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

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

________________________________________________________________________________________

Rubi [A]  time = 0.44, antiderivative size = 437, normalized size of antiderivative = 1.00, 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} \begin {gather*} \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} \end {gather*}

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 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 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 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 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])

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]

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 = 3.25, size = 376, normalized size = 0.86 \begin {gather*} \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} \end {gather*}

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)

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.86, size = 592, normalized size = 1.35 \begin {gather*} \frac {\sqrt {a+b x} (b c-a d)^5 \left (525 a^2 b^6 d^2-\frac {3500 a^2 b^5 d^3 (a+b x)}{c+d x}+\frac {9905 a^2 b^4 d^4 (a+b x)^2}{(c+d x)^2}+\frac {15360 a^2 b^3 d^5 (a+b x)^3}{(c+d x)^3}-\frac {9905 a^2 b^2 d^6 (a+b x)^4}{(c+d x)^4}-\frac {525 a^2 d^8 (a+b x)^6}{(c+d x)^6}+\frac {3500 a^2 b d^7 (a+b x)^5}{(c+d x)^5}-\frac {6300 b^7 c^2 d (a+b x)}{c+d x}+1050 a b^7 c d+\frac {17829 b^6 c^2 d^2 (a+b x)^2}{(c+d x)^2}-\frac {7000 a b^6 c d^2 (a+b x)}{c+d x}-\frac {27648 b^5 c^2 d^3 (a+b x)^3}{(c+d x)^3}+\frac {19810 a b^5 c d^3 (a+b x)^2}{(c+d x)^2}+\frac {25179 b^4 c^2 d^4 (a+b x)^4}{(c+d x)^4}-\frac {30720 a b^4 c d^4 (a+b x)^3}{(c+d x)^3}+\frac {6300 b^3 c^2 d^5 (a+b x)^5}{(c+d x)^5}-\frac {19810 a b^3 c d^5 (a+b x)^4}{(c+d x)^4}-\frac {945 b^2 c^2 d^6 (a+b x)^6}{(c+d x)^6}+\frac {7000 a b^2 c d^6 (a+b x)^5}{(c+d x)^5}-\frac {1050 a b c d^7 (a+b x)^6}{(c+d x)^6}+945 b^8 c^2\right )}{107520 b^4 d^5 \sqrt {c+d x} \left (b-\frac {d (a+b x)}{c+d x}\right )^7}-\frac {(b c-a d)^5 \left (5 a^2 d^2+10 a b c d+9 b^2 c^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 {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((b*c - a*d)^5*Sqrt[a + b*x]*(945*b^8*c^2 + 1050*a*b^7*c*d + 525*a^2*b^6*d^2 - (945*b^2*c^2*d^6*(a + b*x)^6)/(
c + d*x)^6 - (1050*a*b*c*d^7*(a + b*x)^6)/(c + d*x)^6 - (525*a^2*d^8*(a + b*x)^6)/(c + d*x)^6 + (6300*b^3*c^2*
d^5*(a + b*x)^5)/(c + d*x)^5 + (7000*a*b^2*c*d^6*(a + b*x)^5)/(c + d*x)^5 + (3500*a^2*b*d^7*(a + b*x)^5)/(c +
d*x)^5 + (25179*b^4*c^2*d^4*(a + b*x)^4)/(c + d*x)^4 - (19810*a*b^3*c*d^5*(a + b*x)^4)/(c + d*x)^4 - (9905*a^2
*b^2*d^6*(a + b*x)^4)/(c + d*x)^4 - (27648*b^5*c^2*d^3*(a + b*x)^3)/(c + d*x)^3 - (30720*a*b^4*c*d^4*(a + b*x)
^3)/(c + d*x)^3 + (15360*a^2*b^3*d^5*(a + b*x)^3)/(c + d*x)^3 + (17829*b^6*c^2*d^2*(a + b*x)^2)/(c + d*x)^2 +
(19810*a*b^5*c*d^3*(a + b*x)^2)/(c + d*x)^2 + (9905*a^2*b^4*d^4*(a + b*x)^2)/(c + d*x)^2 - (6300*b^7*c^2*d*(a
+ b*x))/(c + d*x) - (7000*a*b^6*c*d^2*(a + b*x))/(c + d*x) - (3500*a^2*b^5*d^3*(a + b*x))/(c + d*x)))/(107520*
b^4*d^5*Sqrt[c + d*x]*(b - (d*(a + b*x))/(c + d*x))^7) - ((b*c - a*d)^5*(9*b^2*c^2 + 10*a*b*c*d + 5*a^2*d^2)*A
rcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(1024*b^(9/2)*d^(11/2))

________________________________________________________________________________________

fricas [A]  time = 1.56, size = 1110, normalized size = 2.54 \begin {gather*} \left [-\frac {105 \, {\left (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}\right )} \sqrt {b d} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \, {\left (2 \, b d x + b c + a d\right )} \sqrt {b d} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) - 4 \, {\left (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 \, {\left (15 \, b^{7} c d^{6} + 29 \, a b^{6} d^{7}\right )} x^{5} + 128 \, {\left (3 \, b^{7} c^{2} d^{5} + 380 \, a b^{6} c d^{6} + 185 \, a^{2} b^{5} d^{7}\right )} x^{4} - 16 \, {\left (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}\right )} x^{3} + 8 \, {\left (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}\right )} x^{2} - 2 \, {\left (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}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{430080 \, b^{5} d^{6}}, \frac {105 \, {\left (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}\right )} \sqrt {-b d} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {-b d} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )}}\right ) + 2 \, {\left (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 \, {\left (15 \, b^{7} c d^{6} + 29 \, a b^{6} d^{7}\right )} x^{5} + 128 \, {\left (3 \, b^{7} c^{2} d^{5} + 380 \, a b^{6} c d^{6} + 185 \, a^{2} b^{5} d^{7}\right )} x^{4} - 16 \, {\left (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}\right )} x^{3} + 8 \, {\left (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}\right )} x^{2} - 2 \, {\left (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}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{215040 \, b^{5} d^{6}}\right ] \end {gather*}

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)]

________________________________________________________________________________________

giac [B]  time = 5.37, size = 3120, normalized size = 7.14

result too large to display

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*(168*(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^4 + (b^20*c*d^7
- 41*a*b^19*d^8)/(b^23*d^8)) - (7*b^21*c^2*d^6 + 26*a*b^20*c*d^7 - 513*a^2*b^19*d^8)/(b^23*d^8)) + 5*(7*b^22*
c^3*d^5 + 19*a*b^21*c^2*d^6 + 37*a^2*b^20*c*d^7 - 447*a^3*b^19*d^8)/(b^23*d^8))*(b*x + a) - 15*(7*b^23*c^4*d^4
+ 12*a*b^22*c^3*d^5 + 18*a^2*b^21*c^2*d^6 + 28*a^3*b^20*c*d^7 - 193*a^4*b^19*d^8)/(b^23*d^8))*sqrt(b*x + a) -
15*(7*b^5*c^5 + 5*a*b^4*c^4*d + 6*a^2*b^3*c^3*d^2 + 10*a^3*b^2*c^2*d^3 + 35*a^4*b*c*d^4 - 63*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^3*d^4))*a*c*abs(b) + 4480*(sqrt
(b^2*c + (b*x + a)*b*d - a*b*d)*sqrt(b*x + a)*(2*(b*x + a)*(4*(b*x + a)/b^2 + (b^6*c*d^3 - 13*a*b^5*d^4)/(b^7*
d^4)) - 3*(b^7*c^2*d^2 + 2*a*b^6*c*d^3 - 11*a^2*b^5*d^4)/(b^7*d^4)) - 3*(b^3*c^3 + a*b^2*c^2*d + 3*a^2*b*c*d^2
- 5*a^3*d^3)*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^2))*a^3*
c*abs(b)/b^2 + 1680*(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^3 + (b^12*c*
d^5 - 25*a*b^11*d^6)/(b^14*d^6)) - (5*b^13*c^2*d^4 + 14*a*b^12*c*d^5 - 163*a^2*b^11*d^6)/(b^14*d^6)) + 3*(5*b^
14*c^3*d^3 + 9*a*b^13*c^2*d^4 + 15*a^2*b^12*c*d^5 - 93*a^3*b^11*d^6)/(b^14*d^6))*sqrt(b*x + a) + 3*(5*b^4*c^4
+ 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 + 20*a^3*b*c*d^3 - 35*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^2*d^3))*a^2*c*abs(b)/b + 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^5 + (b^30*c*d^9 - 61*a*b^29*d^10)/(b^34*d^10)) - 3*(3*b^31*c^2*d
^8 + 14*a*b^30*c*d^9 - 417*a^2*b^29*d^10)/(b^34*d^10)) + (21*b^32*c^3*d^7 + 77*a*b^31*c^2*d^8 + 183*a^2*b^30*c
*d^9 - 3481*a^3*b^29*d^10)/(b^34*d^10))*(b*x + a) - 5*(21*b^33*c^4*d^6 + 56*a*b^32*c^3*d^7 + 106*a^2*b^31*c^2*
d^8 + 176*a^3*b^30*c*d^9 - 2279*a^4*b^29*d^10)/(b^34*d^10))*(b*x + a) + 15*(21*b^34*c^5*d^5 + 35*a*b^33*c^4*d^
6 + 50*a^2*b^32*c^3*d^7 + 70*a^3*b^31*c^2*d^8 + 105*a^4*b^30*c*d^9 - 793*a^5*b^29*d^10)/(b^34*d^10))*sqrt(b*x
+ a) + 15*(21*b^6*c^6 + 14*a*b^5*c^5*d + 15*a^2*b^4*c^4*d^2 + 20*a^3*b^3*c^3*d^3 + 35*a^4*b^2*c^2*d^4 + 126*a^
5*b*c*d^5 - 231*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
^4*d^5))*b*c*abs(b) + 42*(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^
5 + (b^30*c*d^9 - 61*a*b^29*d^10)/(b^34*d^10)) - 3*(3*b^31*c^2*d^8 + 14*a*b^30*c*d^9 - 417*a^2*b^29*d^10)/(b^3
4*d^10)) + (21*b^32*c^3*d^7 + 77*a*b^31*c^2*d^8 + 183*a^2*b^30*c*d^9 - 3481*a^3*b^29*d^10)/(b^34*d^10))*(b*x +
a) - 5*(21*b^33*c^4*d^6 + 56*a*b^32*c^3*d^7 + 106*a^2*b^31*c^2*d^8 + 176*a^3*b^30*c*d^9 - 2279*a^4*b^29*d^10)
/(b^34*d^10))*(b*x + a) + 15*(21*b^34*c^5*d^5 + 35*a*b^33*c^4*d^6 + 50*a^2*b^32*c^3*d^7 + 70*a^3*b^31*c^2*d^8
+ 105*a^4*b^30*c*d^9 - 793*a^5*b^29*d^10)/(b^34*d^10))*sqrt(b*x + a) + 15*(21*b^6*c^6 + 14*a*b^5*c^5*d + 15*a^
2*b^4*c^4*d^2 + 20*a^3*b^3*c^3*d^3 + 35*a^4*b^2*c^2*d^4 + 126*a^5*b*c*d^5 - 231*a^6*d^6)*log(abs(-sqrt(b*d)*sq
rt(b*x + a) + sqrt(b^2*c + (b*x + a)*b*d - a*b*d)))/(sqrt(b*d)*b^4*d^5))*a*d*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^3 + (b^12*c*d^5 - 25*a*b^11*d^6)/(b^14*d^6)) - (5*b^
13*c^2*d^4 + 14*a*b^12*c*d^5 - 163*a^2*b^11*d^6)/(b^14*d^6)) + 3*(5*b^14*c^3*d^3 + 9*a*b^13*c^2*d^4 + 15*a^2*b
^12*c*d^5 - 93*a^3*b^11*d^6)/(b^14*d^6))*sqrt(b*x + a) + 3*(5*b^4*c^4 + 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 + 20
*a^3*b*c*d^3 - 35*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^2*d^3))*a^3*d*abs(b)/b^2 + 168*(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^4 + (b^20*c*d^7 - 41*a*b^19*d^8)/(b^23*d^8)) - (7*b^21*c^2*d^6 + 26*a*b^20*c*d^7 - 513*a^2*b^19*d^8)/(b^2
3*d^8)) + 5*(7*b^22*c^3*d^5 + 19*a*b^21*c^2*d^6 + 37*a^2*b^20*c*d^7 - 447*a^3*b^19*d^8)/(b^23*d^8))*(b*x + a)
- 15*(7*b^23*c^4*d^4 + 12*a*b^22*c^3*d^5 + 18*a^2*b^21*c^2*d^6 + 28*a^3*b^20*c*d^7 - 193*a^4*b^19*d^8)/(b^23*d
^8))*sqrt(b*x + a) - 15*(7*b^5*c^5 + 5*a*b^4*c^4*d + 6*a^2*b^3*c^3*d^2 + 10*a^3*b^2*c^2*d^3 + 35*a^4*b*c*d^4 -
63*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^3*d^4))*a^2
*d*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^6 + (b^
42*c*d^11 - 85*a*b^41*d^12)/(b^47*d^12)) - (11*b^43*c^2*d^10 + 62*a*b^42*c*d^11 - 2593*a^2*b^41*d^12)/(b^47*d^
12)) + 3*(33*b^44*c^3*d^9 + 153*a*b^43*c^2*d^10 + 435*a^2*b^42*c*d^11 - 11821*a^3*b^41*d^12)/(b^47*d^12))*(b*x
+ a) - 7*(33*b^45*c^4*d^8 + 120*a*b^44*c^3*d^9 + 282*a^2*b^43*c^2*d^10 + 544*a^3*b^42*c*d^11 - 10579*a^4*b^41
*d^12)/(b^47*d^12))*(b*x + a) + 35*(33*b^46*c^5*d^7 + 87*a*b^45*c^4*d^8 + 162*a^2*b^44*c^3*d^9 + 262*a^3*b^43*
c^2*d^10 + 397*a^4*b^42*c*d^11 - 5549*a^5*b^41*d^12)/(b^47*d^12))*(b*x + a) - 105*(33*b^47*c^6*d^6 + 54*a*b^46
*c^5*d^7 + 75*a^2*b^45*c^4*d^8 + 100*a^3*b^44*c^3*d^9 + 135*a^4*b^43*c^2*d^10 + 198*a^5*b^42*c*d^11 - 1619*a^6
*b^41*d^12)/(b^47*d^12))*sqrt(b*x + a) - 105*(33*b^7*c^7 + 21*a*b^6*c^6*d + 21*a^2*b^5*c^5*d^2 + 25*a^3*b^4*c^
4*d^3 + 35*a^4*b^3*c^3*d^4 + 63*a^5*b^2*c^2*d^5 + 231*a^6*b*c*d^6 - 429*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^5*d^6))*b*d*abs(b))/b

________________________________________________________________________________________

maple [B]  time = 0.02, size = 1580, normalized size = 3.62

result too large to display

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)*(2800*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)*a^5*b*c*d^5-945*b^7*c^7
*ln(1/2*(2*b*d*x+a*d+b*c+2*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2))/(b*d)^(1/2))+525*a^7*d^7*ln(1/2*(2*b*d
*x+a*d+b*c+2*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2))/(b*d)^(1/2))-1575*a^6*b*c*d^6*ln(1/2*(2*b*d*x+a*d+b*
c+2*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2))/(b*d)^(1/2))+47360*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2
)*a^2*b^4*d^6*x^4+768*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)*b^6*c^2*d^4*x^4+480*(b*d*x^2+a*d*x+b*c*x+a*c
)^(1/2)*(b*d)^(1/2)*a^3*b^3*d^6*x^3-864*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)*b^6*c^3*d^3*x^3-560*(b*d*x
^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)*a^4*b^2*d^6*x^2+1008*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)*b^6*c^4
*d^2*x^2-6720*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)*a*b^5*c^5*d-1050*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*
d)^(1/2)*a^6*d^6+1890*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)*b^6*c^6-1050*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)
*(b*d)^(1/2)*a^4*b^2*c^2*d^4-1200*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)*a^3*b^3*c^3*d^3+7378*(b*d*x^2+a*
d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)*a^2*b^4*c^4*d^2+700*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)*a^5*b*d^6*x-1
260*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)*b^6*c^5*d*x+74240*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)*
a*b^5*d^6*x^5+38400*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)*b^6*c*d^5*x^5+945*a^5*b^2*c^2*d^5*ln(1/2*(2*b*
d*x+a*d+b*c+2*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2))/(b*d)^(1/2))+525*a^4*b^3*c^3*d^4*ln(1/2*(2*b*d*x+a*
d+b*c+2*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2))/(b*d)^(1/2))+1575*a^3*b^4*c^4*d^3*ln(1/2*(2*b*d*x+a*d+b*c
+2*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2))/(b*d)^(1/2))-4725*a^2*b^5*c^5*d^2*ln(1/2*(2*b*d*x+a*d+b*c+2*(b
*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2))/(b*d)^(1/2))+3675*a*b^6*c^6*d*ln(1/2*(2*b*d*x+a*d+b*c+2*(b*d*x^2+a*
d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2))/(b*d)^(1/2))+30720*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)*b^6*d^6*x^6+4
396*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)*a*b^5*c^4*d^2*x-1820*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/
2)*a^4*b^2*c*d^5*x+600*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)*a^3*b^3*c^2*d^4*x-4664*(b*d*x^2+a*d*x+b*c*x
+a*c)^(1/2)*(b*d)^(1/2)*a^2*b^4*c^3*d^3*x+97280*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)*a*b^5*c*d^5*x^4+67
040*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)*a^2*b^4*c*d^5*x^3+2976*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(
1/2)*a*b^5*c^2*d^4*x^3+1440*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)*a^3*b^3*c*d^5*x^2+3648*(b*d*x^2+a*d*x+
b*c*x+a*c)^(1/2)*(b*d)^(1/2)*a^2*b^4*c^2*d^4*x^2-3488*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)*a*b^5*c^3*d^
3*x^2)/b^4/d^5/(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)/(b*d)^(1/2)

________________________________________________________________________________________

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(x^2*(b*x+a)^(5/2)*(d*x+c)^(3/2),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(a*d-b*c>0)', see `assume?` for
more details)Is a*d-b*c zero or nonzero?

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^2\,{\left (a+b\,x\right )}^{5/2}\,{\left (c+d\,x\right )}^{3/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{2} \left (a + b x\right )^{\frac {5}{2}} \left (c + d x\right )^{\frac {3}{2}}\, dx \end {gather*}

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]

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

________________________________________________________________________________________