\(\int (b d+2 c d x)^2 (a+b x+c x^2)^{5/2} \, dx\) [1220]

Optimal result

Integrand size = 26, antiderivative size = 207 \[ \int (b d+2 c d x)^2 \left (a+b x+c x^2\right )^{5/2} \, dx=-\frac {5 \left (b^2-4 a c\right )^3 d^2 (b+2 c x) \sqrt {a+b x+c x^2}}{4096 c^3}+\frac {5 \left (b^2-4 a c\right )^2 d^2 (b+2 c x)^3 \sqrt {a+b x+c x^2}}{2048 c^3}-\frac {5 \left (b^2-4 a c\right ) d^2 (b+2 c x)^3 \left (a+b x+c x^2\right )^{3/2}}{384 c^2}+\frac {d^2 (b+2 c x)^3 \left (a+b x+c x^2\right )^{5/2}}{16 c}-\frac {5 \left (b^2-4 a c\right )^4 d^2 \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{8192 c^{7/2}} \]

[Out]

-5/384*(-4*a*c+b^2)*d^2*(2*c*x+b)^3*(c*x^2+b*x+a)^(3/2)/c^2+1/16*d^2*(2*c*x+b)^3*(c*x^2+b*x+a)^(5/2)/c-5/8192*
(-4*a*c+b^2)^4*d^2*arctanh(1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b*x+a)^(1/2))/c^(7/2)-5/4096*(-4*a*c+b^2)^3*d^2*(2*c*x
+b)*(c*x^2+b*x+a)^(1/2)/c^3+5/2048*(-4*a*c+b^2)^2*d^2*(2*c*x+b)^3*(c*x^2+b*x+a)^(1/2)/c^3

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {699, 706, 635, 212} \[ \int (b d+2 c d x)^2 \left (a+b x+c x^2\right )^{5/2} \, dx=-\frac {5 d^2 \left (b^2-4 a c\right )^4 \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{8192 c^{7/2}}-\frac {5 d^2 \left (b^2-4 a c\right )^3 (b+2 c x) \sqrt {a+b x+c x^2}}{4096 c^3}+\frac {5 d^2 \left (b^2-4 a c\right )^2 (b+2 c x)^3 \sqrt {a+b x+c x^2}}{2048 c^3}-\frac {5 d^2 \left (b^2-4 a c\right ) (b+2 c x)^3 \left (a+b x+c x^2\right )^{3/2}}{384 c^2}+\frac {d^2 (b+2 c x)^3 \left (a+b x+c x^2\right )^{5/2}}{16 c} \]

[In]

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

[Out]

(-5*(b^2 - 4*a*c)^3*d^2*(b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(4096*c^3) + (5*(b^2 - 4*a*c)^2*d^2*(b + 2*c*x)^3*S
qrt[a + b*x + c*x^2])/(2048*c^3) - (5*(b^2 - 4*a*c)*d^2*(b + 2*c*x)^3*(a + b*x + c*x^2)^(3/2))/(384*c^2) + (d^
2*(b + 2*c*x)^3*(a + b*x + c*x^2)^(5/2))/(16*c) - (5*(b^2 - 4*a*c)^4*d^2*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a
+ b*x + c*x^2])])/(8192*c^(7/2))

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 635

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(4*c - x^2), x], x, (b + 2*c*x)
/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 699

Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*((
a + b*x + c*x^2)^p/(e*(m + 2*p + 1))), x] - Dist[d*p*((b^2 - 4*a*c)/(b*e*(m + 2*p + 1))), Int[(d + e*x)^m*(a +
b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[2*c*d - b*e, 0] &&
NeQ[m + 2*p + 3, 0] && GtQ[p, 0] &&  !LtQ[m, -1] &&  !(IGtQ[(m - 1)/2, 0] && ( !IntegerQ[p] || LtQ[m, 2*p]))
&& RationalQ[m] && IntegerQ[2*p]

Rule 706

Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[2*d*(d + e*x)^(m - 1
)*((a + b*x + c*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] + Dist[d^2*(m - 1)*((b^2 - 4*a*c)/(b^2*(m + 2*p + 1))), In
t[(d + e*x)^(m - 2)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[
2*c*d - b*e, 0] && NeQ[m + 2*p + 3, 0] && GtQ[m, 1] && NeQ[m + 2*p + 1, 0] && (IntegerQ[2*p] || (IntegerQ[m] &
& RationalQ[p]) || OddQ[m])

Rubi steps \begin{align*} \text {integral}& = \frac {d^2 (b+2 c x)^3 \left (a+b x+c x^2\right )^{5/2}}{16 c}-\frac {\left (5 \left (b^2-4 a c\right )\right ) \int (b d+2 c d x)^2 \left (a+b x+c x^2\right )^{3/2} \, dx}{32 c} \\ & = -\frac {5 \left (b^2-4 a c\right ) d^2 (b+2 c x)^3 \left (a+b x+c x^2\right )^{3/2}}{384 c^2}+\frac {d^2 (b+2 c x)^3 \left (a+b x+c x^2\right )^{5/2}}{16 c}+\frac {\left (5 \left (b^2-4 a c\right )^2\right ) \int (b d+2 c d x)^2 \sqrt {a+b x+c x^2} \, dx}{256 c^2} \\ & = \frac {5 \left (b^2-4 a c\right )^2 d^2 (b+2 c x)^3 \sqrt {a+b x+c x^2}}{2048 c^3}-\frac {5 \left (b^2-4 a c\right ) d^2 (b+2 c x)^3 \left (a+b x+c x^2\right )^{3/2}}{384 c^2}+\frac {d^2 (b+2 c x)^3 \left (a+b x+c x^2\right )^{5/2}}{16 c}-\frac {\left (5 \left (b^2-4 a c\right )^3\right ) \int \frac {(b d+2 c d x)^2}{\sqrt {a+b x+c x^2}} \, dx}{4096 c^3} \\ & = -\frac {5 \left (b^2-4 a c\right )^3 d^2 (b+2 c x) \sqrt {a+b x+c x^2}}{4096 c^3}+\frac {5 \left (b^2-4 a c\right )^2 d^2 (b+2 c x)^3 \sqrt {a+b x+c x^2}}{2048 c^3}-\frac {5 \left (b^2-4 a c\right ) d^2 (b+2 c x)^3 \left (a+b x+c x^2\right )^{3/2}}{384 c^2}+\frac {d^2 (b+2 c x)^3 \left (a+b x+c x^2\right )^{5/2}}{16 c}-\frac {\left (5 \left (b^2-4 a c\right )^4 d^2\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{8192 c^3} \\ & = -\frac {5 \left (b^2-4 a c\right )^3 d^2 (b+2 c x) \sqrt {a+b x+c x^2}}{4096 c^3}+\frac {5 \left (b^2-4 a c\right )^2 d^2 (b+2 c x)^3 \sqrt {a+b x+c x^2}}{2048 c^3}-\frac {5 \left (b^2-4 a c\right ) d^2 (b+2 c x)^3 \left (a+b x+c x^2\right )^{3/2}}{384 c^2}+\frac {d^2 (b+2 c x)^3 \left (a+b x+c x^2\right )^{5/2}}{16 c}-\frac {\left (5 \left (b^2-4 a c\right )^4 d^2\right ) \text {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c x}{\sqrt {a+b x+c x^2}}\right )}{4096 c^3} \\ & = -\frac {5 \left (b^2-4 a c\right )^3 d^2 (b+2 c x) \sqrt {a+b x+c x^2}}{4096 c^3}+\frac {5 \left (b^2-4 a c\right )^2 d^2 (b+2 c x)^3 \sqrt {a+b x+c x^2}}{2048 c^3}-\frac {5 \left (b^2-4 a c\right ) d^2 (b+2 c x)^3 \left (a+b x+c x^2\right )^{3/2}}{384 c^2}+\frac {d^2 (b+2 c x)^3 \left (a+b x+c x^2\right )^{5/2}}{16 c}-\frac {5 \left (b^2-4 a c\right )^4 d^2 \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{8192 c^{7/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 2.15 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.08 \[ \int (b d+2 c d x)^2 \left (a+b x+c x^2\right )^{5/2} \, dx=\frac {d^2 \left (\sqrt {c} (b+2 c x) \sqrt {a+x (b+c x)} \left (15 b^6-40 b^5 c x+44 b^4 c \left (-5 a+2 c x^2\right )+64 b^3 c^2 x \left (9 a+52 c x^2\right )+128 b c^3 x \left (59 a^2+136 a c x^2+72 c^2 x^4\right )+16 b^2 c^2 \left (73 a^2+580 a c x^2+584 c^2 x^4\right )+64 c^3 \left (15 a^3+118 a^2 c x^2+136 a c^2 x^4+48 c^3 x^6\right )\right )-15 \left (b^2-4 a c\right )^4 \text {arctanh}\left (\frac {\sqrt {c} x}{-\sqrt {a}+\sqrt {a+x (b+c x)}}\right )\right )}{12288 c^{7/2}} \]

[In]

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

[Out]

(d^2*(Sqrt[c]*(b + 2*c*x)*Sqrt[a + x*(b + c*x)]*(15*b^6 - 40*b^5*c*x + 44*b^4*c*(-5*a + 2*c*x^2) + 64*b^3*c^2*
x*(9*a + 52*c*x^2) + 128*b*c^3*x*(59*a^2 + 136*a*c*x^2 + 72*c^2*x^4) + 16*b^2*c^2*(73*a^2 + 580*a*c*x^2 + 584*
c^2*x^4) + 64*c^3*(15*a^3 + 118*a^2*c*x^2 + 136*a*c^2*x^4 + 48*c^3*x^6)) - 15*(b^2 - 4*a*c)^4*ArcTanh[(Sqrt[c]
*x)/(-Sqrt[a] + Sqrt[a + x*(b + c*x)])]))/(12288*c^(7/2))

Maple [A] (verified)

Time = 2.70 (sec) , antiderivative size = 294, normalized size of antiderivative = 1.42

method	result	size
risch	\(\frac {\left (6144 c^{7} x^{7}+21504 b \,c^{6} x^{6}+17408 a \,c^{6} x^{5}+27904 b^{2} c^{5} x^{5}+43520 a b \,c^{5} x^{4}+16000 b^{3} c^{4} x^{4}+15104 a^{2} c^{5} x^{3}+35968 a \,b^{2} c^{4} x^{3}+3504 b^{4} c^{3} x^{3}+22656 a^{2} b \,c^{4} x^{2}+10432 a \,b^{3} c^{3} x^{2}+8 b^{5} c^{2} x^{2}+1920 a^{3} c^{4} x +9888 a^{2} b^{2} c^{3} x +136 c^{2} a \,b^{4} x -10 b^{6} c x +960 a^{3} c^{3} b +1168 a^{2} c^{2} b^{3}-220 a \,b^{5} c +15 b^{7}\right ) \sqrt {c \,x^{2}+b x +a}\, d^{2}}{12288 c^{3}}-\frac {5 \left (256 a^{4} c^{4}-256 a^{3} b^{2} c^{3}+96 a^{2} b^{4} c^{2}-16 a \,b^{6} c +b^{8}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) d^{2}}{8192 c^{\frac {7}{2}}}\)	\(294\)
default	\(d^{2} \left (b^{2} \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}{12 c}+\frac {5 \left (4 a c -b^{2}\right ) \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{8 c}+\frac {3 \left (4 a c -b^{2}\right ) \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{24 c}\right )+4 c^{2} \left (\frac {x \left (c \,x^{2}+b x +a \right )^{\frac {7}{2}}}{8 c}-\frac {9 b \left (\frac {\left (c \,x^{2}+b x +a \right )^{\frac {7}{2}}}{7 c}-\frac {b \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}{12 c}+\frac {5 \left (4 a c -b^{2}\right ) \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{8 c}+\frac {3 \left (4 a c -b^{2}\right ) \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{24 c}\right )}{2 c}\right )}{16 c}-\frac {a \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}{12 c}+\frac {5 \left (4 a c -b^{2}\right ) \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{8 c}+\frac {3 \left (4 a c -b^{2}\right ) \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{24 c}\right )}{8 c}\right )+4 b c \left (\frac {\left (c \,x^{2}+b x +a \right )^{\frac {7}{2}}}{7 c}-\frac {b \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}{12 c}+\frac {5 \left (4 a c -b^{2}\right ) \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{8 c}+\frac {3 \left (4 a c -b^{2}\right ) \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{24 c}\right )}{2 c}\right )\right )\)	\(666\)

[In]

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

[Out]

1/12288/c^3*(6144*c^7*x^7+21504*b*c^6*x^6+17408*a*c^6*x^5+27904*b^2*c^5*x^5+43520*a*b*c^5*x^4+16000*b^3*c^4*x^
4+15104*a^2*c^5*x^3+35968*a*b^2*c^4*x^3+3504*b^4*c^3*x^3+22656*a^2*b*c^4*x^2+10432*a*b^3*c^3*x^2+8*b^5*c^2*x^2
+1920*a^3*c^4*x+9888*a^2*b^2*c^3*x+136*a*b^4*c^2*x-10*b^6*c*x+960*a^3*b*c^3+1168*a^2*b^3*c^2-220*a*b^5*c+15*b^
7)*(c*x^2+b*x+a)^(1/2)*d^2-5/8192*(256*a^4*c^4-256*a^3*b^2*c^3+96*a^2*b^4*c^2-16*a*b^6*c+b^8)/c^(7/2)*ln((1/2*
b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*d^2

Fricas [A] (verification not implemented)

none

Time = 0.43 (sec) , antiderivative size = 675, normalized size of antiderivative = 3.26 \[ \int (b d+2 c d x)^2 \left (a+b x+c x^2\right )^{5/2} \, dx=\left [\frac {15 \, {\left (b^{8} - 16 \, a b^{6} c + 96 \, a^{2} b^{4} c^{2} - 256 \, a^{3} b^{2} c^{3} + 256 \, a^{4} c^{4}\right )} \sqrt {c} d^{2} \log \left (-8 \, c^{2} x^{2} - 8 \, b c x - b^{2} + 4 \, \sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {c} - 4 \, a c\right ) + 4 \, {\left (6144 \, c^{8} d^{2} x^{7} + 21504 \, b c^{7} d^{2} x^{6} + 256 \, {\left (109 \, b^{2} c^{6} + 68 \, a c^{7}\right )} d^{2} x^{5} + 640 \, {\left (25 \, b^{3} c^{5} + 68 \, a b c^{6}\right )} d^{2} x^{4} + 16 \, {\left (219 \, b^{4} c^{4} + 2248 \, a b^{2} c^{5} + 944 \, a^{2} c^{6}\right )} d^{2} x^{3} + 8 \, {\left (b^{5} c^{3} + 1304 \, a b^{3} c^{4} + 2832 \, a^{2} b c^{5}\right )} d^{2} x^{2} - 2 \, {\left (5 \, b^{6} c^{2} - 68 \, a b^{4} c^{3} - 4944 \, a^{2} b^{2} c^{4} - 960 \, a^{3} c^{5}\right )} d^{2} x + {\left (15 \, b^{7} c - 220 \, a b^{5} c^{2} + 1168 \, a^{2} b^{3} c^{3} + 960 \, a^{3} b c^{4}\right )} d^{2}\right )} \sqrt {c x^{2} + b x + a}}{49152 \, c^{4}}, \frac {15 \, {\left (b^{8} - 16 \, a b^{6} c + 96 \, a^{2} b^{4} c^{2} - 256 \, a^{3} b^{2} c^{3} + 256 \, a^{4} c^{4}\right )} \sqrt {-c} d^{2} \arctan \left (\frac {\sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{2} + b c x + a c\right )}}\right ) + 2 \, {\left (6144 \, c^{8} d^{2} x^{7} + 21504 \, b c^{7} d^{2} x^{6} + 256 \, {\left (109 \, b^{2} c^{6} + 68 \, a c^{7}\right )} d^{2} x^{5} + 640 \, {\left (25 \, b^{3} c^{5} + 68 \, a b c^{6}\right )} d^{2} x^{4} + 16 \, {\left (219 \, b^{4} c^{4} + 2248 \, a b^{2} c^{5} + 944 \, a^{2} c^{6}\right )} d^{2} x^{3} + 8 \, {\left (b^{5} c^{3} + 1304 \, a b^{3} c^{4} + 2832 \, a^{2} b c^{5}\right )} d^{2} x^{2} - 2 \, {\left (5 \, b^{6} c^{2} - 68 \, a b^{4} c^{3} - 4944 \, a^{2} b^{2} c^{4} - 960 \, a^{3} c^{5}\right )} d^{2} x + {\left (15 \, b^{7} c - 220 \, a b^{5} c^{2} + 1168 \, a^{2} b^{3} c^{3} + 960 \, a^{3} b c^{4}\right )} d^{2}\right )} \sqrt {c x^{2} + b x + a}}{24576 \, c^{4}}\right ] \]

[In]

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

[Out]

[1/49152*(15*(b^8 - 16*a*b^6*c + 96*a^2*b^4*c^2 - 256*a^3*b^2*c^3 + 256*a^4*c^4)*sqrt(c)*d^2*log(-8*c^2*x^2 -
8*b*c*x - b^2 + 4*sqrt(c*x^2 + b*x + a)*(2*c*x + b)*sqrt(c) - 4*a*c) + 4*(6144*c^8*d^2*x^7 + 21504*b*c^7*d^2*x
^6 + 256*(109*b^2*c^6 + 68*a*c^7)*d^2*x^5 + 640*(25*b^3*c^5 + 68*a*b*c^6)*d^2*x^4 + 16*(219*b^4*c^4 + 2248*a*b
^2*c^5 + 944*a^2*c^6)*d^2*x^3 + 8*(b^5*c^3 + 1304*a*b^3*c^4 + 2832*a^2*b*c^5)*d^2*x^2 - 2*(5*b^6*c^2 - 68*a*b^
4*c^3 - 4944*a^2*b^2*c^4 - 960*a^3*c^5)*d^2*x + (15*b^7*c - 220*a*b^5*c^2 + 1168*a^2*b^3*c^3 + 960*a^3*b*c^4)*
d^2)*sqrt(c*x^2 + b*x + a))/c^4, 1/24576*(15*(b^8 - 16*a*b^6*c + 96*a^2*b^4*c^2 - 256*a^3*b^2*c^3 + 256*a^4*c^
4)*sqrt(-c)*d^2*arctan(1/2*sqrt(c*x^2 + b*x + a)*(2*c*x + b)*sqrt(-c)/(c^2*x^2 + b*c*x + a*c)) + 2*(6144*c^8*d
^2*x^7 + 21504*b*c^7*d^2*x^6 + 256*(109*b^2*c^6 + 68*a*c^7)*d^2*x^5 + 640*(25*b^3*c^5 + 68*a*b*c^6)*d^2*x^4 +
16*(219*b^4*c^4 + 2248*a*b^2*c^5 + 944*a^2*c^6)*d^2*x^3 + 8*(b^5*c^3 + 1304*a*b^3*c^4 + 2832*a^2*b*c^5)*d^2*x^
2 - 2*(5*b^6*c^2 - 68*a*b^4*c^3 - 4944*a^2*b^2*c^4 - 960*a^3*c^5)*d^2*x + (15*b^7*c - 220*a*b^5*c^2 + 1168*a^2
*b^3*c^3 + 960*a^3*b*c^4)*d^2)*sqrt(c*x^2 + b*x + a))/c^4]

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2958 vs. \(2 (202) = 404\).

Time = 0.81 (sec) , antiderivative size = 2958, normalized size of antiderivative = 14.29 \[ \int (b d+2 c d x)^2 \left (a+b x+c x^2\right )^{5/2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Piecewise((sqrt(a + b*x + c*x**2)*(7*b*c**3*d**2*x**6/4 + c**4*d**2*x**7/2 + x**5*(17*a*c**4*d**2/2 + 109*b**2
*c**3*d**2/8)/(6*c) + x**4*(51*a*b*c**3*d**2/2 + 19*b**3*c**2*d**2 - 11*b*(17*a*c**4*d**2/2 + 109*b**2*c**3*d*
*2/8)/(12*c))/(5*c) + x**3*(12*a**2*c**3*d**2 + 39*a*b**2*c**2*d**2 - 5*a*(17*a*c**4*d**2/2 + 109*b**2*c**3*d*
*2/8)/(6*c) + 7*b**4*c*d**2 - 9*b*(51*a*b*c**3*d**2/2 + 19*b**3*c**2*d**2 - 11*b*(17*a*c**4*d**2/2 + 109*b**2*
c**3*d**2/8)/(12*c))/(10*c))/(4*c) + x**2*(24*a**2*b*c**2*d**2 + 18*a*b**3*c*d**2 - 4*a*(51*a*b*c**3*d**2/2 +
19*b**3*c**2*d**2 - 11*b*(17*a*c**4*d**2/2 + 109*b**2*c**3*d**2/8)/(12*c))/(5*c) + b**5*d**2 - 7*b*(12*a**2*c*
*3*d**2 + 39*a*b**2*c**2*d**2 - 5*a*(17*a*c**4*d**2/2 + 109*b**2*c**3*d**2/8)/(6*c) + 7*b**4*c*d**2 - 9*b*(51*
a*b*c**3*d**2/2 + 19*b**3*c**2*d**2 - 11*b*(17*a*c**4*d**2/2 + 109*b**2*c**3*d**2/8)/(12*c))/(10*c))/(8*c))/(3
*c) + x*(4*a**3*c**2*d**2 + 15*a**2*b**2*c*d**2 + 3*a*b**4*d**2 - 3*a*(12*a**2*c**3*d**2 + 39*a*b**2*c**2*d**2
- 5*a*(17*a*c**4*d**2/2 + 109*b**2*c**3*d**2/8)/(6*c) + 7*b**4*c*d**2 - 9*b*(51*a*b*c**3*d**2/2 + 19*b**3*c**
2*d**2 - 11*b*(17*a*c**4*d**2/2 + 109*b**2*c**3*d**2/8)/(12*c))/(10*c))/(4*c) - 5*b*(24*a**2*b*c**2*d**2 + 18*
a*b**3*c*d**2 - 4*a*(51*a*b*c**3*d**2/2 + 19*b**3*c**2*d**2 - 11*b*(17*a*c**4*d**2/2 + 109*b**2*c**3*d**2/8)/(
12*c))/(5*c) + b**5*d**2 - 7*b*(12*a**2*c**3*d**2 + 39*a*b**2*c**2*d**2 - 5*a*(17*a*c**4*d**2/2 + 109*b**2*c**
3*d**2/8)/(6*c) + 7*b**4*c*d**2 - 9*b*(51*a*b*c**3*d**2/2 + 19*b**3*c**2*d**2 - 11*b*(17*a*c**4*d**2/2 + 109*b
**2*c**3*d**2/8)/(12*c))/(10*c))/(8*c))/(6*c))/(2*c) + (4*a**3*b*c*d**2 + 3*a**2*b**3*d**2 - 2*a*(24*a**2*b*c*
*2*d**2 + 18*a*b**3*c*d**2 - 4*a*(51*a*b*c**3*d**2/2 + 19*b**3*c**2*d**2 - 11*b*(17*a*c**4*d**2/2 + 109*b**2*c
**3*d**2/8)/(12*c))/(5*c) + b**5*d**2 - 7*b*(12*a**2*c**3*d**2 + 39*a*b**2*c**2*d**2 - 5*a*(17*a*c**4*d**2/2 +
109*b**2*c**3*d**2/8)/(6*c) + 7*b**4*c*d**2 - 9*b*(51*a*b*c**3*d**2/2 + 19*b**3*c**2*d**2 - 11*b*(17*a*c**4*d
**2/2 + 109*b**2*c**3*d**2/8)/(12*c))/(10*c))/(8*c))/(3*c) - 3*b*(4*a**3*c**2*d**2 + 15*a**2*b**2*c*d**2 + 3*a
*b**4*d**2 - 3*a*(12*a**2*c**3*d**2 + 39*a*b**2*c**2*d**2 - 5*a*(17*a*c**4*d**2/2 + 109*b**2*c**3*d**2/8)/(6*c
) + 7*b**4*c*d**2 - 9*b*(51*a*b*c**3*d**2/2 + 19*b**3*c**2*d**2 - 11*b*(17*a*c**4*d**2/2 + 109*b**2*c**3*d**2/
8)/(12*c))/(10*c))/(4*c) - 5*b*(24*a**2*b*c**2*d**2 + 18*a*b**3*c*d**2 - 4*a*(51*a*b*c**3*d**2/2 + 19*b**3*c**
2*d**2 - 11*b*(17*a*c**4*d**2/2 + 109*b**2*c**3*d**2/8)/(12*c))/(5*c) + b**5*d**2 - 7*b*(12*a**2*c**3*d**2 + 3
9*a*b**2*c**2*d**2 - 5*a*(17*a*c**4*d**2/2 + 109*b**2*c**3*d**2/8)/(6*c) + 7*b**4*c*d**2 - 9*b*(51*a*b*c**3*d*
*2/2 + 19*b**3*c**2*d**2 - 11*b*(17*a*c**4*d**2/2 + 109*b**2*c**3*d**2/8)/(12*c))/(10*c))/(8*c))/(6*c))/(4*c))
/c) + (a**3*b**2*d**2 - a*(4*a**3*c**2*d**2 + 15*a**2*b**2*c*d**2 + 3*a*b**4*d**2 - 3*a*(12*a**2*c**3*d**2 + 3
9*a*b**2*c**2*d**2 - 5*a*(17*a*c**4*d**2/2 + 109*b**2*c**3*d**2/8)/(6*c) + 7*b**4*c*d**2 - 9*b*(51*a*b*c**3*d*
*2/2 + 19*b**3*c**2*d**2 - 11*b*(17*a*c**4*d**2/2 + 109*b**2*c**3*d**2/8)/(12*c))/(10*c))/(4*c) - 5*b*(24*a**2
*b*c**2*d**2 + 18*a*b**3*c*d**2 - 4*a*(51*a*b*c**3*d**2/2 + 19*b**3*c**2*d**2 - 11*b*(17*a*c**4*d**2/2 + 109*b
**2*c**3*d**2/8)/(12*c))/(5*c) + b**5*d**2 - 7*b*(12*a**2*c**3*d**2 + 39*a*b**2*c**2*d**2 - 5*a*(17*a*c**4*d**
2/2 + 109*b**2*c**3*d**2/8)/(6*c) + 7*b**4*c*d**2 - 9*b*(51*a*b*c**3*d**2/2 + 19*b**3*c**2*d**2 - 11*b*(17*a*c
**4*d**2/2 + 109*b**2*c**3*d**2/8)/(12*c))/(10*c))/(8*c))/(6*c))/(2*c) - b*(4*a**3*b*c*d**2 + 3*a**2*b**3*d**2
- 2*a*(24*a**2*b*c**2*d**2 + 18*a*b**3*c*d**2 - 4*a*(51*a*b*c**3*d**2/2 + 19*b**3*c**2*d**2 - 11*b*(17*a*c**4
*d**2/2 + 109*b**2*c**3*d**2/8)/(12*c))/(5*c) + b**5*d**2 - 7*b*(12*a**2*c**3*d**2 + 39*a*b**2*c**2*d**2 - 5*a
*(17*a*c**4*d**2/2 + 109*b**2*c**3*d**2/8)/(6*c) + 7*b**4*c*d**2 - 9*b*(51*a*b*c**3*d**2/2 + 19*b**3*c**2*d**2
- 11*b*(17*a*c**4*d**2/2 + 109*b**2*c**3*d**2/8)/(12*c))/(10*c))/(8*c))/(3*c) - 3*b*(4*a**3*c**2*d**2 + 15*a*
*2*b**2*c*d**2 + 3*a*b**4*d**2 - 3*a*(12*a**2*c**3*d**2 + 39*a*b**2*c**2*d**2 - 5*a*(17*a*c**4*d**2/2 + 109*b*
*2*c**3*d**2/8)/(6*c) + 7*b**4*c*d**2 - 9*b*(51*a*b*c**3*d**2/2 + 19*b**3*c**2*d**2 - 11*b*(17*a*c**4*d**2/2 +
109*b**2*c**3*d**2/8)/(12*c))/(10*c))/(4*c) - 5*b*(24*a**2*b*c**2*d**2 + 18*a*b**3*c*d**2 - 4*a*(51*a*b*c**3*
d**2/2 + 19*b**3*c**2*d**2 - 11*b*(17*a*c**4*d**2/2 + 109*b**2*c**3*d**2/8)/(12*c))/(5*c) + b**5*d**2 - 7*b*(1
2*a**2*c**3*d**2 + 39*a*b**2*c**2*d**2 - 5*a*(17*a*c**4*d**2/2 + 109*b**2*c**3*d**2/8)/(6*c) + 7*b**4*c*d**2 -
9*b*(51*a*b*c**3*d**2/2 + 19*b**3*c**2*d**2 - 11*b*(17*a*c**4*d**2/2 + 109*b**2*c**3*d**2/8)/(12*c))/(10*c))/
(8*c))/(6*c))/(4*c))/(2*c))*Piecewise((log(b + 2*sqrt(c)*sqrt(a + b*x + c*x**2) + 2*c*x)/sqrt(c), Ne(a - b**2/
(4*c), 0)), ((b/(2*c) + x)*log(b/(2*c) + x)/sqrt(c*(b/(2*c) + x)**2), True)), Ne(c, 0)), (2*(4*c**2*d**2*(a +
b*x)**(11/2)/(11*b**2) + (a + b*x)**(9/2)*(-8*a*c**2*d**2 + 4*b**2*c*d**2)/(9*b**2) + (a + b*x)**(7/2)*(4*a**2
*c**2*d**2 - 4*a*b**2*c*d**2 + b**4*d**2)/(7*b**2))/b, Ne(b, 0)), (4*a**(5/2)*c**2*d**2*x**3/3, True))

Maxima [F(-2)]

Exception generated. \[ \int (b d+2 c d x)^2 \left (a+b x+c x^2\right )^{5/2} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((2*c*d*x+b*d)^2*(c*x^2+b*x+a)^(5/2),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(4*a*c-b^2>0)', see `assume?` f
or more deta

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 387 vs. \(2 (181) = 362\).

Time = 0.29 (sec) , antiderivative size = 387, normalized size of antiderivative = 1.87 \[ \int (b d+2 c d x)^2 \left (a+b x+c x^2\right )^{5/2} \, dx=\frac {1}{12288} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (2 \, {\left (12 \, {\left (2 \, c^{4} d^{2} x + 7 \, b c^{3} d^{2}\right )} x + \frac {109 \, b^{2} c^{9} d^{2} + 68 \, a c^{10} d^{2}}{c^{7}}\right )} x + \frac {5 \, {\left (25 \, b^{3} c^{8} d^{2} + 68 \, a b c^{9} d^{2}\right )}}{c^{7}}\right )} x + \frac {219 \, b^{4} c^{7} d^{2} + 2248 \, a b^{2} c^{8} d^{2} + 944 \, a^{2} c^{9} d^{2}}{c^{7}}\right )} x + \frac {b^{5} c^{6} d^{2} + 1304 \, a b^{3} c^{7} d^{2} + 2832 \, a^{2} b c^{8} d^{2}}{c^{7}}\right )} x - \frac {5 \, b^{6} c^{5} d^{2} - 68 \, a b^{4} c^{6} d^{2} - 4944 \, a^{2} b^{2} c^{7} d^{2} - 960 \, a^{3} c^{8} d^{2}}{c^{7}}\right )} x + \frac {15 \, b^{7} c^{4} d^{2} - 220 \, a b^{5} c^{5} d^{2} + 1168 \, a^{2} b^{3} c^{6} d^{2} + 960 \, a^{3} b c^{7} d^{2}}{c^{7}}\right )} + \frac {5 \, {\left (b^{8} d^{2} - 16 \, a b^{6} c d^{2} + 96 \, a^{2} b^{4} c^{2} d^{2} - 256 \, a^{3} b^{2} c^{3} d^{2} + 256 \, a^{4} c^{4} d^{2}\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} + b \right |}\right )}{8192 \, c^{\frac {7}{2}}} \]

[In]

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

[Out]

1/12288*sqrt(c*x^2 + b*x + a)*(2*(4*(2*(8*(2*(12*(2*c^4*d^2*x + 7*b*c^3*d^2)*x + (109*b^2*c^9*d^2 + 68*a*c^10*
d^2)/c^7)*x + 5*(25*b^3*c^8*d^2 + 68*a*b*c^9*d^2)/c^7)*x + (219*b^4*c^7*d^2 + 2248*a*b^2*c^8*d^2 + 944*a^2*c^9
*d^2)/c^7)*x + (b^5*c^6*d^2 + 1304*a*b^3*c^7*d^2 + 2832*a^2*b*c^8*d^2)/c^7)*x - (5*b^6*c^5*d^2 - 68*a*b^4*c^6*
d^2 - 4944*a^2*b^2*c^7*d^2 - 960*a^3*c^8*d^2)/c^7)*x + (15*b^7*c^4*d^2 - 220*a*b^5*c^5*d^2 + 1168*a^2*b^3*c^6*
d^2 + 960*a^3*b*c^7*d^2)/c^7) + 5/8192*(b^8*d^2 - 16*a*b^6*c*d^2 + 96*a^2*b^4*c^2*d^2 - 256*a^3*b^2*c^3*d^2 +
256*a^4*c^4*d^2)*log(abs(2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) + b))/c^(7/2)

Mupad [F(-1)]

Timed out. \[ \int (b d+2 c d x)^2 \left (a+b x+c x^2\right )^{5/2} \, dx=\int {\left (b\,d+2\,c\,d\,x\right )}^2\,{\left (c\,x^2+b\,x+a\right )}^{5/2} \,d x \]

[In]

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

[Out]

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