3.656 \(\int x^2 (a+b x)^{5/2} (c+d x)^{3/2} \, dx\)

Optimal. Leaf size=437 \[ \frac{\sqrt{a+b x} \sqrt{c+d x} \left (5 a^2 d^2+10 a b c d+9 b^2 c^2\right ) (b c-a d)^4}{1024 b^4 d^5}-\frac{(a+b x)^{3/2} \sqrt{c+d x} \left (5 a^2 d^2+10 a b c d+9 b^2 c^2\right ) (b c-a d)^3}{1536 b^4 d^4}+\frac{(a+b x)^{5/2} \sqrt{c+d x} \left (5 a^2 d^2+10 a b c d+9 b^2 c^2\right ) (b c-a d)^2}{1920 b^4 d^3}+\frac{(a+b x)^{7/2} \sqrt{c+d x} \left (5 a^2 d^2+10 a b c d+9 b^2 c^2\right ) (b c-a d)}{320 b^4 d^2}+\frac{(a+b x)^{7/2} (c+d x)^{3/2} \left (5 a^2 d^2+10 a b c d+9 b^2 c^2\right )}{120 b^3 d^2}-\frac{\left (5 a^2 d^2+10 a b c d+9 b^2 c^2\right ) (b c-a d)^5 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{1024 b^{9/2} d^{11/2}}-\frac{(a+b x)^{7/2} (c+d x)^{5/2} (7 a d+9 b c)}{84 b^2 d^2}+\frac{x (a+b x)^{7/2} (c+d x)^{5/2}}{7 b d} \]

[Out]

((b*c - a*d)^4*(9*b^2*c^2 + 10*a*b*c*d + 5*a^2*d^2)*Sqrt[a + b*x]*Sqrt[c + d*x])/(1024*b^4*d^5) - ((b*c - a*d)
^3*(9*b^2*c^2 + 10*a*b*c*d + 5*a^2*d^2)*(a + b*x)^(3/2)*Sqrt[c + d*x])/(1536*b^4*d^4) + ((b*c - a*d)^2*(9*b^2*
c^2 + 10*a*b*c*d + 5*a^2*d^2)*(a + b*x)^(5/2)*Sqrt[c + d*x])/(1920*b^4*d^3) + ((b*c - a*d)*(9*b^2*c^2 + 10*a*b
*c*d + 5*a^2*d^2)*(a + b*x)^(7/2)*Sqrt[c + d*x])/(320*b^4*d^2) + ((9*b^2*c^2 + 10*a*b*c*d + 5*a^2*d^2)*(a + b*
x)^(7/2)*(c + d*x)^(3/2))/(120*b^3*d^2) - ((9*b*c + 7*a*d)*(a + b*x)^(7/2)*(c + d*x)^(5/2))/(84*b^2*d^2) + (x*
(a + b*x)^(7/2)*(c + d*x)^(5/2))/(7*b*d) - ((b*c - a*d)^5*(9*b^2*c^2 + 10*a*b*c*d + 5*a^2*d^2)*ArcTanh[(Sqrt[d
]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(1024*b^(9/2)*d^(11/2))

________________________________________________________________________________________

Rubi [A]  time = 0.4419, antiderivative size = 437, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {90, 80, 50, 63, 217, 206} \[ \frac{\sqrt{a+b x} \sqrt{c+d x} \left (5 a^2 d^2+10 a b c d+9 b^2 c^2\right ) (b c-a d)^4}{1024 b^4 d^5}-\frac{(a+b x)^{3/2} \sqrt{c+d x} \left (5 a^2 d^2+10 a b c d+9 b^2 c^2\right ) (b c-a d)^3}{1536 b^4 d^4}+\frac{(a+b x)^{5/2} \sqrt{c+d x} \left (5 a^2 d^2+10 a b c d+9 b^2 c^2\right ) (b c-a d)^2}{1920 b^4 d^3}+\frac{(a+b x)^{7/2} \sqrt{c+d x} \left (5 a^2 d^2+10 a b c d+9 b^2 c^2\right ) (b c-a d)}{320 b^4 d^2}+\frac{(a+b x)^{7/2} (c+d x)^{3/2} \left (5 a^2 d^2+10 a b c d+9 b^2 c^2\right )}{120 b^3 d^2}-\frac{\left (5 a^2 d^2+10 a b c d+9 b^2 c^2\right ) (b c-a d)^5 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{1024 b^{9/2} d^{11/2}}-\frac{(a+b x)^{7/2} (c+d x)^{5/2} (7 a d+9 b c)}{84 b^2 d^2}+\frac{x (a+b x)^{7/2} (c+d x)^{5/2}}{7 b d} \]

Antiderivative was successfully verified.

[In]

Int[x^2*(a + b*x)^(5/2)*(c + d*x)^(3/2),x]

[Out]

((b*c - a*d)^4*(9*b^2*c^2 + 10*a*b*c*d + 5*a^2*d^2)*Sqrt[a + b*x]*Sqrt[c + d*x])/(1024*b^4*d^5) - ((b*c - a*d)
^3*(9*b^2*c^2 + 10*a*b*c*d + 5*a^2*d^2)*(a + b*x)^(3/2)*Sqrt[c + d*x])/(1536*b^4*d^4) + ((b*c - a*d)^2*(9*b^2*
c^2 + 10*a*b*c*d + 5*a^2*d^2)*(a + b*x)^(5/2)*Sqrt[c + d*x])/(1920*b^4*d^3) + ((b*c - a*d)*(9*b^2*c^2 + 10*a*b
*c*d + 5*a^2*d^2)*(a + b*x)^(7/2)*Sqrt[c + d*x])/(320*b^4*d^2) + ((9*b^2*c^2 + 10*a*b*c*d + 5*a^2*d^2)*(a + b*
x)^(7/2)*(c + d*x)^(3/2))/(120*b^3*d^2) - ((9*b*c + 7*a*d)*(a + b*x)^(7/2)*(c + d*x)^(5/2))/(84*b^2*d^2) + (x*
(a + b*x)^(7/2)*(c + d*x)^(5/2))/(7*b*d) - ((b*c - a*d)^5*(9*b^2*c^2 + 10*a*b*c*d + 5*a^2*d^2)*ArcTanh[(Sqrt[d
]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(1024*b^(9/2)*d^(11/2))

Rule 90

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a + b*
x)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 3)), x] + Dist[1/(d*f*(n + p + 3)), Int[(c + d*x)^n*(e +
f*x)^p*Simp[a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(n + p + 4) - b*(d*e*(
n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 3, 0]

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 x^2 (a+b x)^{5/2} (c+d x)^{3/2} \, dx &=\frac{x (a+b x)^{7/2} (c+d x)^{5/2}}{7 b d}+\frac{\int (a+b x)^{5/2} (c+d x)^{3/2} \left (-a c-\frac{1}{2} (9 b c+7 a d) x\right ) \, dx}{7 b d}\\ &=-\frac{(9 b c+7 a d) (a+b x)^{7/2} (c+d x)^{5/2}}{84 b^2 d^2}+\frac{x (a+b x)^{7/2} (c+d x)^{5/2}}{7 b d}+\frac{\left (9 b^2 c^2+10 a b c d+5 a^2 d^2\right ) \int (a+b x)^{5/2} (c+d x)^{3/2} \, dx}{24 b^2 d^2}\\ &=\frac{\left (9 b^2 c^2+10 a b c d+5 a^2 d^2\right ) (a+b x)^{7/2} (c+d x)^{3/2}}{120 b^3 d^2}-\frac{(9 b c+7 a d) (a+b x)^{7/2} (c+d x)^{5/2}}{84 b^2 d^2}+\frac{x (a+b x)^{7/2} (c+d x)^{5/2}}{7 b d}+\frac{\left ((b c-a d) \left (9 b^2 c^2+10 a b c d+5 a^2 d^2\right )\right ) \int (a+b x)^{5/2} \sqrt{c+d x} \, dx}{80 b^3 d^2}\\ &=\frac{(b c-a d) \left (9 b^2 c^2+10 a b c d+5 a^2 d^2\right ) (a+b x)^{7/2} \sqrt{c+d x}}{320 b^4 d^2}+\frac{\left (9 b^2 c^2+10 a b c d+5 a^2 d^2\right ) (a+b x)^{7/2} (c+d x)^{3/2}}{120 b^3 d^2}-\frac{(9 b c+7 a d) (a+b x)^{7/2} (c+d x)^{5/2}}{84 b^2 d^2}+\frac{x (a+b x)^{7/2} (c+d x)^{5/2}}{7 b d}+\frac{\left ((b c-a d)^2 \left (9 b^2 c^2+10 a b c d+5 a^2 d^2\right )\right ) \int \frac{(a+b x)^{5/2}}{\sqrt{c+d x}} \, dx}{640 b^4 d^2}\\ &=\frac{(b c-a d)^2 \left (9 b^2 c^2+10 a b c d+5 a^2 d^2\right ) (a+b x)^{5/2} \sqrt{c+d x}}{1920 b^4 d^3}+\frac{(b c-a d) \left (9 b^2 c^2+10 a b c d+5 a^2 d^2\right ) (a+b x)^{7/2} \sqrt{c+d x}}{320 b^4 d^2}+\frac{\left (9 b^2 c^2+10 a b c d+5 a^2 d^2\right ) (a+b x)^{7/2} (c+d x)^{3/2}}{120 b^3 d^2}-\frac{(9 b c+7 a d) (a+b x)^{7/2} (c+d x)^{5/2}}{84 b^2 d^2}+\frac{x (a+b x)^{7/2} (c+d x)^{5/2}}{7 b d}-\frac{\left ((b c-a d)^3 \left (9 b^2 c^2+10 a b c d+5 a^2 d^2\right )\right ) \int \frac{(a+b x)^{3/2}}{\sqrt{c+d x}} \, dx}{768 b^4 d^3}\\ &=-\frac{(b c-a d)^3 \left (9 b^2 c^2+10 a b c d+5 a^2 d^2\right ) (a+b x)^{3/2} \sqrt{c+d x}}{1536 b^4 d^4}+\frac{(b c-a d)^2 \left (9 b^2 c^2+10 a b c d+5 a^2 d^2\right ) (a+b x)^{5/2} \sqrt{c+d x}}{1920 b^4 d^3}+\frac{(b c-a d) \left (9 b^2 c^2+10 a b c d+5 a^2 d^2\right ) (a+b x)^{7/2} \sqrt{c+d x}}{320 b^4 d^2}+\frac{\left (9 b^2 c^2+10 a b c d+5 a^2 d^2\right ) (a+b x)^{7/2} (c+d x)^{3/2}}{120 b^3 d^2}-\frac{(9 b c+7 a d) (a+b x)^{7/2} (c+d x)^{5/2}}{84 b^2 d^2}+\frac{x (a+b x)^{7/2} (c+d x)^{5/2}}{7 b d}+\frac{\left ((b c-a d)^4 \left (9 b^2 c^2+10 a b c d+5 a^2 d^2\right )\right ) \int \frac{\sqrt{a+b x}}{\sqrt{c+d x}} \, dx}{1024 b^4 d^4}\\ &=\frac{(b c-a d)^4 \left (9 b^2 c^2+10 a b c d+5 a^2 d^2\right ) \sqrt{a+b x} \sqrt{c+d x}}{1024 b^4 d^5}-\frac{(b c-a d)^3 \left (9 b^2 c^2+10 a b c d+5 a^2 d^2\right ) (a+b x)^{3/2} \sqrt{c+d x}}{1536 b^4 d^4}+\frac{(b c-a d)^2 \left (9 b^2 c^2+10 a b c d+5 a^2 d^2\right ) (a+b x)^{5/2} \sqrt{c+d x}}{1920 b^4 d^3}+\frac{(b c-a d) \left (9 b^2 c^2+10 a b c d+5 a^2 d^2\right ) (a+b x)^{7/2} \sqrt{c+d x}}{320 b^4 d^2}+\frac{\left (9 b^2 c^2+10 a b c d+5 a^2 d^2\right ) (a+b x)^{7/2} (c+d x)^{3/2}}{120 b^3 d^2}-\frac{(9 b c+7 a d) (a+b x)^{7/2} (c+d x)^{5/2}}{84 b^2 d^2}+\frac{x (a+b x)^{7/2} (c+d x)^{5/2}}{7 b d}-\frac{\left ((b c-a d)^5 \left (9 b^2 c^2+10 a b c d+5 a^2 d^2\right )\right ) \int \frac{1}{\sqrt{a+b x} \sqrt{c+d x}} \, dx}{2048 b^4 d^5}\\ &=\frac{(b c-a d)^4 \left (9 b^2 c^2+10 a b c d+5 a^2 d^2\right ) \sqrt{a+b x} \sqrt{c+d x}}{1024 b^4 d^5}-\frac{(b c-a d)^3 \left (9 b^2 c^2+10 a b c d+5 a^2 d^2\right ) (a+b x)^{3/2} \sqrt{c+d x}}{1536 b^4 d^4}+\frac{(b c-a d)^2 \left (9 b^2 c^2+10 a b c d+5 a^2 d^2\right ) (a+b x)^{5/2} \sqrt{c+d x}}{1920 b^4 d^3}+\frac{(b c-a d) \left (9 b^2 c^2+10 a b c d+5 a^2 d^2\right ) (a+b x)^{7/2} \sqrt{c+d x}}{320 b^4 d^2}+\frac{\left (9 b^2 c^2+10 a b c d+5 a^2 d^2\right ) (a+b x)^{7/2} (c+d x)^{3/2}}{120 b^3 d^2}-\frac{(9 b c+7 a d) (a+b x)^{7/2} (c+d x)^{5/2}}{84 b^2 d^2}+\frac{x (a+b x)^{7/2} (c+d x)^{5/2}}{7 b d}-\frac{\left ((b c-a d)^5 \left (9 b^2 c^2+10 a b c d+5 a^2 d^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{c-\frac{a d}{b}+\frac{d x^2}{b}}} \, dx,x,\sqrt{a+b x}\right )}{1024 b^5 d^5}\\ &=\frac{(b c-a d)^4 \left (9 b^2 c^2+10 a b c d+5 a^2 d^2\right ) \sqrt{a+b x} \sqrt{c+d x}}{1024 b^4 d^5}-\frac{(b c-a d)^3 \left (9 b^2 c^2+10 a b c d+5 a^2 d^2\right ) (a+b x)^{3/2} \sqrt{c+d x}}{1536 b^4 d^4}+\frac{(b c-a d)^2 \left (9 b^2 c^2+10 a b c d+5 a^2 d^2\right ) (a+b x)^{5/2} \sqrt{c+d x}}{1920 b^4 d^3}+\frac{(b c-a d) \left (9 b^2 c^2+10 a b c d+5 a^2 d^2\right ) (a+b x)^{7/2} \sqrt{c+d x}}{320 b^4 d^2}+\frac{\left (9 b^2 c^2+10 a b c d+5 a^2 d^2\right ) (a+b x)^{7/2} (c+d x)^{3/2}}{120 b^3 d^2}-\frac{(9 b c+7 a d) (a+b x)^{7/2} (c+d x)^{5/2}}{84 b^2 d^2}+\frac{x (a+b x)^{7/2} (c+d x)^{5/2}}{7 b d}-\frac{\left ((b c-a d)^5 \left (9 b^2 c^2+10 a b c d+5 a^2 d^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{1-\frac{d x^2}{b}} \, dx,x,\frac{\sqrt{a+b x}}{\sqrt{c+d x}}\right )}{1024 b^5 d^5}\\ &=\frac{(b c-a d)^4 \left (9 b^2 c^2+10 a b c d+5 a^2 d^2\right ) \sqrt{a+b x} \sqrt{c+d x}}{1024 b^4 d^5}-\frac{(b c-a d)^3 \left (9 b^2 c^2+10 a b c d+5 a^2 d^2\right ) (a+b x)^{3/2} \sqrt{c+d x}}{1536 b^4 d^4}+\frac{(b c-a d)^2 \left (9 b^2 c^2+10 a b c d+5 a^2 d^2\right ) (a+b x)^{5/2} \sqrt{c+d x}}{1920 b^4 d^3}+\frac{(b c-a d) \left (9 b^2 c^2+10 a b c d+5 a^2 d^2\right ) (a+b x)^{7/2} \sqrt{c+d x}}{320 b^4 d^2}+\frac{\left (9 b^2 c^2+10 a b c d+5 a^2 d^2\right ) (a+b x)^{7/2} (c+d x)^{3/2}}{120 b^3 d^2}-\frac{(9 b c+7 a d) (a+b x)^{7/2} (c+d x)^{5/2}}{84 b^2 d^2}+\frac{x (a+b x)^{7/2} (c+d x)^{5/2}}{7 b d}-\frac{(b c-a d)^5 \left (9 b^2 c^2+10 a b c d+5 a^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{1024 b^{9/2} d^{11/2}}\\ \end{align*}

Mathematica [A]  time = 3.4767, size = 376, normalized size = 0.86 \[ \frac{(a+b x)^{7/2} (c+d x)^{5/2} \left (\frac{49 \sqrt{b c-a d} \left (5 a^2 d^2+10 a b c d+9 b^2 c^2\right ) \left (\frac{b (c+d x)}{b c-a d}\right )^{3/2} \left (-10 d^{3/2} (a+b x)^2 (b c-a d)^{9/2} \sqrt{\frac{b (c+d x)}{b c-a d}}+8 d^{5/2} (a+b x)^3 (b c-a d)^{7/2} \sqrt{\frac{b (c+d x)}{b c-a d}}+16 d^{7/2} (a+b x)^4 (b c-a d)^{3/2} \sqrt{\frac{b (c+d x)}{b c-a d}} (-3 a d+11 b c+8 b d x)+15 \sqrt{d} (a+b x) (b c-a d)^{11/2} \sqrt{\frac{b (c+d x)}{b c-a d}}-15 \sqrt{a+b x} (b c-a d)^6 \sinh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b c-a d}}\right )\right )}{1280 b^5 d^{9/2} (a+b x)^4 (c+d x)^4}-\frac{49 a}{b}-\frac{63 c}{d}+84 x\right )}{588 b d} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*(a + b*x)^(5/2)*(c + d*x)^(3/2),x]

[Out]

((a + b*x)^(7/2)*(c + d*x)^(5/2)*((-49*a)/b - (63*c)/d + 84*x + (49*Sqrt[b*c - a*d]*(9*b^2*c^2 + 10*a*b*c*d +
5*a^2*d^2)*((b*(c + d*x))/(b*c - a*d))^(3/2)*(15*Sqrt[d]*(b*c - a*d)^(11/2)*(a + b*x)*Sqrt[(b*(c + d*x))/(b*c
- a*d)] - 10*d^(3/2)*(b*c - a*d)^(9/2)*(a + b*x)^2*Sqrt[(b*(c + d*x))/(b*c - a*d)] + 8*d^(5/2)*(b*c - a*d)^(7/
2)*(a + b*x)^3*Sqrt[(b*(c + d*x))/(b*c - a*d)] + 16*d^(7/2)*(b*c - a*d)^(3/2)*(a + b*x)^4*Sqrt[(b*(c + d*x))/(
b*c - a*d)]*(11*b*c - 3*a*d + 8*b*d*x) - 15*(b*c - a*d)^6*Sqrt[a + b*x]*ArcSinh[(Sqrt[d]*Sqrt[a + b*x])/Sqrt[b
*c - a*d]]))/(1280*b^5*d^(9/2)*(a + b*x)^4*(c + d*x)^4)))/(588*b*d)

________________________________________________________________________________________

Maple [B]  time = 0.019, size = 1580, normalized size = 3.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(b*x+a)^(5/2)*(d*x+c)^(3/2),x)

[Out]

1/215040*(b*x+a)^(1/2)*(d*x+c)^(1/2)*(700*(b*d)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x*a^5*b*d^6-1050*(b*d)^(
1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^6*d^6+1890*(b*d)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*b^6*c^6+30720*x^
6*b^6*d^6*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)-1260*(b*d)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x*b^6*c
^5*d+2800*(b*d)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^5*b*c*d^5-1050*(b*d)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(
1/2)*a^4*b^2*c^2*d^4-1200*(b*d)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^3*b^3*c^3*d^3+7378*(b*d)^(1/2)*(b*d*x^
2+a*d*x+b*c*x+a*c)^(1/2)*a^2*b^4*c^4*d^2-6720*a*b^5*c^5*d*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)+74240*x^
5*a*b^5*d^6*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)+38400*x^5*b^6*c*d^5*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b
*d)^(1/2)+47360*x^4*a^2*b^4*d^6*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)+768*x^4*b^6*c^2*d^4*(b*d*x^2+a*d*x
+b*c*x+a*c)^(1/2)*(b*d)^(1/2)+480*x^3*a^3*b^3*d^6*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)-864*x^3*b^6*c^3*
d^3*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)-560*x^2*a^4*b^2*d^6*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2
)+1008*x^2*b^6*c^4*d^2*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)-1575*ln(1/2*(2*b*d*x+2*(b*d*x^2+a*d*x+b*c*x
+a*c)^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^6*b*c*d^6+945*ln(1/2*(2*b*d*x+2*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2
)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^5*b^2*c^2*d^5+525*ln(1/2*(2*b*d*x+2*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d
)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^4*b^3*c^3*d^4+1575*ln(1/2*(2*b*d*x+2*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/
2)+a*d+b*c)/(b*d)^(1/2))*a^3*b^4*c^4*d^3-4725*ln(1/2*(2*b*d*x+2*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)+a*
d+b*c)/(b*d)^(1/2))*a^2*b^5*c^5*d^2+3675*ln(1/2*(2*b*d*x+2*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)+a*d+b*c
)/(b*d)^(1/2))*a*b^6*c^6*d+525*ln(1/2*(2*b*d*x+2*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1
/2))*a^7*d^7-945*ln(1/2*(2*b*d*x+2*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*b^7*c^7-1
820*(b*d)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x*a^4*b^2*c*d^5+600*(b*d)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2
)*x*a^3*b^3*c^2*d^4-4664*(b*d)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x*a^2*b^4*c^3*d^3+4396*(b*d)^(1/2)*(b*d*x
^2+a*d*x+b*c*x+a*c)^(1/2)*x*a*b^5*c^4*d^2+97280*x^4*a*b^5*c*d^5*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)+67
040*x^3*a^2*b^4*c*d^5*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)+2976*x^3*a*b^5*c^2*d^4*(b*d*x^2+a*d*x+b*c*x+
a*c)^(1/2)*(b*d)^(1/2)+1440*x^2*a^3*b^3*c*d^5*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)+3648*x^2*a^2*b^4*c^2
*d^4*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)-3488*x^2*a*b^5*c^3*d^3*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^
(1/2))/b^4/d^5/(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)/(b*d)^(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(x^2*(b*x+a)^(5/2)*(d*x+c)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 3.025, size = 2484, normalized size = 5.68 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(b*x+a)^(5/2)*(d*x+c)^(3/2),x, algorithm="fricas")

[Out]

[-1/430080*(105*(9*b^7*c^7 - 35*a*b^6*c^6*d + 45*a^2*b^5*c^5*d^2 - 15*a^3*b^4*c^4*d^3 - 5*a^4*b^3*c^3*d^4 - 9*
a^5*b^2*c^2*d^5 + 15*a^6*b*c*d^6 - 5*a^7*d^7)*sqrt(b*d)*log(8*b^2*d^2*x^2 + b^2*c^2 + 6*a*b*c*d + a^2*d^2 + 4*
(2*b*d*x + b*c + a*d)*sqrt(b*d)*sqrt(b*x + a)*sqrt(d*x + c) + 8*(b^2*c*d + a*b*d^2)*x) - 4*(15360*b^7*d^7*x^6
+ 945*b^7*c^6*d - 3360*a*b^6*c^5*d^2 + 3689*a^2*b^5*c^4*d^3 - 600*a^3*b^4*c^3*d^4 - 525*a^4*b^3*c^2*d^5 + 1400
*a^5*b^2*c*d^6 - 525*a^6*b*d^7 + 1280*(15*b^7*c*d^6 + 29*a*b^6*d^7)*x^5 + 128*(3*b^7*c^2*d^5 + 380*a*b^6*c*d^6
+ 185*a^2*b^5*d^7)*x^4 - 16*(27*b^7*c^3*d^4 - 93*a*b^6*c^2*d^5 - 2095*a^2*b^5*c*d^6 - 15*a^3*b^4*d^7)*x^3 + 8
*(63*b^7*c^4*d^3 - 218*a*b^6*c^3*d^4 + 228*a^2*b^5*c^2*d^5 + 90*a^3*b^4*c*d^6 - 35*a^4*b^3*d^7)*x^2 - 2*(315*b
^7*c^5*d^2 - 1099*a*b^6*c^4*d^3 + 1166*a^2*b^5*c^3*d^4 - 150*a^3*b^4*c^2*d^5 + 455*a^4*b^3*c*d^6 - 175*a^5*b^2
*d^7)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(b^5*d^6), 1/215040*(105*(9*b^7*c^7 - 35*a*b^6*c^6*d + 45*a^2*b^5*c^5*d^
2 - 15*a^3*b^4*c^4*d^3 - 5*a^4*b^3*c^3*d^4 - 9*a^5*b^2*c^2*d^5 + 15*a^6*b*c*d^6 - 5*a^7*d^7)*sqrt(-b*d)*arctan
(1/2*(2*b*d*x + b*c + a*d)*sqrt(-b*d)*sqrt(b*x + a)*sqrt(d*x + c)/(b^2*d^2*x^2 + a*b*c*d + (b^2*c*d + a*b*d^2)
*x)) + 2*(15360*b^7*d^7*x^6 + 945*b^7*c^6*d - 3360*a*b^6*c^5*d^2 + 3689*a^2*b^5*c^4*d^3 - 600*a^3*b^4*c^3*d^4
- 525*a^4*b^3*c^2*d^5 + 1400*a^5*b^2*c*d^6 - 525*a^6*b*d^7 + 1280*(15*b^7*c*d^6 + 29*a*b^6*d^7)*x^5 + 128*(3*b
^7*c^2*d^5 + 380*a*b^6*c*d^6 + 185*a^2*b^5*d^7)*x^4 - 16*(27*b^7*c^3*d^4 - 93*a*b^6*c^2*d^5 - 2095*a^2*b^5*c*d
^6 - 15*a^3*b^4*d^7)*x^3 + 8*(63*b^7*c^4*d^3 - 218*a*b^6*c^3*d^4 + 228*a^2*b^5*c^2*d^5 + 90*a^3*b^4*c*d^6 - 35
*a^4*b^3*d^7)*x^2 - 2*(315*b^7*c^5*d^2 - 1099*a*b^6*c^4*d^3 + 1166*a^2*b^5*c^3*d^4 - 150*a^3*b^4*c^2*d^5 + 455
*a^4*b^3*c*d^6 - 175*a^5*b^2*d^7)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(b^5*d^6)]

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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(x**2*(b*x+a)**(5/2)*(d*x+c)**(3/2),x)

[Out]

Timed out

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Giac [B]  time = 1.91232, size = 3482, normalized size = 7.97 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(b*x+a)^(5/2)*(d*x+c)^(3/2),x, algorithm="giac")

[Out]

1/107520*(14*(sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*(2*(4*(2*(b*x + a)*(8*(b*x + a)*(10*(b*x + a)/b^4 + (b^21*c*
d^9 - 49*a*b^20*d^10)/(b^24*d^10)) - 3*(3*b^22*c^2*d^8 + 10*a*b^21*c*d^9 - 253*a^2*b^20*d^10)/(b^24*d^10)) + (
21*b^23*c^3*d^7 + 49*a*b^22*c^2*d^8 + 79*a^2*b^21*c*d^9 - 1429*a^3*b^20*d^10)/(b^24*d^10))*(b*x + a) - 5*(21*b
^24*c^4*d^6 + 28*a*b^23*c^3*d^7 + 30*a^2*b^22*c^2*d^8 + 28*a^3*b^21*c*d^9 - 491*a^4*b^20*d^10)/(b^24*d^10))*(b
*x + a) + 15*(21*b^25*c^5*d^5 + 7*a*b^24*c^4*d^6 + 2*a^2*b^23*c^3*d^7 - 2*a^3*b^22*c^2*d^8 - 7*a^4*b^21*c*d^9
- 21*a^5*b^20*d^10)/(b^24*d^10))*sqrt(b*x + a) + 15*(21*b^6*c^6 - 14*a*b^5*c^5*d - 5*a^2*b^4*c^4*d^2 - 4*a^3*b
^3*c^3*d^3 - 5*a^4*b^2*c^2*d^4 - 14*a^5*b*c*d^5 + 21*a^6*d^6)*log(abs(-sqrt(b*d)*sqrt(b*x + a) + sqrt(b^2*c +
(b*x + a)*b*d - a*b*d)))/(sqrt(b*d)*b^3*d^5))*c*abs(b) + 560*(sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*(2*(b*x + a)
*(4*(b*x + a)*(6*(b*x + a)/b^2 + (b^7*c*d^5 - 17*a*b^6*d^6)/(b^8*d^6)) - (5*b^8*c^2*d^4 + 6*a*b^7*c*d^5 - 59*a
^2*b^6*d^6)/(b^8*d^6)) + 3*(5*b^9*c^3*d^3 + a*b^8*c^2*d^4 - a^2*b^7*c*d^5 - 5*a^3*b^6*d^6)/(b^8*d^6))*sqrt(b*x
+ a) + 3*(5*b^4*c^4 - 4*a*b^3*c^3*d - 2*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + 5*a^4*d^4)*log(abs(-sqrt(b*d)*sqrt(
b*x + a) + sqrt(b^2*c + (b*x + a)*b*d - a*b*d)))/(sqrt(b*d)*b*d^3))*a^2*c*abs(b)/b^2 + 112*(sqrt(b^2*c + (b*x
+ a)*b*d - a*b*d)*(2*(4*(b*x + a)*(6*(b*x + a)*(8*(b*x + a)/b^3 + (b^13*c*d^7 - 31*a*b^12*d^8)/(b^15*d^8)) - (
7*b^14*c^2*d^6 + 16*a*b^13*c*d^7 - 263*a^2*b^12*d^8)/(b^15*d^8)) + 5*(7*b^15*c^3*d^5 + 9*a*b^14*c^2*d^6 + 9*a^
2*b^13*c*d^7 - 121*a^3*b^12*d^8)/(b^15*d^8))*(b*x + a) - 15*(7*b^16*c^4*d^4 + 2*a*b^15*c^3*d^5 - 2*a^3*b^13*c*
d^7 - 7*a^4*b^12*d^8)/(b^15*d^8))*sqrt(b*x + a) - 15*(7*b^5*c^5 - 5*a*b^4*c^4*d - 2*a^2*b^3*c^3*d^2 - 2*a^3*b^
2*c^2*d^3 - 5*a^4*b*c*d^4 + 7*a^5*d^5)*log(abs(-sqrt(b*d)*sqrt(b*x + a) + sqrt(b^2*c + (b*x + a)*b*d - a*b*d))
)/(sqrt(b*d)*b^2*d^4))*a*c*abs(b)/b + (sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*(2*(4*(2*(8*(b*x + a)*(10*(b*x + a)
*(12*(b*x + a)/b^5 + (b^31*c*d^11 - 71*a*b^30*d^12)/(b^35*d^12)) - (11*b^32*c^2*d^10 + 48*a*b^31*c*d^11 - 1739
*a^2*b^30*d^12)/(b^35*d^12)) + 3*(33*b^33*c^3*d^9 + 111*a*b^32*c^2*d^10 + 239*a^2*b^31*c*d^11 - 5983*a^3*b^30*
d^12)/(b^35*d^12))*(b*x + a) - 7*(33*b^34*c^4*d^8 + 78*a*b^33*c^3*d^9 + 128*a^2*b^32*c^2*d^10 + 178*a^3*b^31*c
*d^11 - 3617*a^4*b^30*d^12)/(b^35*d^12))*(b*x + a) + 35*(33*b^35*c^5*d^7 + 45*a*b^34*c^4*d^8 + 50*a^2*b^33*c^3
*d^9 + 50*a^3*b^32*c^2*d^10 + 45*a^4*b^31*c*d^11 - 991*a^5*b^30*d^12)/(b^35*d^12))*(b*x + a) - 105*(33*b^36*c^
6*d^6 + 12*a*b^35*c^5*d^7 + 5*a^2*b^34*c^4*d^8 - 5*a^4*b^32*c^2*d^10 - 12*a^5*b^31*c*d^11 - 33*a^6*b^30*d^12)/
(b^35*d^12))*sqrt(b*x + a) - 105*(33*b^7*c^7 - 21*a*b^6*c^6*d - 7*a^2*b^5*c^5*d^2 - 5*a^3*b^4*c^4*d^3 - 5*a^4*
b^3*c^3*d^4 - 7*a^5*b^2*c^2*d^5 - 21*a^6*b*c*d^6 + 33*a^7*d^7)*log(abs(-sqrt(b*d)*sqrt(b*x + a) + sqrt(b^2*c +
(b*x + a)*b*d - a*b*d)))/(sqrt(b*d)*b^4*d^6))*d*abs(b) + 56*(sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*(2*(4*(b*x +
a)*(6*(b*x + a)*(8*(b*x + a)/b^3 + (b^13*c*d^7 - 31*a*b^12*d^8)/(b^15*d^8)) - (7*b^14*c^2*d^6 + 16*a*b^13*c*d
^7 - 263*a^2*b^12*d^8)/(b^15*d^8)) + 5*(7*b^15*c^3*d^5 + 9*a*b^14*c^2*d^6 + 9*a^2*b^13*c*d^7 - 121*a^3*b^12*d^
8)/(b^15*d^8))*(b*x + a) - 15*(7*b^16*c^4*d^4 + 2*a*b^15*c^3*d^5 - 2*a^3*b^13*c*d^7 - 7*a^4*b^12*d^8)/(b^15*d^
8))*sqrt(b*x + a) - 15*(7*b^5*c^5 - 5*a*b^4*c^4*d - 2*a^2*b^3*c^3*d^2 - 2*a^3*b^2*c^2*d^3 - 5*a^4*b*c*d^4 + 7*
a^5*d^5)*log(abs(-sqrt(b*d)*sqrt(b*x + a) + sqrt(b^2*c + (b*x + a)*b*d - a*b*d)))/(sqrt(b*d)*b^2*d^4))*a^2*d*a
bs(b)/b^2 + 28*(sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*(2*(4*(2*(b*x + a)*(8*(b*x + a)*(10*(b*x + a)/b^4 + (b^21*
c*d^9 - 49*a*b^20*d^10)/(b^24*d^10)) - 3*(3*b^22*c^2*d^8 + 10*a*b^21*c*d^9 - 253*a^2*b^20*d^10)/(b^24*d^10)) +
(21*b^23*c^3*d^7 + 49*a*b^22*c^2*d^8 + 79*a^2*b^21*c*d^9 - 1429*a^3*b^20*d^10)/(b^24*d^10))*(b*x + a) - 5*(21
*b^24*c^4*d^6 + 28*a*b^23*c^3*d^7 + 30*a^2*b^22*c^2*d^8 + 28*a^3*b^21*c*d^9 - 491*a^4*b^20*d^10)/(b^24*d^10))*
(b*x + a) + 15*(21*b^25*c^5*d^5 + 7*a*b^24*c^4*d^6 + 2*a^2*b^23*c^3*d^7 - 2*a^3*b^22*c^2*d^8 - 7*a^4*b^21*c*d^
9 - 21*a^5*b^20*d^10)/(b^24*d^10))*sqrt(b*x + a) + 15*(21*b^6*c^6 - 14*a*b^5*c^5*d - 5*a^2*b^4*c^4*d^2 - 4*a^3
*b^3*c^3*d^3 - 5*a^4*b^2*c^2*d^4 - 14*a^5*b*c*d^5 + 21*a^6*d^6)*log(abs(-sqrt(b*d)*sqrt(b*x + a) + sqrt(b^2*c
+ (b*x + a)*b*d - a*b*d)))/(sqrt(b*d)*b^3*d^5))*a*d*abs(b)/b)/b