3.2200 \(\int \sqrt{a+b x} (A+B x) (d+e x)^{5/2} \, dx\)

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

[Out]

-((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) - ((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))

________________________________________________________________________________________

Rubi [A]  time = 0.266139, antiderivative size = 304, normalized size of antiderivative = 1., 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 = {80, 50, 63, 217, 206} \[ -\frac{\sqrt{a+b x} \sqrt{d+e x} (b d-a e)^3 (7 a B e-10 A b e+3 b B d)}{128 b^4 e^2}+\frac{(b d-a e)^4 (7 a B e-10 A b e+3 b B d) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{128 b^{9/2} e^{5/2}}-\frac{(a+b x)^{3/2} \sqrt{d+e x} (b d-a e)^2 (7 a B e-10 A b e+3 b B d)}{64 b^4 e}-\frac{(a+b x)^{3/2} (d+e x)^{3/2} (b d-a e) (7 a B e-10 A b e+3 b B d)}{48 b^3 e}-\frac{(a+b x)^{3/2} (d+e x)^{5/2} (7 a B e-10 A b e+3 b B d)}{40 b^2 e}+\frac{B (a+b x)^{3/2} (d+e x)^{7/2}}{5 b e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((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) - ((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 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 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 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]

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])

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 ) \operatorname{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 ) \operatorname{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*}

Mathematica [A]  time = 3.01475, size = 301, normalized size = 0.99 \[ \frac{\sqrt{d+e x} \left (\frac{\left (-\frac{7 a B e}{2}+5 A b e-\frac{3}{2} b B d\right ) \left (2 b^5 e^2 (a+b x)^2 (b d-a e)^{3/2} \sqrt{\frac{b (d+e x)}{b d-a e}} \left (15 a^2 e^2-10 a b e (5 d+2 e x)+b^2 \left (59 d^2+68 d e x+24 e^2 x^2\right )\right )+15 b^5 e (a+b x) (b d-a e)^{9/2} \sqrt{\frac{b (d+e x)}{b d-a e}}-15 b^5 \sqrt{e} \sqrt{a+b x} (b d-a e)^5 \sinh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b d-a e}}\right )\right )}{(b d-a e)^{3/2} \sqrt{\frac{b (d+e x)}{b d-a e}}}+192 b^8 B e^2 (a+b x)^2 (d+e x)^3\right )}{960 b^9 e^3 \sqrt{a+b x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[d + e*x]*(192*b^8*B*e^2*(a + b*x)^2*(d + e*x)^3 + (((-3*b*B*d)/2 + 5*A*b*e - (7*a*B*e)/2)*(15*b^5*e*(b*d
- a*e)^(9/2)*(a + b*x)*Sqrt[(b*(d + e*x))/(b*d - a*e)] + 2*b^5*e^2*(b*d - a*e)^(3/2)*(a + b*x)^2*Sqrt[(b*(d +
e*x))/(b*d - a*e)]*(15*a^2*e^2 - 10*a*b*e*(5*d + 2*e*x) + b^2*(59*d^2 + 68*d*e*x + 24*e^2*x^2)) - 15*b^5*Sqrt
[e]*(b*d - a*e)^5*Sqrt[a + b*x]*ArcSinh[(Sqrt[e]*Sqrt[a + b*x])/Sqrt[b*d - a*e]]))/((b*d - a*e)^(3/2)*Sqrt[(b*
(d + e*x))/(b*d - a*e)])))/(960*b^9*e^3*Sqrt[a + b*x])

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Maple [B]  time = 0.027, size = 1631, normalized size = 5.4 \begin{align*} \text{result too large to display} \end{align*}

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)

[Out]

-1/3840*(e*x+d)^(1/2)*(b*x+a)^(1/2)*(75*B*ln(1/2*(2*b*x*e+2*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)+a*e+b*
d)/(b*e)^(1/2))*a*b^4*d^4*e-96*B*x^3*a*b^3*e^4*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)-1460*A*(b*e)^(1/2)*
(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*a*b^3*d^2*e^2-768*B*x^4*b^4*e^4*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)-96
0*A*x^3*b^4*e^4*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)-300*A*(b*e)^(1/2)*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*
a^3*b*e^4-300*A*(b*e)^(1/2)*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*b^4*d^3*e+150*A*ln(1/2*(2*b*x*e+2*(b*e*x^2+a*e*x+b
*d*x+a*d)^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a^4*b*e^5+150*B*ln(1/2*(2*b*x*e+2*(b*e*x^2+a*e*x+b*d*x+a*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*x*e+2*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2
)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a^4*b*d*e^4-450*B*ln(1/2*(2*b*x*e+2*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^
(1/2)+a*e+b*d)/(b*e)^(1/2))*a^3*b^2*d^2*e^3-2016*B*x^3*b^4*d*e^3*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)-6
0*B*(b*e)^(1/2)*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*x*b^4*d^3*e-680*B*(b*e)^(1/2)*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*
a^3*b*d*e^3-1488*B*x^2*b^4*d^2*e^2*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)-160*A*x^2*a*b^3*e^4*(b*e*x^2+a*
e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)-2720*A*x^2*b^4*d*e^3*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)+112*B*x^2*a^
2*b^2*e^4*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)-600*A*ln(1/2*(2*b*x*e+2*(b*e*x^2+a*e*x+b*d*x+a*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*x*e+2*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^
(1/2)+a*e+b*d)/(b*e)^(1/2))*a^2*b^3*d^2*e^3+200*A*(b*e)^(1/2)*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*x*a^2*b^2*e^4-23
60*A*(b*e)^(1/2)*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*x*b^4*d^2*e^2-140*B*(b*e)^(1/2)*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/
2)*x*a^3*b*e^4-436*B*(b*e)^(1/2)*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*x*a*b^3*d^2*e^2-720*A*(b*e)^(1/2)*(b*e*x^2+a*
e*x+b*d*x+a*d)^(1/2)*x*a*b^3*d*e^3+444*B*(b*e)^(1/2)*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*x*a^2*b^2*d*e^3-352*B*x^2
*a*b^3*d*e^3*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)-105*B*ln(1/2*(2*b*x*e+2*(b*e*x^2+a*e*x+b*d*x+a*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*x*e+2*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2
)+a*e+b*d)/(b*e)^(1/2))*b^5*d^5-120*B*(b*e)^(1/2)*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*a*b^3*d^3*e+692*B*(b*e)^(1/2
)*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*a^2*b^2*d^2*e^2+1100*A*(b*e)^(1/2)*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*a^2*b^2*d
*e^3+210*B*(b*e)^(1/2)*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*a^4*e^4+90*B*(b*e)^(1/2)*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2
)*b^4*d^4+150*A*ln(1/2*(2*b*x*e+2*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*b^5*d^4*e-
600*A*ln(1/2*(2*b*x*e+2*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a*b^4*d^3*e^2)/e^2/(
b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)/b^4/(b*e)^(1/2)

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

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

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Fricas [A]  time = 1.47378, size = 2318, normalized size = 7.62 \begin{align*} \text{result too large to display} \end{align*}

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)*log(8*b^2*e^
2*x^2 + b^2*d^2 + 6*a*b*d*e + a^2*e^2 - 4*(2*b*e*x + b*d + a*e)*sqrt(b*e)*sqrt(b*x + a)*sqrt(e*x + d) + 8*(b^2
*d*e + a*b*e^2)*x) - 4*(384*B*b^5*e^5*x^4 - 45*B*b^5*d^4*e + 30*(2*B*a*b^4 + 5*A*b^5)*d^3*e^2 - 2*(173*B*a^2*b
^3 - 365*A*a*b^4)*d^2*e^3 + 10*(34*B*a^3*b^2 - 55*A*a^2*b^3)*d*e^4 - 15*(7*B*a^4*b - 10*A*a^3*b^2)*e^5 + 48*(2
1*B*b^5*d*e^4 + (B*a*b^4 + 10*A*b^5)*e^5)*x^3 + 8*(93*B*b^5*d^2*e^3 + 2*(11*B*a*b^4 + 85*A*b^5)*d*e^4 - (7*B*a
^2*b^3 - 10*A*a*b^4)*e^5)*x^2 + 2*(15*B*b^5*d^3*e^2 + (109*B*a*b^4 + 590*A*b^5)*d^2*e^3 - 3*(37*B*a^2*b^3 - 60
*A*a*b^4)*d*e^4 + 5*(7*B*a^3*b^2 - 10*A*a^2*b^3)*e^5)*x)*sqrt(b*x + a)*sqrt(e*x + d))/(b^5*e^3), -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*(2*b*e*x + b*d
+ a*e)*sqrt(-b*e)*sqrt(b*x + a)*sqrt(e*x + d)/(b^2*e^2*x^2 + a*b*d*e + (b^2*d*e + a*b*e^2)*x)) - 2*(384*B*b^5
*e^5*x^4 - 45*B*b^5*d^4*e + 30*(2*B*a*b^4 + 5*A*b^5)*d^3*e^2 - 2*(173*B*a^2*b^3 - 365*A*a*b^4)*d^2*e^3 + 10*(3
4*B*a^3*b^2 - 55*A*a^2*b^3)*d*e^4 - 15*(7*B*a^4*b - 10*A*a^3*b^2)*e^5 + 48*(21*B*b^5*d*e^4 + (B*a*b^4 + 10*A*b
^5)*e^5)*x^3 + 8*(93*B*b^5*d^2*e^3 + 2*(11*B*a*b^4 + 85*A*b^5)*d*e^4 - (7*B*a^2*b^3 - 10*A*a*b^4)*e^5)*x^2 + 2
*(15*B*b^5*d^3*e^2 + (109*B*a*b^4 + 590*A*b^5)*d^2*e^3 - 3*(37*B*a^2*b^3 - 60*A*a*b^4)*d*e^4 + 5*(7*B*a^3*b^2
- 10*A*a^2*b^3)*e^5)*x)*sqrt(b*x + a)*sqrt(e*x + d))/(b^5*e^3)]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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]  time = 2.53115, size = 1901, normalized size = 6.25 \begin{align*} \text{result too large to display} \end{align*}

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*(20*(sqrt(b^2*d + (b*x + a)*b*e - a*b*e)*sqrt(b*x + a)*(2*(b*x + a)*e^(-2)/b^4 + (b*d*e - a*e^2)*e^(-4)
/b^4) + (b^2*d^2 - 2*a*b*d*e + a^2*e^2)*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^(7/2))*A*d^2*abs(b)/b^2 + 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^2 + (b^7*d*e^5 - 17*a*b^6*e^6)*e^(-6)/b^8) - (5*b^8*d^2*e^4 + 6*a*b^7*d*e^5 - 59*a^2*b^6*e^6)
*e^(-6)/b^8) + 3*(5*b^9*d^3*e^3 + a*b^8*d^2*e^4 - a^2*b^7*d*e^5 - 5*a^3*b^6*e^6)*e^(-6)/b^8)*sqrt(b*x + a) + 3
*(5*b^4*d^4 - 4*a*b^3*d^3*e - 2*a^2*b^2*d^2*e^2 - 4*a^3*b*d*e^3 + 5*a^4*e^4)*e^(-7/2)*log(abs(-sqrt(b*x + a)*s
qrt(b)*e^(1/2) + sqrt(b^2*d + (b*x + a)*b*e - a*b*e)))/b^(3/2))*B*d*abs(b)*e/b^2 + (sqrt(b^2*d + (b*x + a)*b*e
- a*b*e)*sqrt(b*x + a)*(2*(b*x + a)*(4*(b*x + a)*e^(-2)/b^6 + (b*d*e^3 - 7*a*e^4)*e^(-6)/b^6) - 3*(b^2*d^2*e^
2 - a^2*e^4)*e^(-6)/b^6) - 3*(b^3*d^3 - a*b^2*d^2*e - a^2*b*d*e^2 + a^3*e^3)*e^(-9/2)*log(abs(-sqrt(b*x + a)*s
qrt(b)*e^(1/2) + sqrt(b^2*d + (b*x + a)*b*e - a*b*e)))/b^(11/2))*B*d^2*abs(b)/b^3 + 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^2 + (b^7*d*e^5 - 17*a*b^6*e^6)*e^(-6)/b^8) - (5*b^8*d^2
*e^4 + 6*a*b^7*d*e^5 - 59*a^2*b^6*e^6)*e^(-6)/b^8) + 3*(5*b^9*d^3*e^3 + a*b^8*d^2*e^4 - a^2*b^7*d*e^5 - 5*a^3*
b^6*e^6)*e^(-6)/b^8)*sqrt(b*x + a) + 3*(5*b^4*d^4 - 4*a*b^3*d^3*e - 2*a^2*b^2*d^2*e^2 - 4*a^3*b*d*e^3 + 5*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^(3/2))*A*abs(b)
*e^2/b^2 + (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^3 + (b^13*d*e^7 -
31*a*b^12*e^8)*e^(-8)/b^15) - (7*b^14*d^2*e^6 + 16*a*b^13*d*e^7 - 263*a^2*b^12*e^8)*e^(-8)/b^15) + 5*(7*b^15*d
^3*e^5 + 9*a*b^14*d^2*e^6 + 9*a^2*b^13*d*e^7 - 121*a^3*b^12*e^8)*e^(-8)/b^15)*(b*x + a) - 15*(7*b^16*d^4*e^4 +
2*a*b^15*d^3*e^5 - 2*a^3*b^13*d*e^7 - 7*a^4*b^12*e^8)*e^(-8)/b^15)*sqrt(b*x + a) - 15*(7*b^5*d^5 - 5*a*b^4*d^
4*e - 2*a^2*b^3*d^3*e^2 - 2*a^3*b^2*d^2*e^3 - 5*a^4*b*d*e^4 + 7*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^(5/2))*B*abs(b)*e^2/b^2 + 2*(sqrt(b^2*d + (b*x + a)*b*e -
a*b*e)*sqrt(b*x + a)*(2*(b*x + a)*(4*(b*x + a)*e^(-2)/b^6 + (b*d*e^3 - 7*a*e^4)*e^(-6)/b^6) - 3*(b^2*d^2*e^2
- a^2*e^4)*e^(-6)/b^6) - 3*(b^3*d^3 - a*b^2*d^2*e - a^2*b*d*e^2 + a^3*e^3)*e^(-9/2)*log(abs(-sqrt(b*x + a)*sqr
t(b)*e^(1/2) + sqrt(b^2*d + (b*x + a)*b*e - a*b*e)))/b^(11/2))*A*d*abs(b)*e/b^3)/b