150 \, A a^{3} b^{2} + 48 \, {\left (B a b^{4} + 10 \, A b^{5}\right )} x^{3} - 8 \, {\left (7 \, B a^{2} b^{3} - 10 \, A a b^{4}\right )} x^{2} + 10 \, {\left (7 \, B a^{3} b^{2} - 10 \, A a^{2} b^{3}\right )} x\right )} e^{5} - 2 \, {\left (504 \, B b^{5} d x^{3} + 8 \, {\left (11 \, B a b^{4} + 85 \, A b^{5}\right )} d x^{2} - 3 \, {\left (37 \, B a^{2} b^{3} - 60 \, A a b^{4}\right )} d x + 5 \, {\left (34 \, B a^{3} b^{2} - 55 \, A a^{2} b^{3}\right )} d\right )} e^{4} - 2 \, {\left (372 \, B b^{5} d^{2} x^{2} + {\left (109 \, B a b^{4} + 590 \, A b^{5}\right )} d^{2} x - {\left (173 \, B a^{2} b^{3} - 365 \, A a b^{4}\right )} d^{2}\right )} e^{3} - 30 \, {\left (B b^{5} d^{3} x + {\left (2 \, B a b^{4} + 5 \, A b^{5}\right )} d^{3}\right )} e^{2}\right )} \sqrt {b x + a} \sqrt {x e + d}\right )} e^{\left (-3\right )}}{7680 \, b^{5}}, -\frac {{\left (15 \, {\left (3 \, B b^{5} d^{5} - 5 \, {\left (B a b^{4} + 2 \, A b^{5}\right )} d^{4} e - 10 \, {\left (B a^{2} b^{3} - 4 \, A a b^{4}\right )} d^{3} e^{2} + 30 \, {\left (B a^{3} b^{2} - 2 \, A a^{2} b^{3}\right )} d^{2} e^{3} - 5 \, {\left (5 \, B a^{4} b - 8 \, A a^{3} b^{2}\right )} d e^{4} + {\left (7 \, 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 (45 \, B b^{5} d^{4} e - {\left (384 \, B b^{5} x^{4} - 105 \, B a^{4} b + 150 \, A a^{3} b^{2} + 48 \, {\left (B a b^{4} + 10 \, A b^{5}\right )} x^{3} - 8 \, {\left (7 \, B a^{2} b^{3} - 10 \, A a b^{4}\right )} x^{2} + 10 \, {\left (7 \, B a^{3} b^{2} - 10 \, A a^{2} b^{3}\right )} x\right )} e^{5} - 2 \, {\left (504 \, B b^{5} d x^{3} + 8 \, {\left (11 \, B a b^{4} + 85 \, A b^{5}\right )} d x^{2} - 3 \, {\left (37 \, B a^{2} b^{3} - 60 \, A a b^{4}\right )} d x + 5 \, {\left (34 \, B a^{3} b^{2} - 55 \, A a^{2} b^{3}\right )} d\right )} e^{4} - 2 \, {\left (372 \, B b^{5} d^{2} x^{2} + {\left (109 \, B a b^{4} + 590 \, A b^{5}\right )} d^{2} x - {\left (173 \, B a^{2} b^{3} - 365 \, A a b^{4}\right )} d^{2}\right )} e^{3} - 30 \, {\left (B b^{5} d^{3} x + {\left (2 \, B a b^{4} + 5 \, A b^{5}\right )} d^{3}\right )} e^{2}\right )} \sqrt {b x + a} \sqrt {x e + d}\right )} e^{\left (-3\right )}}{3840 \, b^{5}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/7680*(15*(3*B*b^5*d^5 - 5*(B*a*b^4 + 2*A*b^5)*d^4*e - 10*(B*a^2*b^3 - 4*A*a*b^4)*d^3*e^2 + 30*(B*a^3*b^2 -
2*A*a^2*b^3)*d^2*e^3 - 5*(5*B*a^4*b - 8*A*a^3*b^2)*d*e^4 + (7*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*(45*B*b^5*d^4*e - (384*B*b^5*x^4 - 105*B*a^4*b + 150*A*a^3*b^2 + 48*(B*a*b^4
+ 10*A*b^5)*x^3 - 8*(7*B*a^2*b^3 - 10*A*a*b^4)*x^2 + 10*(7*B*a^3*b^2 - 10*A*a^2*b^3)*x)*e^5 - 2*(504*B*b^5*d*x
^3 + 8*(11*B*a*b^4 + 85*A*b^5)*d*x^2 - 3*(37*B*a^2*b^3 - 60*A*a*b^4)*d*x + 5*(34*B*a^3*b^2 - 55*A*a^2*b^3)*d)*
e^4 - 2*(372*B*b^5*d^2*x^2 + (109*B*a*b^4 + 590*A*b^5)*d^2*x - (173*B*a^2*b^3 - 365*A*a*b^4)*d^2)*e^3 - 30*(B*
b^5*d^3*x + (2*B*a*b^4 + 5*A*b^5)*d^3)*e^2)*sqrt(b*x + a)*sqrt(x*e + d))*e^(-3)/b^5, -1/3840*(15*(3*B*b^5*d^5
- 5*(B*a*b^4 + 2*A*b^5)*d^4*e - 10*(B*a^2*b^3 - 4*A*a*b^4)*d^3*e^2 + 30*(B*a^3*b^2 - 2*A*a^2*b^3)*d^2*e^3 - 5*
(5*B*a^4*b - 8*A*a^3*b^2)*d*e^4 + (7*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*(45*B*b^5*d^4*e - (384*B
*b^5*x^4 - 105*B*a^4*b + 150*A*a^3*b^2 + 48*(B*a*b^4 + 10*A*b^5)*x^3 - 8*(7*B*a^2*b^3 - 10*A*a*b^4)*x^2 + 10*(
7*B*a^3*b^2 - 10*A*a^2*b^3)*x)*e^5 - 2*(504*B*b^5*d*x^3 + 8*(11*B*a*b^4 + 85*A*b^5)*d*x^2 - 3*(37*B*a^2*b^3 -
60*A*a*b^4)*d*x + 5*(34*B*a^3*b^2 - 55*A*a^2*b^3)*d)*e^4 - 2*(372*B*b^5*d^2*x^2 + (109*B*a*b^4 + 590*A*b^5)*d^
2*x - (173*B*a^2*b^3 - 365*A*a*b^4)*d^2)*e^3 - 30*(B*b^5*d^3*x + (2*B*a*b^4 + 5*A*b^5)*d^3)*e^2)*sqrt(b*x + a)
*sqrt(x*e + d))*e^(-3)/b^5]

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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)*(e*x+d)**(5/2)*(b*x+a)**(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. 2506 vs. \(2 (273) = 546\).
time = 1.31, size = 2506, normalized size = 8.24 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/1920*(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*d^2*abs(b)/b^2 - 80*(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*d^2*ab
s(b)/b - 160*(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 - 1
3*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*d*abs(b)*e/b^2 - 160*(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*d*abs(b)*e/b - 20*(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*d*abs(b)*e/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*d^2*abs(b)/b^3 - 480*((b^3*d^2 + 2*a*b^2*d*e - 3*a^2*b*e^2)*e^(-3/2)*log(ab
s(-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*d^2*abs(b)/b^2 - 80*(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)*e^2/
b^2 - 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^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) + s
qrt(b^2*d + (b*x + a)*b*e - a*b*e)))/b^(5/2))*B*a*abs(b)*e^2/b^2 - 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^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))*A*a
bs(b)*e^2/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^2
2*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) - 15*(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^23)*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 - 63*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*abs(b)*e^2/b -
960*((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)*s
qrt(b*x + a))*A*a*d*abs(b)*e/b^3)/b

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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