3.2212 \(\int (a+b x)^{3/2} (A+B x) (d+e x)^{3/2} \, dx\)

Optimal. Leaf size=304 \[ -\frac{3 \sqrt{a+b x} \sqrt{d+e x} (b d-a e)^3 (2 A b e-B (a e+b d))}{128 b^3 e^3}+\frac{(a+b x)^{3/2} \sqrt{d+e x} (b d-a e)^2 (2 A b e-B (a e+b d))}{64 b^3 e^2}+\frac{3 (b d-a e)^4 (2 A b e-B (a e+b d)) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{128 b^{7/2} e^{7/2}}+\frac{(a+b x)^{5/2} \sqrt{d+e x} (b d-a e) (2 A b e-B (a e+b d))}{16 b^3 e}+\frac{(a+b x)^{5/2} (d+e x)^{3/2} (2 A b e-B (a e+b d))}{8 b^2 e}+\frac{B (a+b x)^{5/2} (d+e x)^{5/2}}{5 b e} \]

[Out]

(-3*(b*d - a*e)^3*(2*A*b*e - B*(b*d + a*e))*Sqrt[a + b*x]*Sqrt[d + e*x])/(128*b^3*e^3) + ((b*d - a*e)^2*(2*A*b
*e - B*(b*d + a*e))*(a + b*x)^(3/2)*Sqrt[d + e*x])/(64*b^3*e^2) + ((b*d - a*e)*(2*A*b*e - B*(b*d + a*e))*(a +
b*x)^(5/2)*Sqrt[d + e*x])/(16*b^3*e) + ((2*A*b*e - B*(b*d + a*e))*(a + b*x)^(5/2)*(d + e*x)^(3/2))/(8*b^2*e) +
(B*(a + b*x)^(5/2)*(d + e*x)^(5/2))/(5*b*e) + (3*(b*d - a*e)^4*(2*A*b*e - B*(b*d + a*e))*ArcTanh[(Sqrt[e]*Sqr
t[a + b*x])/(Sqrt[b]*Sqrt[d + e*x])])/(128*b^(7/2)*e^(7/2))

________________________________________________________________________________________

Rubi [A]  time = 0.241204, 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} \[ -\frac{3 \sqrt{a+b x} \sqrt{d+e x} (b d-a e)^3 (2 A b e-B (a e+b d))}{128 b^3 e^3}+\frac{(a+b x)^{3/2} \sqrt{d+e x} (b d-a e)^2 (2 A b e-B (a e+b d))}{64 b^3 e^2}+\frac{3 (b d-a e)^4 (2 A b e-B (a e+b d)) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{128 b^{7/2} e^{7/2}}+\frac{(a+b x)^{5/2} \sqrt{d+e x} (b d-a e) (2 A b e-B (a e+b d))}{16 b^3 e}+\frac{(a+b x)^{5/2} (d+e x)^{3/2} (2 A b e-B (a e+b d))}{8 b^2 e}+\frac{B (a+b x)^{5/2} (d+e x)^{5/2}}{5 b e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*(b*d - a*e)^3*(2*A*b*e - B*(b*d + a*e))*Sqrt[a + b*x]*Sqrt[d + e*x])/(128*b^3*e^3) + ((b*d - a*e)^2*(2*A*b
*e - B*(b*d + a*e))*(a + b*x)^(3/2)*Sqrt[d + e*x])/(64*b^3*e^2) + ((b*d - a*e)*(2*A*b*e - B*(b*d + a*e))*(a +
b*x)^(5/2)*Sqrt[d + e*x])/(16*b^3*e) + ((2*A*b*e - B*(b*d + a*e))*(a + b*x)^(5/2)*(d + e*x)^(3/2))/(8*b^2*e) +
(B*(a + b*x)^(5/2)*(d + e*x)^(5/2))/(5*b*e) + (3*(b*d - a*e)^4*(2*A*b*e - B*(b*d + a*e))*ArcTanh[(Sqrt[e]*Sqr
t[a + b*x])/(Sqrt[b]*Sqrt[d + e*x])])/(128*b^(7/2)*e^(7/2))

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 (a+b x)^{3/2} (A+B x) (d+e x)^{3/2} \, dx &=\frac{B (a+b x)^{5/2} (d+e x)^{5/2}}{5 b e}+\frac{\left (5 A b e-B \left (\frac{5 b d}{2}+\frac{5 a e}{2}\right )\right ) \int (a+b x)^{3/2} (d+e x)^{3/2} \, dx}{5 b e}\\ &=\frac{(2 A b e-B (b d+a e)) (a+b x)^{5/2} (d+e x)^{3/2}}{8 b^2 e}+\frac{B (a+b x)^{5/2} (d+e x)^{5/2}}{5 b e}+\frac{\left (3 (b d-a e) \left (5 A b e-B \left (\frac{5 b d}{2}+\frac{5 a e}{2}\right )\right )\right ) \int (a+b x)^{3/2} \sqrt{d+e x} \, dx}{40 b^2 e}\\ &=\frac{(b d-a e) (2 A b e-B (b d+a e)) (a+b x)^{5/2} \sqrt{d+e x}}{16 b^3 e}+\frac{(2 A b e-B (b d+a e)) (a+b x)^{5/2} (d+e x)^{3/2}}{8 b^2 e}+\frac{B (a+b x)^{5/2} (d+e x)^{5/2}}{5 b e}+\frac{\left ((b d-a e)^2 \left (5 A b e-B \left (\frac{5 b d}{2}+\frac{5 a e}{2}\right )\right )\right ) \int \frac{(a+b x)^{3/2}}{\sqrt{d+e x}} \, dx}{80 b^3 e}\\ &=\frac{(b d-a e)^2 (2 A b e-B (b d+a e)) (a+b x)^{3/2} \sqrt{d+e x}}{64 b^3 e^2}+\frac{(b d-a e) (2 A b e-B (b d+a e)) (a+b x)^{5/2} \sqrt{d+e x}}{16 b^3 e}+\frac{(2 A b e-B (b d+a e)) (a+b x)^{5/2} (d+e x)^{3/2}}{8 b^2 e}+\frac{B (a+b x)^{5/2} (d+e x)^{5/2}}{5 b e}-\frac{\left (3 (b d-a e)^3 \left (5 A b e-B \left (\frac{5 b d}{2}+\frac{5 a e}{2}\right )\right )\right ) \int \frac{\sqrt{a+b x}}{\sqrt{d+e x}} \, dx}{320 b^3 e^2}\\ &=-\frac{3 (b d-a e)^3 (2 A b e-B (b d+a e)) \sqrt{a+b x} \sqrt{d+e x}}{128 b^3 e^3}+\frac{(b d-a e)^2 (2 A b e-B (b d+a e)) (a+b x)^{3/2} \sqrt{d+e x}}{64 b^3 e^2}+\frac{(b d-a e) (2 A b e-B (b d+a e)) (a+b x)^{5/2} \sqrt{d+e x}}{16 b^3 e}+\frac{(2 A b e-B (b d+a e)) (a+b x)^{5/2} (d+e x)^{3/2}}{8 b^2 e}+\frac{B (a+b x)^{5/2} (d+e x)^{5/2}}{5 b e}+\frac{\left (3 (b d-a e)^4 \left (5 A b e-B \left (\frac{5 b d}{2}+\frac{5 a e}{2}\right )\right )\right ) \int \frac{1}{\sqrt{a+b x} \sqrt{d+e x}} \, dx}{640 b^3 e^3}\\ &=-\frac{3 (b d-a e)^3 (2 A b e-B (b d+a e)) \sqrt{a+b x} \sqrt{d+e x}}{128 b^3 e^3}+\frac{(b d-a e)^2 (2 A b e-B (b d+a e)) (a+b x)^{3/2} \sqrt{d+e x}}{64 b^3 e^2}+\frac{(b d-a e) (2 A b e-B (b d+a e)) (a+b x)^{5/2} \sqrt{d+e x}}{16 b^3 e}+\frac{(2 A b e-B (b d+a e)) (a+b x)^{5/2} (d+e x)^{3/2}}{8 b^2 e}+\frac{B (a+b x)^{5/2} (d+e x)^{5/2}}{5 b e}+\frac{\left (3 (b d-a e)^4 \left (5 A b e-B \left (\frac{5 b d}{2}+\frac{5 a e}{2}\right )\right )\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 )}{320 b^4 e^3}\\ &=-\frac{3 (b d-a e)^3 (2 A b e-B (b d+a e)) \sqrt{a+b x} \sqrt{d+e x}}{128 b^3 e^3}+\frac{(b d-a e)^2 (2 A b e-B (b d+a e)) (a+b x)^{3/2} \sqrt{d+e x}}{64 b^3 e^2}+\frac{(b d-a e) (2 A b e-B (b d+a e)) (a+b x)^{5/2} \sqrt{d+e x}}{16 b^3 e}+\frac{(2 A b e-B (b d+a e)) (a+b x)^{5/2} (d+e x)^{3/2}}{8 b^2 e}+\frac{B (a+b x)^{5/2} (d+e x)^{5/2}}{5 b e}+\frac{\left (3 (b d-a e)^4 \left (5 A b e-B \left (\frac{5 b d}{2}+\frac{5 a e}{2}\right )\right )\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 )}{320 b^4 e^3}\\ &=-\frac{3 (b d-a e)^3 (2 A b e-B (b d+a e)) \sqrt{a+b x} \sqrt{d+e x}}{128 b^3 e^3}+\frac{(b d-a e)^2 (2 A b e-B (b d+a e)) (a+b x)^{3/2} \sqrt{d+e x}}{64 b^3 e^2}+\frac{(b d-a e) (2 A b e-B (b d+a e)) (a+b x)^{5/2} \sqrt{d+e x}}{16 b^3 e}+\frac{(2 A b e-B (b d+a e)) (a+b x)^{5/2} (d+e x)^{3/2}}{8 b^2 e}+\frac{B (a+b x)^{5/2} (d+e x)^{5/2}}{5 b e}+\frac{3 (b d-a e)^4 (2 A b e-B (b d+a e)) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{128 b^{7/2} e^{7/2}}\\ \end{align*}

Mathematica [A]  time = 2.80261, size = 313, normalized size = 1.03 \[ \frac{128 b^9 B e^3 (a+b x)^3 (d+e x)^4-5 \sqrt{b d-a e} \left (\frac{b (d+e x)}{b d-a e}\right )^{3/2} (a B e-2 A b e+b B d) \left (8 b^5 e^3 (a+b x)^3 (b d-a e)^{3/2} \sqrt{\frac{b (d+e x)}{b d-a e}} (-a e+3 b d+2 b e x)+2 b^5 e^2 (a+b x)^2 (b d-a e)^{7/2} \sqrt{\frac{b (d+e x)}{b d-a e}}-3 b^5 e (a+b x) (b d-a e)^{9/2} \sqrt{\frac{b (d+e x)}{b d-a e}}+3 b^5 \sqrt{e} \sqrt{a+b x} (b d-a e)^5 \sinh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b d-a e}}\right )\right )}{640 b^{10} e^4 \sqrt{a+b x} (d+e x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(128*b^9*B*e^3*(a + b*x)^3*(d + e*x)^4 - 5*Sqrt[b*d - a*e]*(b*B*d - 2*A*b*e + a*B*e)*((b*(d + e*x))/(b*d - a*e
))^(3/2)*(-3*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)^(7/2)*(
a + b*x)^2*Sqrt[(b*(d + e*x))/(b*d - a*e)] + 8*b^5*e^3*(b*d - a*e)^(3/2)*(a + b*x)^3*Sqrt[(b*(d + e*x))/(b*d -
a*e)]*(3*b*d - a*e + 2*b*e*x) + 3*b^5*Sqrt[e]*(b*d - a*e)^5*Sqrt[a + b*x]*ArcSinh[(Sqrt[e]*Sqrt[a + b*x])/Sqr
t[b*d - a*e]]))/(640*b^10*e^4*Sqrt[a + b*x]*(d + e*x)^(3/2))

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Maple [B]  time = 0.016, 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)^(3/2)*(B*x+A)*(e*x+d)^(3/2),x)

[Out]

1/1280*(b*x+a)^(1/2)*(e*x+d)^(1/2)*(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))*a*b^4*d^4*e+352*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)+220*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+256*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)+320
*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)-60*A*(b*e)^(1/2)*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*a^
3*b*e^4-60*A*(b*e)^(1/2)*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*b^4*d^3*e+30*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^5-30*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+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))*a^4*b*d*e^4-30*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+352*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)-20*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-80*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+16*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)+480*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)+480*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)+16*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)-120*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+180*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+40*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+40*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^2-20*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+52
*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+880*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+52*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+544*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)-15*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^5-15*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^5-80*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+36*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+220*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+30*B*(b*e)^(1/2)*(
b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*a^4*e^4+30*B*(b*e)^(1/2)*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*b^4*d^4+30*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-120*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)/b^3/e^3/(b*e*x^2+a*e*x+b*d*x+a
*d)^(1/2)/(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)^(3/2)*(B*x+A)*(e*x+d)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.92067, size = 2248, normalized size = 7.39 \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)^(3/2)*(B*x+A)*(e*x+d)^(3/2),x, algorithm="fricas")

[Out]

[1/2560*(15*(B*b^5*d^5 - (3*B*a*b^4 + 2*A*b^5)*d^4*e + 2*(B*a^2*b^3 + 4*A*a*b^4)*d^3*e^2 + 2*(B*a^3*b^2 - 6*A*
a^2*b^3)*d^2*e^3 - (3*B*a^4*b - 8*A*a^3*b^2)*d*e^4 + (B*a^5 - 2*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*(128*B*b^5*e^5*x^4 + 15*B*b^5*d^4*e - 10*(4*B*a*b^4 + 3*A*b^5)*d^3*e^2 + 2*(9*B*a^2*b^3 + 55*A*a*
b^4)*d^2*e^3 - 10*(4*B*a^3*b^2 - 11*A*a^2*b^3)*d*e^4 + 15*(B*a^4*b - 2*A*a^3*b^2)*e^5 + 16*(11*B*b^5*d*e^4 + (
11*B*a*b^4 + 10*A*b^5)*e^5)*x^3 + 8*(B*b^5*d^2*e^3 + 2*(17*B*a*b^4 + 15*A*b^5)*d*e^4 + (B*a^2*b^3 + 30*A*a*b^4
)*e^5)*x^2 - 2*(5*B*b^5*d^3*e^2 - (13*B*a*b^4 + 10*A*b^5)*d^2*e^3 - (13*B*a^2*b^3 + 220*A*a*b^4)*d*e^4 + 5*(B*
a^3*b^2 - 2*A*a^2*b^3)*e^5)*x)*sqrt(b*x + a)*sqrt(e*x + d))/(b^4*e^4), 1/1280*(15*(B*b^5*d^5 - (3*B*a*b^4 + 2*
A*b^5)*d^4*e + 2*(B*a^2*b^3 + 4*A*a*b^4)*d^3*e^2 + 2*(B*a^3*b^2 - 6*A*a^2*b^3)*d^2*e^3 - (3*B*a^4*b - 8*A*a^3*
b^2)*d*e^4 + (B*a^5 - 2*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)*sqr
t(e*x + d)/(b^2*e^2*x^2 + a*b*d*e + (b^2*d*e + a*b*e^2)*x)) + 2*(128*B*b^5*e^5*x^4 + 15*B*b^5*d^4*e - 10*(4*B*
a*b^4 + 3*A*b^5)*d^3*e^2 + 2*(9*B*a^2*b^3 + 55*A*a*b^4)*d^2*e^3 - 10*(4*B*a^3*b^2 - 11*A*a^2*b^3)*d*e^4 + 15*(
B*a^4*b - 2*A*a^3*b^2)*e^5 + 16*(11*B*b^5*d*e^4 + (11*B*a*b^4 + 10*A*b^5)*e^5)*x^3 + 8*(B*b^5*d^2*e^3 + 2*(17*
B*a*b^4 + 15*A*b^5)*d*e^4 + (B*a^2*b^3 + 30*A*a*b^4)*e^5)*x^2 - 2*(5*B*b^5*d^3*e^2 - (13*B*a*b^4 + 10*A*b^5)*d
^2*e^3 - (13*B*a^2*b^3 + 220*A*a*b^4)*d*e^4 + 5*(B*a^3*b^2 - 2*A*a^2*b^3)*e^5)*x)*sqrt(b*x + a)*sqrt(e*x + d))
/(b^4*e^4)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (A + B x\right ) \left (a + b x\right )^{\frac{3}{2}} \left (d + e x\right )^{\frac{3}{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**(3/2)*(B*x+A)*(e*x+d)**(3/2),x)

[Out]

Integral((A + B*x)*(a + b*x)**(3/2)*(d + e*x)**(3/2), x)

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Giac [B]  time = 2.52015, size = 2511, normalized size = 8.26 \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)^(3/2)*(B*x+A)*(e*x+d)^(3/2),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*a*d*abs(b)/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^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)/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^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))*B*a*abs(b)*e/
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^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/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^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/b + (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)*sqrt(b)*e^(1/2) + sqrt(b^2*d + (b*x + a)*b*e - a*b*e)))/b^(11/2))*B*a*d*abs(b)/b^3 + (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)*sqrt(b)*e^(1/2) + sqrt(b^2*d + (b*x + a)*b*e - a*b*e)))/b^(11/2))*A*d*abs(b)/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)*sqrt(b)*e^(1/2) + sqrt(b^2*d + (b*x + a)*b*e - a*b*e)))/b^(11/2))*A*a*abs(b)*e/b^3)/b