3.6.66 \(\int x^2 \sqrt {a+b x} (c+d x)^{5/2} \, dx\) [566]

Optimal. Leaf size=376 \[ \frac {(b c-a d)^3 \left (5 b^2 c^2+14 a b c d+21 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{512 b^5 d^3}+\frac {(b c-a d)^2 \left (5 b^2 c^2+14 a b c d+21 a^2 d^2\right ) (a+b x)^{3/2} \sqrt {c+d x}}{256 b^5 d^2}+\frac {(b c-a d) \left (5 b^2 c^2+14 a b c d+21 a^2 d^2\right ) (a+b x)^{3/2} (c+d x)^{3/2}}{192 b^4 d^2}+\frac {\left (5 b^2 c^2+14 a b c d+21 a^2 d^2\right ) (a+b x)^{3/2} (c+d x)^{5/2}}{160 b^3 d^2}-\frac {(5 b c+9 a d) (a+b x)^{3/2} (c+d x)^{7/2}}{60 b^2 d^2}+\frac {x (a+b x)^{3/2} (c+d x)^{7/2}}{6 b d}-\frac {(b c-a d)^4 \left (5 b^2 c^2+14 a b c d+21 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{512 b^{11/2} d^{7/2}} \]

[Out]

1/192*(-a*d+b*c)*(21*a^2*d^2+14*a*b*c*d+5*b^2*c^2)*(b*x+a)^(3/2)*(d*x+c)^(3/2)/b^4/d^2+1/160*(21*a^2*d^2+14*a*
b*c*d+5*b^2*c^2)*(b*x+a)^(3/2)*(d*x+c)^(5/2)/b^3/d^2-1/60*(9*a*d+5*b*c)*(b*x+a)^(3/2)*(d*x+c)^(7/2)/b^2/d^2+1/
6*x*(b*x+a)^(3/2)*(d*x+c)^(7/2)/b/d-1/512*(-a*d+b*c)^4*(21*a^2*d^2+14*a*b*c*d+5*b^2*c^2)*arctanh(d^(1/2)*(b*x+
a)^(1/2)/b^(1/2)/(d*x+c)^(1/2))/b^(11/2)/d^(7/2)+1/256*(-a*d+b*c)^2*(21*a^2*d^2+14*a*b*c*d+5*b^2*c^2)*(b*x+a)^
(3/2)*(d*x+c)^(1/2)/b^5/d^2+1/512*(-a*d+b*c)^3*(21*a^2*d^2+14*a*b*c*d+5*b^2*c^2)*(b*x+a)^(1/2)*(d*x+c)^(1/2)/b
^5/d^3

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Rubi [A]
time = 0.25, antiderivative size = 376, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {92, 81, 52, 65, 223, 212} \begin {gather*} -\frac {\left (21 a^2 d^2+14 a b c d+5 b^2 c^2\right ) (b c-a d)^4 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{512 b^{11/2} d^{7/2}}+\frac {(a+b x)^{3/2} \sqrt {c+d x} \left (21 a^2 d^2+14 a b c d+5 b^2 c^2\right ) (b c-a d)^2}{256 b^5 d^2}+\frac {\sqrt {a+b x} \sqrt {c+d x} \left (21 a^2 d^2+14 a b c d+5 b^2 c^2\right ) (b c-a d)^3}{512 b^5 d^3}+\frac {(a+b x)^{3/2} (c+d x)^{3/2} \left (21 a^2 d^2+14 a b c d+5 b^2 c^2\right ) (b c-a d)}{192 b^4 d^2}+\frac {(a+b x)^{3/2} (c+d x)^{5/2} \left (21 a^2 d^2+14 a b c d+5 b^2 c^2\right )}{160 b^3 d^2}-\frac {(a+b x)^{3/2} (c+d x)^{7/2} (9 a d+5 b c)}{60 b^2 d^2}+\frac {x (a+b x)^{3/2} (c+d x)^{7/2}}{6 b d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((b*c - a*d)^3*(5*b^2*c^2 + 14*a*b*c*d + 21*a^2*d^2)*Sqrt[a + b*x]*Sqrt[c + d*x])/(512*b^5*d^3) + ((b*c - a*d)
^2*(5*b^2*c^2 + 14*a*b*c*d + 21*a^2*d^2)*(a + b*x)^(3/2)*Sqrt[c + d*x])/(256*b^5*d^2) + ((b*c - a*d)*(5*b^2*c^
2 + 14*a*b*c*d + 21*a^2*d^2)*(a + b*x)^(3/2)*(c + d*x)^(3/2))/(192*b^4*d^2) + ((5*b^2*c^2 + 14*a*b*c*d + 21*a^
2*d^2)*(a + b*x)^(3/2)*(c + d*x)^(5/2))/(160*b^3*d^2) - ((5*b*c + 9*a*d)*(a + b*x)^(3/2)*(c + d*x)^(7/2))/(60*
b^2*d^2) + (x*(a + b*x)^(3/2)*(c + d*x)^(7/2))/(6*b*d) - ((b*c - a*d)^4*(5*b^2*c^2 + 14*a*b*c*d + 21*a^2*d^2)*
ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(512*b^(11/2)*d^(7/2))

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 92

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 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 x^2 \sqrt {a+b x} (c+d x)^{5/2} \, dx &=\frac {x (a+b x)^{3/2} (c+d x)^{7/2}}{6 b d}+\frac {\int \sqrt {a+b x} (c+d x)^{5/2} \left (-a c-\frac {1}{2} (5 b c+9 a d) x\right ) \, dx}{6 b d}\\ &=-\frac {(5 b c+9 a d) (a+b x)^{3/2} (c+d x)^{7/2}}{60 b^2 d^2}+\frac {x (a+b x)^{3/2} (c+d x)^{7/2}}{6 b d}+\frac {\left (5 b^2 c^2+14 a b c d+21 a^2 d^2\right ) \int \sqrt {a+b x} (c+d x)^{5/2} \, dx}{40 b^2 d^2}\\ &=\frac {\left (5 b^2 c^2+14 a b c d+21 a^2 d^2\right ) (a+b x)^{3/2} (c+d x)^{5/2}}{160 b^3 d^2}-\frac {(5 b c+9 a d) (a+b x)^{3/2} (c+d x)^{7/2}}{60 b^2 d^2}+\frac {x (a+b x)^{3/2} (c+d x)^{7/2}}{6 b d}+\frac {\left ((b c-a d) \left (5 b^2 c^2+14 a b c d+21 a^2 d^2\right )\right ) \int \sqrt {a+b x} (c+d x)^{3/2} \, dx}{64 b^3 d^2}\\ &=\frac {(b c-a d) \left (5 b^2 c^2+14 a b c d+21 a^2 d^2\right ) (a+b x)^{3/2} (c+d x)^{3/2}}{192 b^4 d^2}+\frac {\left (5 b^2 c^2+14 a b c d+21 a^2 d^2\right ) (a+b x)^{3/2} (c+d x)^{5/2}}{160 b^3 d^2}-\frac {(5 b c+9 a d) (a+b x)^{3/2} (c+d x)^{7/2}}{60 b^2 d^2}+\frac {x (a+b x)^{3/2} (c+d x)^{7/2}}{6 b d}+\frac {\left ((b c-a d)^2 \left (5 b^2 c^2+14 a b c d+21 a^2 d^2\right )\right ) \int \sqrt {a+b x} \sqrt {c+d x} \, dx}{128 b^4 d^2}\\ &=\frac {(b c-a d)^2 \left (5 b^2 c^2+14 a b c d+21 a^2 d^2\right ) (a+b x)^{3/2} \sqrt {c+d x}}{256 b^5 d^2}+\frac {(b c-a d) \left (5 b^2 c^2+14 a b c d+21 a^2 d^2\right ) (a+b x)^{3/2} (c+d x)^{3/2}}{192 b^4 d^2}+\frac {\left (5 b^2 c^2+14 a b c d+21 a^2 d^2\right ) (a+b x)^{3/2} (c+d x)^{5/2}}{160 b^3 d^2}-\frac {(5 b c+9 a d) (a+b x)^{3/2} (c+d x)^{7/2}}{60 b^2 d^2}+\frac {x (a+b x)^{3/2} (c+d x)^{7/2}}{6 b d}+\frac {\left ((b c-a d)^3 \left (5 b^2 c^2+14 a b c d+21 a^2 d^2\right )\right ) \int \frac {\sqrt {a+b x}}{\sqrt {c+d x}} \, dx}{512 b^5 d^2}\\ &=\frac {(b c-a d)^3 \left (5 b^2 c^2+14 a b c d+21 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{512 b^5 d^3}+\frac {(b c-a d)^2 \left (5 b^2 c^2+14 a b c d+21 a^2 d^2\right ) (a+b x)^{3/2} \sqrt {c+d x}}{256 b^5 d^2}+\frac {(b c-a d) \left (5 b^2 c^2+14 a b c d+21 a^2 d^2\right ) (a+b x)^{3/2} (c+d x)^{3/2}}{192 b^4 d^2}+\frac {\left (5 b^2 c^2+14 a b c d+21 a^2 d^2\right ) (a+b x)^{3/2} (c+d x)^{5/2}}{160 b^3 d^2}-\frac {(5 b c+9 a d) (a+b x)^{3/2} (c+d x)^{7/2}}{60 b^2 d^2}+\frac {x (a+b x)^{3/2} (c+d x)^{7/2}}{6 b d}-\frac {\left ((b c-a d)^4 \left (5 b^2 c^2+14 a b c d+21 a^2 d^2\right )\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{1024 b^5 d^3}\\ &=\frac {(b c-a d)^3 \left (5 b^2 c^2+14 a b c d+21 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{512 b^5 d^3}+\frac {(b c-a d)^2 \left (5 b^2 c^2+14 a b c d+21 a^2 d^2\right ) (a+b x)^{3/2} \sqrt {c+d x}}{256 b^5 d^2}+\frac {(b c-a d) \left (5 b^2 c^2+14 a b c d+21 a^2 d^2\right ) (a+b x)^{3/2} (c+d x)^{3/2}}{192 b^4 d^2}+\frac {\left (5 b^2 c^2+14 a b c d+21 a^2 d^2\right ) (a+b x)^{3/2} (c+d x)^{5/2}}{160 b^3 d^2}-\frac {(5 b c+9 a d) (a+b x)^{3/2} (c+d x)^{7/2}}{60 b^2 d^2}+\frac {x (a+b x)^{3/2} (c+d x)^{7/2}}{6 b d}-\frac {\left ((b c-a d)^4 \left (5 b^2 c^2+14 a b c d+21 a^2 d^2\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {c-\frac {a d}{b}+\frac {d x^2}{b}}} \, dx,x,\sqrt {a+b x}\right )}{512 b^6 d^3}\\ &=\frac {(b c-a d)^3 \left (5 b^2 c^2+14 a b c d+21 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{512 b^5 d^3}+\frac {(b c-a d)^2 \left (5 b^2 c^2+14 a b c d+21 a^2 d^2\right ) (a+b x)^{3/2} \sqrt {c+d x}}{256 b^5 d^2}+\frac {(b c-a d) \left (5 b^2 c^2+14 a b c d+21 a^2 d^2\right ) (a+b x)^{3/2} (c+d x)^{3/2}}{192 b^4 d^2}+\frac {\left (5 b^2 c^2+14 a b c d+21 a^2 d^2\right ) (a+b x)^{3/2} (c+d x)^{5/2}}{160 b^3 d^2}-\frac {(5 b c+9 a d) (a+b x)^{3/2} (c+d x)^{7/2}}{60 b^2 d^2}+\frac {x (a+b x)^{3/2} (c+d x)^{7/2}}{6 b d}-\frac {\left ((b c-a d)^4 \left (5 b^2 c^2+14 a b c d+21 a^2 d^2\right )\right ) \text {Subst}\left (\int \frac {1}{1-\frac {d x^2}{b}} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{512 b^6 d^3}\\ &=\frac {(b c-a d)^3 \left (5 b^2 c^2+14 a b c d+21 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{512 b^5 d^3}+\frac {(b c-a d)^2 \left (5 b^2 c^2+14 a b c d+21 a^2 d^2\right ) (a+b x)^{3/2} \sqrt {c+d x}}{256 b^5 d^2}+\frac {(b c-a d) \left (5 b^2 c^2+14 a b c d+21 a^2 d^2\right ) (a+b x)^{3/2} (c+d x)^{3/2}}{192 b^4 d^2}+\frac {\left (5 b^2 c^2+14 a b c d+21 a^2 d^2\right ) (a+b x)^{3/2} (c+d x)^{5/2}}{160 b^3 d^2}-\frac {(5 b c+9 a d) (a+b x)^{3/2} (c+d x)^{7/2}}{60 b^2 d^2}+\frac {x (a+b x)^{3/2} (c+d x)^{7/2}}{6 b d}-\frac {(b c-a d)^4 \left (5 b^2 c^2+14 a b c d+21 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{512 b^{11/2} d^{7/2}}\\ \end {align*}

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Mathematica [A]
time = 0.89, size = 307, normalized size = 0.82 \begin {gather*} \frac {\sqrt {a+b x} \sqrt {c+d x} \left (315 a^5 d^5-105 a^4 b d^4 (9 c+2 d x)+2 a^3 b^2 d^3 \left (419 c^2+308 c d x+84 d^2 x^2\right )-2 a^2 b^3 d^2 \left (45 c^3+262 c^2 d x+244 c d^2 x^2+72 d^3 x^3\right )+a b^4 d \left (-65 c^4+40 c^3 d x+408 c^2 d^2 x^2+416 c d^3 x^3+128 d^4 x^4\right )+5 b^5 \left (15 c^5-10 c^4 d x+8 c^3 d^2 x^2+432 c^2 d^3 x^3+640 c d^4 x^4+256 d^5 x^5\right )\right )}{7680 b^5 d^3}-\frac {(b c-a d)^4 \left (5 b^2 c^2+14 a b c d+21 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{512 b^{11/2} d^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[a + b*x]*Sqrt[c + d*x]*(315*a^5*d^5 - 105*a^4*b*d^4*(9*c + 2*d*x) + 2*a^3*b^2*d^3*(419*c^2 + 308*c*d*x +
84*d^2*x^2) - 2*a^2*b^3*d^2*(45*c^3 + 262*c^2*d*x + 244*c*d^2*x^2 + 72*d^3*x^3) + a*b^4*d*(-65*c^4 + 40*c^3*d
*x + 408*c^2*d^2*x^2 + 416*c*d^3*x^3 + 128*d^4*x^4) + 5*b^5*(15*c^5 - 10*c^4*d*x + 8*c^3*d^2*x^2 + 432*c^2*d^3
*x^3 + 640*c*d^4*x^4 + 256*d^5*x^5)))/(7680*b^5*d^3) - ((b*c - a*d)^4*(5*b^2*c^2 + 14*a*b*c*d + 21*a^2*d^2)*Ar
cTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(512*b^(11/2)*d^(7/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1036\) vs. \(2(326)=652\).
time = 0.07, size = 1037, normalized size = 2.76

method	result	size
default	\(-\frac {\sqrt {d x +c}\, \sqrt {b x +a}\, \left (288 a^{2} b^{3} d^{5} x^{3} \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {b d}-4320 b^{5} c^{2} d^{3} x^{3} \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {b d}-336 a^{3} b^{2} d^{5} x^{2} \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {b d}-80 b^{5} c^{3} d^{2} x^{2} \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {b d}-256 a \,b^{4} d^{5} x^{4} \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {b d}-6400 b^{5} c \,d^{4} x^{4} \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {b d}-630 \sqrt {b d}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{5} d^{5}-150 \sqrt {b d}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, b^{5} c^{5}+420 \sqrt {b d}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{4} b \,d^{5} x +100 \sqrt {b d}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, b^{5} c^{4} d x +1890 \sqrt {b d}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{4} b c \,d^{4}-1676 \sqrt {b d}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{3} b^{2} c^{2} d^{3}+180 \sqrt {b d}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{2} b^{3} c^{3} d^{2}+130 \sqrt {b d}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a \,b^{4} c^{4} d -1232 \sqrt {b d}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{3} b^{2} c \,d^{4} x +1048 \sqrt {b d}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{2} b^{3} c^{2} d^{3} x -80 \sqrt {b d}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a \,b^{4} c^{3} d^{2} x +315 \ln \left (\frac {2 b d x +2 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{6} d^{6}+75 \ln \left (\frac {2 b d x +2 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) b^{6} c^{6}-1050 \ln \left (\frac {2 b d x +2 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{5} b c \,d^{5}-2560 b^{5} d^{5} x^{5} \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {b d}+1125 \ln \left (\frac {2 b d x +2 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{4} b^{2} c^{2} d^{4}-300 \ln \left (\frac {2 b d x +2 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{3} b^{3} c^{3} d^{3}-75 \ln \left (\frac {2 b d x +2 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{2} b^{4} c^{4} d^{2}-90 \ln \left (\frac {2 b d x +2 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a \,b^{5} c^{5} d -832 a \,b^{4} c \,d^{4} x^{3} \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {b d}+976 a^{2} b^{3} c \,d^{4} x^{2} \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {b d}-816 a \,b^{4} c^{2} d^{3} x^{2} \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {b d}\right )}{15360 d^{3} \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, b^{5} \sqrt {b d}}\)	\(1037\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(d*x+c)^(5/2)*(b*x+a)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/15360*(d*x+c)^(1/2)*(b*x+a)^(1/2)*(288*a^2*b^3*d^5*x^3*((d*x+c)*(b*x+a))^(1/2)*(b*d)^(1/2)-4320*b^5*c^2*d^3
*x^3*((d*x+c)*(b*x+a))^(1/2)*(b*d)^(1/2)-336*a^3*b^2*d^5*x^2*((d*x+c)*(b*x+a))^(1/2)*(b*d)^(1/2)-80*b^5*c^3*d^
2*x^2*((d*x+c)*(b*x+a))^(1/2)*(b*d)^(1/2)-256*a*b^4*d^5*x^4*((d*x+c)*(b*x+a))^(1/2)*(b*d)^(1/2)-6400*b^5*c*d^4
*x^4*((d*x+c)*(b*x+a))^(1/2)*(b*d)^(1/2)-630*(b*d)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a^5*d^5-150*(b*d)^(1/2)*((d*x
+c)*(b*x+a))^(1/2)*b^5*c^5+420*(b*d)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a^4*b*d^5*x+100*(b*d)^(1/2)*((d*x+c)*(b*x+a
))^(1/2)*b^5*c^4*d*x+1890*(b*d)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a^4*b*c*d^4-1676*(b*d)^(1/2)*((d*x+c)*(b*x+a))^(
1/2)*a^3*b^2*c^2*d^3+180*(b*d)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a^2*b^3*c^3*d^2+130*(b*d)^(1/2)*((d*x+c)*(b*x+a))
^(1/2)*a*b^4*c^4*d-1232*(b*d)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a^3*b^2*c*d^4*x+1048*(b*d)^(1/2)*((d*x+c)*(b*x+a))
^(1/2)*a^2*b^3*c^2*d^3*x-80*(b*d)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a*b^4*c^3*d^2*x+315*ln(1/2*(2*b*d*x+2*((d*x+c)
*(b*x+a))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^6*d^6+75*ln(1/2*(2*b*d*x+2*((d*x+c)*(b*x+a))^(1/2)*(b*d)^(
1/2)+a*d+b*c)/(b*d)^(1/2))*b^6*c^6-1050*ln(1/2*(2*b*d*x+2*((d*x+c)*(b*x+a))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(
1/2))*a^5*b*c*d^5-2560*b^5*d^5*x^5*((d*x+c)*(b*x+a))^(1/2)*(b*d)^(1/2)+1125*ln(1/2*(2*b*d*x+2*((d*x+c)*(b*x+a)
)^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^4*b^2*c^2*d^4-300*ln(1/2*(2*b*d*x+2*((d*x+c)*(b*x+a))^(1/2)*(b*d)^
(1/2)+a*d+b*c)/(b*d)^(1/2))*a^3*b^3*c^3*d^3-75*ln(1/2*(2*b*d*x+2*((d*x+c)*(b*x+a))^(1/2)*(b*d)^(1/2)+a*d+b*c)/
(b*d)^(1/2))*a^2*b^4*c^4*d^2-90*ln(1/2*(2*b*d*x+2*((d*x+c)*(b*x+a))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a*
b^5*c^5*d-832*a*b^4*c*d^4*x^3*((d*x+c)*(b*x+a))^(1/2)*(b*d)^(1/2)+976*a^2*b^3*c*d^4*x^2*((d*x+c)*(b*x+a))^(1/2
)*(b*d)^(1/2)-816*a*b^4*c^2*d^3*x^2*((d*x+c)*(b*x+a))^(1/2)*(b*d)^(1/2))/d^3/((d*x+c)*(b*x+a))^(1/2)/b^5/(b*d)
^(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(x^2*(d*x+c)^(5/2)*(b*x+a)^(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(a*d-b*c>0)', see `assume?` for
more detail

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Fricas [A]
time = 0.76, size = 892, normalized size = 2.37 \begin {gather*} \left [\frac {15 \, {\left (5 \, b^{6} c^{6} - 6 \, a b^{5} c^{5} d - 5 \, a^{2} b^{4} c^{4} d^{2} - 20 \, a^{3} b^{3} c^{3} d^{3} + 75 \, a^{4} b^{2} c^{2} d^{4} - 70 \, a^{5} b c d^{5} + 21 \, a^{6} d^{6}\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 (1280 \, b^{6} d^{6} x^{5} + 75 \, b^{6} c^{5} d - 65 \, a b^{5} c^{4} d^{2} - 90 \, a^{2} b^{4} c^{3} d^{3} + 838 \, a^{3} b^{3} c^{2} d^{4} - 945 \, a^{4} b^{2} c d^{5} + 315 \, a^{5} b d^{6} + 128 \, {\left (25 \, b^{6} c d^{5} + a b^{5} d^{6}\right )} x^{4} + 16 \, {\left (135 \, b^{6} c^{2} d^{4} + 26 \, a b^{5} c d^{5} - 9 \, a^{2} b^{4} d^{6}\right )} x^{3} + 8 \, {\left (5 \, b^{6} c^{3} d^{3} + 51 \, a b^{5} c^{2} d^{4} - 61 \, a^{2} b^{4} c d^{5} + 21 \, a^{3} b^{3} d^{6}\right )} x^{2} - 2 \, {\left (25 \, b^{6} c^{4} d^{2} - 20 \, a b^{5} c^{3} d^{3} + 262 \, a^{2} b^{4} c^{2} d^{4} - 308 \, a^{3} b^{3} c d^{5} + 105 \, a^{4} b^{2} d^{6}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{30720 \, b^{6} d^{4}}, \frac {15 \, {\left (5 \, b^{6} c^{6} - 6 \, a b^{5} c^{5} d - 5 \, a^{2} b^{4} c^{4} d^{2} - 20 \, a^{3} b^{3} c^{3} d^{3} + 75 \, a^{4} b^{2} c^{2} d^{4} - 70 \, a^{5} b c d^{5} + 21 \, a^{6} d^{6}\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 (1280 \, b^{6} d^{6} x^{5} + 75 \, b^{6} c^{5} d - 65 \, a b^{5} c^{4} d^{2} - 90 \, a^{2} b^{4} c^{3} d^{3} + 838 \, a^{3} b^{3} c^{2} d^{4} - 945 \, a^{4} b^{2} c d^{5} + 315 \, a^{5} b d^{6} + 128 \, {\left (25 \, b^{6} c d^{5} + a b^{5} d^{6}\right )} x^{4} + 16 \, {\left (135 \, b^{6} c^{2} d^{4} + 26 \, a b^{5} c d^{5} - 9 \, a^{2} b^{4} d^{6}\right )} x^{3} + 8 \, {\left (5 \, b^{6} c^{3} d^{3} + 51 \, a b^{5} c^{2} d^{4} - 61 \, a^{2} b^{4} c d^{5} + 21 \, a^{3} b^{3} d^{6}\right )} x^{2} - 2 \, {\left (25 \, b^{6} c^{4} d^{2} - 20 \, a b^{5} c^{3} d^{3} + 262 \, a^{2} b^{4} c^{2} d^{4} - 308 \, a^{3} b^{3} c d^{5} + 105 \, a^{4} b^{2} d^{6}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{15360 \, b^{6} d^{4}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(d*x+c)^(5/2)*(b*x+a)^(1/2),x, algorithm="fricas")

[Out]

[1/30720*(15*(5*b^6*c^6 - 6*a*b^5*c^5*d - 5*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 + 75*a^4*b^2*c^2*d^4 - 70*a^5
*b*c*d^5 + 21*a^6*d^6)*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)*s
qrt(b*d)*sqrt(b*x + a)*sqrt(d*x + c) + 8*(b^2*c*d + a*b*d^2)*x) + 4*(1280*b^6*d^6*x^5 + 75*b^6*c^5*d - 65*a*b^
5*c^4*d^2 - 90*a^2*b^4*c^3*d^3 + 838*a^3*b^3*c^2*d^4 - 945*a^4*b^2*c*d^5 + 315*a^5*b*d^6 + 128*(25*b^6*c*d^5 +
a*b^5*d^6)*x^4 + 16*(135*b^6*c^2*d^4 + 26*a*b^5*c*d^5 - 9*a^2*b^4*d^6)*x^3 + 8*(5*b^6*c^3*d^3 + 51*a*b^5*c^2*
d^4 - 61*a^2*b^4*c*d^5 + 21*a^3*b^3*d^6)*x^2 - 2*(25*b^6*c^4*d^2 - 20*a*b^5*c^3*d^3 + 262*a^2*b^4*c^2*d^4 - 30
8*a^3*b^3*c*d^5 + 105*a^4*b^2*d^6)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(b^6*d^4), 1/15360*(15*(5*b^6*c^6 - 6*a*b^5
*c^5*d - 5*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 + 75*a^4*b^2*c^2*d^4 - 70*a^5*b*c*d^5 + 21*a^6*d^6)*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*(1280*b^6*d^6*x^5 + 75*b^6*c^5*d - 65*a*b^5*c^4*d^2 - 90*a^2*b^4*c^3*d^3 + 838*a^3*b^3*c^2*d^4
- 945*a^4*b^2*c*d^5 + 315*a^5*b*d^6 + 128*(25*b^6*c*d^5 + a*b^5*d^6)*x^4 + 16*(135*b^6*c^2*d^4 + 26*a*b^5*c*d
^5 - 9*a^2*b^4*d^6)*x^3 + 8*(5*b^6*c^3*d^3 + 51*a*b^5*c^2*d^4 - 61*a^2*b^4*c*d^5 + 21*a^3*b^3*d^6)*x^2 - 2*(25
*b^6*c^4*d^2 - 20*a*b^5*c^3*d^3 + 262*a^2*b^4*c^2*d^4 - 308*a^3*b^3*c*d^5 + 105*a^4*b^2*d^6)*x)*sqrt(b*x + a)*
sqrt(d*x + c))/(b^6*d^4)]

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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(x**2*(d*x+c)**(5/2)*(b*x+a)**(1/2),x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 2043 vs. \(2 (326) = 652\).
time = 1.48, size = 2043, normalized size = 5.43 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(d*x+c)^(5/2)*(b*x+a)^(1/2),x, algorithm="giac")

[Out]

1/7680*(320*(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*c^2*abs(b)/b^2 + 40*(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))*c^2*abs(b)/b + 80*(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*c*d*abs(b)/b^2 + 8*(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))*sqr
t(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))*c*d*abs(b)/
b + 4*(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*d^2*abs(b)/b^2 + (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))*d^2*abs(b)/b)/b

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(a + b*x)^(1/2)*(c + d*x)^(5/2),x)

[Out]

int(x^2*(a + b*x)^(1/2)*(c + d*x)^(5/2), x)

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