3.23.26 \(\int \frac {(a+b x)^{5/2} (A+B x)}{\sqrt {d+e x}} \, dx\) [2226]

Optimal. Leaf size=246 \[ -\frac {5 (b d-a e)^2 (7 b B d-8 A b e+a B e) \sqrt {a+b x} \sqrt {d+e x}}{64 b e^4}+\frac {5 (b d-a e) (7 b B d-8 A b e+a B e) (a+b x)^{3/2} \sqrt {d+e x}}{96 b e^3}-\frac {(7 b B d-8 A b e+a B e) (a+b x)^{5/2} \sqrt {d+e x}}{24 b e^2}+\frac {B (a+b x)^{7/2} \sqrt {d+e x}}{4 b e}+\frac {5 (b d-a e)^3 (7 b B d-8 A b e+a B e) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{64 b^{3/2} e^{9/2}} \]

[Out]

5/64*(-a*e+b*d)^3*(-8*A*b*e+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^(3/2)/e^(9/2
)+5/96*(-a*e+b*d)*(-8*A*b*e+B*a*e+7*B*b*d)*(b*x+a)^(3/2)*(e*x+d)^(1/2)/b/e^3-1/24*(-8*A*b*e+B*a*e+7*B*b*d)*(b*
x+a)^(5/2)*(e*x+d)^(1/2)/b/e^2+1/4*B*(b*x+a)^(7/2)*(e*x+d)^(1/2)/b/e-5/64*(-a*e+b*d)^2*(-8*A*b*e+B*a*e+7*B*b*d
)*(b*x+a)^(1/2)*(e*x+d)^(1/2)/b/e^4

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Rubi [A]
time = 0.13, antiderivative size = 246, normalized size of antiderivative = 1.00, number of steps used = 7, 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 {5 (b d-a e)^3 (a B e-8 A b e+7 b B d) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{64 b^{3/2} e^{9/2}}-\frac {5 \sqrt {a+b x} \sqrt {d+e x} (b d-a e)^2 (a B e-8 A b e+7 b B d)}{64 b e^4}+\frac {5 (a+b x)^{3/2} \sqrt {d+e x} (b d-a e) (a B e-8 A b e+7 b B d)}{96 b e^3}-\frac {(a+b x)^{5/2} \sqrt {d+e x} (a B e-8 A b e+7 b B d)}{24 b e^2}+\frac {B (a+b x)^{7/2} \sqrt {d+e x}}{4 b e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-5*(b*d - a*e)^2*(7*b*B*d - 8*A*b*e + a*B*e)*Sqrt[a + b*x]*Sqrt[d + e*x])/(64*b*e^4) + (5*(b*d - a*e)*(7*b*B*
d - 8*A*b*e + a*B*e)*(a + b*x)^(3/2)*Sqrt[d + e*x])/(96*b*e^3) - ((7*b*B*d - 8*A*b*e + a*B*e)*(a + b*x)^(5/2)*
Sqrt[d + e*x])/(24*b*e^2) + (B*(a + b*x)^(7/2)*Sqrt[d + e*x])/(4*b*e) + (5*(b*d - a*e)^3*(7*b*B*d - 8*A*b*e +
a*B*e)*ArcTanh[(Sqrt[e]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[d + e*x])])/(64*b^(3/2)*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 \frac {(a+b x)^{5/2} (A+B x)}{\sqrt {d+e x}} \, dx &=\frac {B (a+b x)^{7/2} \sqrt {d+e x}}{4 b e}+\frac {\left (4 A b e-B \left (\frac {7 b d}{2}+\frac {a e}{2}\right )\right ) \int \frac {(a+b x)^{5/2}}{\sqrt {d+e x}} \, dx}{4 b e}\\ &=-\frac {(7 b B d-8 A b e+a B e) (a+b x)^{5/2} \sqrt {d+e x}}{24 b e^2}+\frac {B (a+b x)^{7/2} \sqrt {d+e x}}{4 b e}+\frac {(5 (b d-a e) (7 b B d-8 A b e+a B e)) \int \frac {(a+b x)^{3/2}}{\sqrt {d+e x}} \, dx}{48 b e^2}\\ &=\frac {5 (b d-a e) (7 b B d-8 A b e+a B e) (a+b x)^{3/2} \sqrt {d+e x}}{96 b e^3}-\frac {(7 b B d-8 A b e+a B e) (a+b x)^{5/2} \sqrt {d+e x}}{24 b e^2}+\frac {B (a+b x)^{7/2} \sqrt {d+e x}}{4 b e}-\frac {\left (5 (b d-a e)^2 (7 b B d-8 A b e+a B e)\right ) \int \frac {\sqrt {a+b x}}{\sqrt {d+e x}} \, dx}{64 b e^3}\\ &=-\frac {5 (b d-a e)^2 (7 b B d-8 A b e+a B e) \sqrt {a+b x} \sqrt {d+e x}}{64 b e^4}+\frac {5 (b d-a e) (7 b B d-8 A b e+a B e) (a+b x)^{3/2} \sqrt {d+e x}}{96 b e^3}-\frac {(7 b B d-8 A b e+a B e) (a+b x)^{5/2} \sqrt {d+e x}}{24 b e^2}+\frac {B (a+b x)^{7/2} \sqrt {d+e x}}{4 b e}+\frac {\left (5 (b d-a e)^3 (7 b B d-8 A b e+a B e)\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {d+e x}} \, dx}{128 b e^4}\\ &=-\frac {5 (b d-a e)^2 (7 b B d-8 A b e+a B e) \sqrt {a+b x} \sqrt {d+e x}}{64 b e^4}+\frac {5 (b d-a e) (7 b B d-8 A b e+a B e) (a+b x)^{3/2} \sqrt {d+e x}}{96 b e^3}-\frac {(7 b B d-8 A b e+a B e) (a+b x)^{5/2} \sqrt {d+e x}}{24 b e^2}+\frac {B (a+b x)^{7/2} \sqrt {d+e x}}{4 b e}+\frac {\left (5 (b d-a e)^3 (7 b B d-8 A b e+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 )}{64 b^2 e^4}\\ &=-\frac {5 (b d-a e)^2 (7 b B d-8 A b e+a B e) \sqrt {a+b x} \sqrt {d+e x}}{64 b e^4}+\frac {5 (b d-a e) (7 b B d-8 A b e+a B e) (a+b x)^{3/2} \sqrt {d+e x}}{96 b e^3}-\frac {(7 b B d-8 A b e+a B e) (a+b x)^{5/2} \sqrt {d+e x}}{24 b e^2}+\frac {B (a+b x)^{7/2} \sqrt {d+e x}}{4 b e}+\frac {\left (5 (b d-a e)^3 (7 b B d-8 A b e+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 )}{64 b^2 e^4}\\ &=-\frac {5 (b d-a e)^2 (7 b B d-8 A b e+a B e) \sqrt {a+b x} \sqrt {d+e x}}{64 b e^4}+\frac {5 (b d-a e) (7 b B d-8 A b e+a B e) (a+b x)^{3/2} \sqrt {d+e x}}{96 b e^3}-\frac {(7 b B d-8 A b e+a B e) (a+b x)^{5/2} \sqrt {d+e x}}{24 b e^2}+\frac {B (a+b x)^{7/2} \sqrt {d+e x}}{4 b e}+\frac {5 (b d-a e)^3 (7 b B d-8 A b e+a B e) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{64 b^{3/2} e^{9/2}}\\ \end {align*}

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Mathematica [A]
time = 0.56, size = 230, normalized size = 0.93 \begin {gather*} \frac {\sqrt {a+b x} \sqrt {d+e x} \left (15 a^3 B e^3+a^2 b e^2 (-191 B d+264 A e+118 B e x)+a b^2 e \left (16 A e (-20 d+13 e x)+B \left (265 d^2-172 d e x+136 e^2 x^2\right )\right )+b^3 \left (8 A e \left (15 d^2-10 d e x+8 e^2 x^2\right )+B \left (-105 d^3+70 d^2 e x-56 d e^2 x^2+48 e^3 x^3\right )\right )\right )}{192 b e^4}+\frac {5 (b d-a e)^3 (7 b B d-8 A b e+a B e) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {e} \sqrt {a+b x}}\right )}{64 b^{3/2} 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]*(15*a^3*B*e^3 + a^2*b*e^2*(-191*B*d + 264*A*e + 118*B*e*x) + a*b^2*e*(16*A*e*(-20
*d + 13*e*x) + B*(265*d^2 - 172*d*e*x + 136*e^2*x^2)) + b^3*(8*A*e*(15*d^2 - 10*d*e*x + 8*e^2*x^2) + B*(-105*d
^3 + 70*d^2*e*x - 56*d*e^2*x^2 + 48*e^3*x^3))))/(192*b*e^4) + (5*(b*d - a*e)^3*(7*b*B*d - 8*A*b*e + a*B*e)*Arc
Tanh[(Sqrt[b]*Sqrt[d + e*x])/(Sqrt[e]*Sqrt[a + b*x])])/(64*b^(3/2)*e^(9/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(967\) vs. \(2(208)=416\).
time = 0.09, size = 968, normalized size = 3.93

method	result	size
default	\(\frac {\sqrt {b x +a}\, \sqrt {e x +d}\, \left (120 A \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) a^{3} b \,e^{4}-120 A \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) b^{4} d^{3} e -382 B \sqrt {b e}\, \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, a^{2} b d \,e^{2}+530 B \sqrt {b e}\, \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, a \,b^{2} d^{2} e +236 B \sqrt {b e}\, \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, a^{2} b \,e^{3} x +140 B \sqrt {b e}\, \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, b^{3} d^{2} e x -160 A \sqrt {b e}\, \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, b^{3} d \,e^{2} x +416 A \sqrt {b e}\, \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, a \,b^{2} e^{3} x +96 B \,b^{3} e^{3} x^{3} \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+128 A \,b^{3} e^{3} x^{2} \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}-15 B \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) a^{4} e^{4}+105 B \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) b^{4} d^{4}+528 A \sqrt {b e}\, \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, a^{2} b \,e^{3}+240 A \sqrt {b e}\, \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, b^{3} d^{2} e -60 B \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) a^{3} b d \,e^{3}+270 B \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) a^{2} b^{2} d^{2} e^{2}-300 B \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) a \,b^{3} d^{3} e -360 A \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) a^{2} b^{2} d \,e^{3}+360 A \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) a \,b^{3} d^{2} e^{2}+30 B \sqrt {b e}\, \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, a^{3} e^{3}-210 B \sqrt {b e}\, \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, b^{3} d^{3}-344 B \sqrt {b e}\, \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, a \,b^{2} d \,e^{2} x +272 B a \,b^{2} e^{3} x^{2} \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}-112 B \,b^{3} d \,e^{2} x^{2} \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}-640 A a \,b^{2} d \,e^{2} \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\right )}{384 b \,e^{4} \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}}\)	\(968\)

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]

1/384*(b*x+a)^(1/2)*(e*x+d)^(1/2)*(120*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*e^4-120*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^4*d^3*e-3
82*B*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)*a^2*b*d*e^2+530*B*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)*a*b^2*d^2*e+236
*B*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)*a^2*b*e^3*x+140*B*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)*b^3*d^2*e*x-160*A
*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)*b^3*d*e^2*x+416*A*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)*a*b^2*e^3*x+96*B*b^
3*e^3*x^3*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+128*A*b^3*e^3*x^2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-15*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*e^4+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^4*d^4+528*A*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)*a^2*b*e^3
+240*A*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)*b^3*d^2*e-60*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*d*e^3+270*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^2*d^2*e^2-300*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^
3*d^3*e-360*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^2*d*e^3+360*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^3*d^2*e^2+30*B*(b*e)^(1/2)*((b
*x+a)*(e*x+d))^(1/2)*a^3*e^3-210*B*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)*b^3*d^3-344*B*(b*e)^(1/2)*((b*x+a)*(e*x
+d))^(1/2)*a*b^2*d*e^2*x+272*B*a*b^2*e^3*x^2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-112*B*b^3*d*e^2*x^2*((b*x+a)*
(e*x+d))^(1/2)*(b*e)^(1/2)-640*A*a*b^2*d*e^2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2))/b/e^4/((b*x+a)*(e*x+d))^(1/2
)/(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 = 0.78, size = 734, normalized size = 2.98 \begin {gather*} \left [-\frac {{\left (15 \, {\left (7 \, B b^{4} d^{4} - 4 \, {\left (5 \, B a b^{3} + 2 \, A b^{4}\right )} d^{3} e + 6 \, {\left (3 \, B a^{2} b^{2} + 4 \, A a b^{3}\right )} d^{2} e^{2} - 4 \, {\left (B a^{3} b + 6 \, A a^{2} b^{2}\right )} d e^{3} - {\left (B a^{4} - 8 \, A a^{3} b\right )} e^{4}\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^{4} d^{3} e - {\left (48 \, B b^{4} x^{3} + 15 \, B a^{3} b + 264 \, A a^{2} b^{2} + 8 \, {\left (17 \, B a b^{3} + 8 \, A b^{4}\right )} x^{2} + 2 \, {\left (59 \, B a^{2} b^{2} + 104 \, A a b^{3}\right )} x\right )} e^{4} + {\left (56 \, B b^{4} d x^{2} + 4 \, {\left (43 \, B a b^{3} + 20 \, A b^{4}\right )} d x + {\left (191 \, B a^{2} b^{2} + 320 \, A a b^{3}\right )} d\right )} e^{3} - 5 \, {\left (14 \, B b^{4} d^{2} x + {\left (53 \, B a b^{3} + 24 \, A b^{4}\right )} d^{2}\right )} e^{2}\right )} \sqrt {b x + a} \sqrt {x e + d}\right )} e^{\left (-5\right )}}{768 \, b^{2}}, -\frac {{\left (15 \, {\left (7 \, B b^{4} d^{4} - 4 \, {\left (5 \, B a b^{3} + 2 \, A b^{4}\right )} d^{3} e + 6 \, {\left (3 \, B a^{2} b^{2} + 4 \, A a b^{3}\right )} d^{2} e^{2} - 4 \, {\left (B a^{3} b + 6 \, A a^{2} b^{2}\right )} d e^{3} - {\left (B a^{4} - 8 \, A a^{3} b\right )} e^{4}\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^{4} d^{3} e - {\left (48 \, B b^{4} x^{3} + 15 \, B a^{3} b + 264 \, A a^{2} b^{2} + 8 \, {\left (17 \, B a b^{3} + 8 \, A b^{4}\right )} x^{2} + 2 \, {\left (59 \, B a^{2} b^{2} + 104 \, A a b^{3}\right )} x\right )} e^{4} + {\left (56 \, B b^{4} d x^{2} + 4 \, {\left (43 \, B a b^{3} + 20 \, A b^{4}\right )} d x + {\left (191 \, B a^{2} b^{2} + 320 \, A a b^{3}\right )} d\right )} e^{3} - 5 \, {\left (14 \, B b^{4} d^{2} x + {\left (53 \, B a b^{3} + 24 \, A b^{4}\right )} d^{2}\right )} e^{2}\right )} \sqrt {b x + a} \sqrt {x e + d}\right )} e^{\left (-5\right )}}{384 \, b^{2}}\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]

[-1/768*(15*(7*B*b^4*d^4 - 4*(5*B*a*b^3 + 2*A*b^4)*d^3*e + 6*(3*B*a^2*b^2 + 4*A*a*b^3)*d^2*e^2 - 4*(B*a^3*b +
6*A*a^2*b^2)*d*e^3 - (B*a^4 - 8*A*a^3*b)*e^4)*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^
4*d^3*e - (48*B*b^4*x^3 + 15*B*a^3*b + 264*A*a^2*b^2 + 8*(17*B*a*b^3 + 8*A*b^4)*x^2 + 2*(59*B*a^2*b^2 + 104*A*
a*b^3)*x)*e^4 + (56*B*b^4*d*x^2 + 4*(43*B*a*b^3 + 20*A*b^4)*d*x + (191*B*a^2*b^2 + 320*A*a*b^3)*d)*e^3 - 5*(14
*B*b^4*d^2*x + (53*B*a*b^3 + 24*A*b^4)*d^2)*e^2)*sqrt(b*x + a)*sqrt(x*e + d))*e^(-5)/b^2, -1/384*(15*(7*B*b^4*
d^4 - 4*(5*B*a*b^3 + 2*A*b^4)*d^3*e + 6*(3*B*a^2*b^2 + 4*A*a*b^3)*d^2*e^2 - 4*(B*a^3*b + 6*A*a^2*b^2)*d*e^3 -
(B*a^4 - 8*A*a^3*b)*e^4)*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^4*d^3*e - (48*B*b^4*x^3 + 15*B*a^3*b + 264*A*a^2*b^2
+ 8*(17*B*a*b^3 + 8*A*b^4)*x^2 + 2*(59*B*a^2*b^2 + 104*A*a*b^3)*x)*e^4 + (56*B*b^4*d*x^2 + 4*(43*B*a*b^3 + 20
*A*b^4)*d*x + (191*B*a^2*b^2 + 320*A*a*b^3)*d)*e^3 - 5*(14*B*b^4*d^2*x + (53*B*a*b^3 + 24*A*b^4)*d^2)*e^2)*sqr
t(b*x + a)*sqrt(x*e + d))*e^(-5)/b^2]

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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 [A]
time = 1.87, size = 390, normalized size = 1.59 \begin {gather*} \frac {{\left (\sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e} {\left (2 \, {\left (b x + a\right )} {\left (4 \, {\left (b x + a\right )} {\left (\frac {6 \, {\left (b x + a\right )} B e^{\left (-1\right )}}{b^{2}} - \frac {{\left (7 \, B b^{3} d e^{5} + B a b^{2} e^{6} - 8 \, A b^{3} e^{6}\right )} e^{\left (-7\right )}}{b^{4}}\right )} + \frac {5 \, {\left (7 \, B b^{4} d^{2} e^{4} - 6 \, B a b^{3} d e^{5} - 8 \, A b^{4} d e^{5} - B a^{2} b^{2} e^{6} + 8 \, A a b^{3} e^{6}\right )} e^{\left (-7\right )}}{b^{4}}\right )} - \frac {15 \, {\left (7 \, B b^{5} d^{3} e^{3} - 13 \, B a b^{4} d^{2} e^{4} - 8 \, A b^{5} d^{2} e^{4} + 5 \, B a^{2} b^{3} d e^{5} + 16 \, A a b^{4} d e^{5} + B a^{3} b^{2} e^{6} - 8 \, A a^{2} b^{3} e^{6}\right )} e^{\left (-7\right )}}{b^{4}}\right )} \sqrt {b x + a} - \frac {15 \, {\left (7 \, B b^{4} d^{4} - 20 \, B a b^{3} d^{3} e - 8 \, A b^{4} d^{3} e + 18 \, B a^{2} b^{2} d^{2} e^{2} + 24 \, A a b^{3} d^{2} e^{2} - 4 \, B a^{3} b d e^{3} - 24 \, A a^{2} b^{2} d e^{3} - B a^{4} e^{4} + 8 \, A a^{3} b e^{4}\right )} e^{\left (-\frac {9}{2}\right )} \log \left ({\left | -\sqrt {b x + a} \sqrt {b} e^{\frac {1}{2}} + \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e} \right |}\right )}{b^{\frac {3}{2}}}\right )} b}{192 \, {\left | b \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="giac")

[Out]

1/192*(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*e^(-1)/b^2 - (7*B*b^3*d*e^
5 + B*a*b^2*e^6 - 8*A*b^3*e^6)*e^(-7)/b^4) + 5*(7*B*b^4*d^2*e^4 - 6*B*a*b^3*d*e^5 - 8*A*b^4*d*e^5 - B*a^2*b^2*
e^6 + 8*A*a*b^3*e^6)*e^(-7)/b^4) - 15*(7*B*b^5*d^3*e^3 - 13*B*a*b^4*d^2*e^4 - 8*A*b^5*d^2*e^4 + 5*B*a^2*b^3*d*
e^5 + 16*A*a*b^4*d*e^5 + B*a^3*b^2*e^6 - 8*A*a^2*b^3*e^6)*e^(-7)/b^4)*sqrt(b*x + a) - 15*(7*B*b^4*d^4 - 20*B*a
*b^3*d^3*e - 8*A*b^4*d^3*e + 18*B*a^2*b^2*d^2*e^2 + 24*A*a*b^3*d^2*e^2 - 4*B*a^3*b*d*e^3 - 24*A*a^2*b^2*d*e^3
- B*a^4*e^4 + 8*A*a^3*b*e^4)*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^(3/2))*b/abs(b)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\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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