3.24.44 \(\int (d+e x)^2 (a+b x+c x^2)^{3/2} \, dx\) [2344]

Optimal. Leaf size=257 \[ -\frac {\left (b^2-4 a c\right ) \left (24 c^2 d^2+7 b^2 e^2-4 c e (6 b d+a e)\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{512 c^4}+\frac {\left (24 c^2 d^2+7 b^2 e^2-4 c e (6 b d+a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{192 c^3}+\frac {7 e (2 c d-b e) \left (a+b x+c x^2\right )^{5/2}}{60 c^2}+\frac {e (d+e x) \left (a+b x+c x^2\right )^{5/2}}{6 c}+\frac {\left (b^2-4 a c\right )^2 \left (24 c^2 d^2+7 b^2 e^2-4 c e (6 b d+a e)\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{1024 c^{9/2}} \]

[Out]

1/192*(24*c^2*d^2+7*b^2*e^2-4*c*e*(a*e+6*b*d))*(2*c*x+b)*(c*x^2+b*x+a)^(3/2)/c^3+7/60*e*(-b*e+2*c*d)*(c*x^2+b*
x+a)^(5/2)/c^2+1/6*e*(e*x+d)*(c*x^2+b*x+a)^(5/2)/c+1/1024*(-4*a*c+b^2)^2*(24*c^2*d^2+7*b^2*e^2-4*c*e*(a*e+6*b*
d))*arctanh(1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b*x+a)^(1/2))/c^(9/2)-1/512*(-4*a*c+b^2)*(24*c^2*d^2+7*b^2*e^2-4*c*e*
(a*e+6*b*d))*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/c^4

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Rubi [A]
time = 0.21, antiderivative size = 257, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {756, 654, 626, 635, 212} \begin {gather*} \frac {\left (b^2-4 a c\right )^2 \left (-4 c e (a e+6 b d)+7 b^2 e^2+24 c^2 d^2\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{1024 c^{9/2}}-\frac {\left (b^2-4 a c\right ) (b+2 c x) \sqrt {a+b x+c x^2} \left (-4 c e (a e+6 b d)+7 b^2 e^2+24 c^2 d^2\right )}{512 c^4}+\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2} \left (-4 c e (a e+6 b d)+7 b^2 e^2+24 c^2 d^2\right )}{192 c^3}+\frac {7 e \left (a+b x+c x^2\right )^{5/2} (2 c d-b e)}{60 c^2}+\frac {e (d+e x) \left (a+b x+c x^2\right )^{5/2}}{6 c} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/512*((b^2 - 4*a*c)*(24*c^2*d^2 + 7*b^2*e^2 - 4*c*e*(6*b*d + a*e))*(b + 2*c*x)*Sqrt[a + b*x + c*x^2])/c^4 +
((24*c^2*d^2 + 7*b^2*e^2 - 4*c*e*(6*b*d + a*e))*(b + 2*c*x)*(a + b*x + c*x^2)^(3/2))/(192*c^3) + (7*e*(2*c*d -
b*e)*(a + b*x + c*x^2)^(5/2))/(60*c^2) + (e*(d + e*x)*(a + b*x + c*x^2)^(5/2))/(6*c) + ((b^2 - 4*a*c)^2*(24*c
^2*d^2 + 7*b^2*e^2 - 4*c*e*(6*b*d + a*e))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(1024*c^(9/2
))

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 626

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

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 654

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

Rule 756

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

Rubi steps

\begin {align*} \int (d+e x)^2 \left (a+b x+c x^2\right )^{3/2} \, dx &=\frac {e (d+e x) \left (a+b x+c x^2\right )^{5/2}}{6 c}+\frac {\int \left (\frac {1}{2} \left (12 c d^2-2 e \left (\frac {5 b d}{2}+a e\right )\right )+\frac {7}{2} e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{3/2} \, dx}{6 c}\\ &=\frac {7 e (2 c d-b e) \left (a+b x+c x^2\right )^{5/2}}{60 c^2}+\frac {e (d+e x) \left (a+b x+c x^2\right )^{5/2}}{6 c}+\frac {\left (-\frac {7}{2} b e (2 c d-b e)+c \left (12 c d^2-2 e \left (\frac {5 b d}{2}+a e\right )\right )\right ) \int \left (a+b x+c x^2\right )^{3/2} \, dx}{12 c^2}\\ &=\frac {\left (24 c^2 d^2+7 b^2 e^2-4 c e (6 b d+a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{192 c^3}+\frac {7 e (2 c d-b e) \left (a+b x+c x^2\right )^{5/2}}{60 c^2}+\frac {e (d+e x) \left (a+b x+c x^2\right )^{5/2}}{6 c}-\frac {\left (\left (b^2-4 a c\right ) \left (24 c^2 d^2+7 b^2 e^2-4 c e (6 b d+a e)\right )\right ) \int \sqrt {a+b x+c x^2} \, dx}{128 c^3}\\ &=-\frac {\left (b^2-4 a c\right ) \left (24 c^2 d^2+7 b^2 e^2-4 c e (6 b d+a e)\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{512 c^4}+\frac {\left (24 c^2 d^2+7 b^2 e^2-4 c e (6 b d+a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{192 c^3}+\frac {7 e (2 c d-b e) \left (a+b x+c x^2\right )^{5/2}}{60 c^2}+\frac {e (d+e x) \left (a+b x+c x^2\right )^{5/2}}{6 c}+\frac {\left (\left (b^2-4 a c\right )^2 \left (24 c^2 d^2+7 b^2 e^2-4 c e (6 b d+a e)\right )\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{1024 c^4}\\ &=-\frac {\left (b^2-4 a c\right ) \left (24 c^2 d^2+7 b^2 e^2-4 c e (6 b d+a e)\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{512 c^4}+\frac {\left (24 c^2 d^2+7 b^2 e^2-4 c e (6 b d+a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{192 c^3}+\frac {7 e (2 c d-b e) \left (a+b x+c x^2\right )^{5/2}}{60 c^2}+\frac {e (d+e x) \left (a+b x+c x^2\right )^{5/2}}{6 c}+\frac {\left (\left (b^2-4 a c\right )^2 \left (24 c^2 d^2+7 b^2 e^2-4 c e (6 b d+a e)\right )\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 )}{512 c^4}\\ &=-\frac {\left (b^2-4 a c\right ) \left (24 c^2 d^2+7 b^2 e^2-4 c e (6 b d+a e)\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{512 c^4}+\frac {\left (24 c^2 d^2+7 b^2 e^2-4 c e (6 b d+a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{192 c^3}+\frac {7 e (2 c d-b e) \left (a+b x+c x^2\right )^{5/2}}{60 c^2}+\frac {e (d+e x) \left (a+b x+c x^2\right )^{5/2}}{6 c}+\frac {\left (b^2-4 a c\right )^2 \left (24 c^2 d^2+7 b^2 e^2-4 c e (6 b d+a e)\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{1024 c^{9/2}}\\ \end {align*}

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Mathematica [A]
time = 1.21, size = 334, normalized size = 1.30 \begin {gather*} \frac {2 \sqrt {c} \sqrt {a+x (b+c x)} \left (-105 b^5 e^2+10 b^4 c e (36 d+7 e x)-48 b^2 c^2 \left (a e (50 d+9 e x)-c x \left (5 d^2+4 d e x+e^2 x^2\right )\right )-8 b^3 c \left (-95 a e^2+c \left (45 d^2+30 d e x+7 e^2 x^2\right )\right )+16 b c^2 \left (-81 a^2 e^2+6 a c \left (25 d^2+14 d e x+3 e^2 x^2\right )+4 c^2 x^2 \left (45 d^2+66 d e x+26 e^2 x^2\right )\right )+32 c^3 \left (3 a^2 e (32 d+5 e x)+4 c^2 x^3 \left (15 d^2+24 d e x+10 e^2 x^2\right )+2 a c x \left (75 d^2+96 d e x+35 e^2 x^2\right )\right )\right )-15 \left (b^2-4 a c\right )^2 \left (24 c^2 d^2+7 b^2 e^2-4 c e (6 b d+a e)\right ) \log \left (b+2 c x-2 \sqrt {c} \sqrt {a+x (b+c x)}\right )}{15360 c^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[c]*Sqrt[a + x*(b + c*x)]*(-105*b^5*e^2 + 10*b^4*c*e*(36*d + 7*e*x) - 48*b^2*c^2*(a*e*(50*d + 9*e*x) -
c*x*(5*d^2 + 4*d*e*x + e^2*x^2)) - 8*b^3*c*(-95*a*e^2 + c*(45*d^2 + 30*d*e*x + 7*e^2*x^2)) + 16*b*c^2*(-81*a^2
*e^2 + 6*a*c*(25*d^2 + 14*d*e*x + 3*e^2*x^2) + 4*c^2*x^2*(45*d^2 + 66*d*e*x + 26*e^2*x^2)) + 32*c^3*(3*a^2*e*(
32*d + 5*e*x) + 4*c^2*x^3*(15*d^2 + 24*d*e*x + 10*e^2*x^2) + 2*a*c*x*(75*d^2 + 96*d*e*x + 35*e^2*x^2))) - 15*(
b^2 - 4*a*c)^2*(24*c^2*d^2 + 7*b^2*e^2 - 4*c*e*(6*b*d + a*e))*Log[b + 2*c*x - 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]]
)/(15360*c^(9/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(504\) vs. \(2(231)=462\).
time = 0.79, size = 505, normalized size = 1.96

method	result	size
default	\(e^{2} \left (\frac {x \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}{6 c}-\frac {7 b \left (\frac {\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}{5 c}-\frac {b \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{8 c}+\frac {3 \left (4 a c -b^{2}\right ) \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{2 c}\right )}{12 c}-\frac {a \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{8 c}+\frac {3 \left (4 a c -b^{2}\right ) \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{6 c}\right )+2 d e \left (\frac {\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}{5 c}-\frac {b \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{8 c}+\frac {3 \left (4 a c -b^{2}\right ) \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{2 c}\right )+d^{2} \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{8 c}+\frac {3 \left (4 a c -b^{2}\right ) \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )\)	\(505\)
risch	\(-\frac {\left (-1280 e^{2} c^{5} x^{5}-1664 b \,c^{4} e^{2} x^{4}-3072 c^{5} d e \,x^{4}-2240 a \,c^{4} e^{2} x^{3}-48 b^{2} c^{3} e^{2} x^{3}-4224 b \,c^{4} d e \,x^{3}-1920 c^{5} d^{2} x^{3}-288 a b \,c^{3} e^{2} x^{2}-6144 a \,c^{4} d e \,x^{2}+56 b^{3} c^{2} e^{2} x^{2}-192 b^{2} c^{3} d e \,x^{2}-2880 c^{4} b \,d^{2} x^{2}-480 a^{2} c^{3} e^{2} x +432 a \,b^{2} c^{2} e^{2} x -1344 a b \,c^{3} d e x -4800 a \,c^{4} d^{2} x -70 b^{4} c \,e^{2} x +240 b^{3} c^{2} d e x -240 b^{2} d^{2} c^{3} x +1296 a^{2} b \,c^{2} e^{2}-3072 a^{2} c^{3} d e -760 a \,b^{3} c \,e^{2}+2400 a \,b^{2} c^{2} d e -2400 a b \,c^{3} d^{2}+105 b^{5} e^{2}-360 b^{4} c d e +360 b^{3} c^{2} d^{2}\right ) \sqrt {c \,x^{2}+b x +a}}{7680 c^{4}}-\frac {\ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) a^{3} e^{2}}{16 c^{\frac {3}{2}}}+\frac {9 \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) a^{2} b^{2} e^{2}}{64 c^{\frac {5}{2}}}-\frac {3 \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) a^{2} b d e}{8 c^{\frac {3}{2}}}+\frac {3 \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) a^{2} d^{2}}{8 \sqrt {c}}-\frac {15 \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) a \,b^{4} e^{2}}{256 c^{\frac {7}{2}}}+\frac {3 \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) a \,b^{3} d e}{16 c^{\frac {5}{2}}}-\frac {3 \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) a \,b^{2} d^{2}}{16 c^{\frac {3}{2}}}+\frac {7 \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) b^{6} e^{2}}{1024 c^{\frac {9}{2}}}-\frac {3 \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) b^{5} d e}{128 c^{\frac {7}{2}}}+\frac {3 \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) b^{4} d^{2}}{128 c^{\frac {5}{2}}}\)	\(686\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^2*(c*x^2+b*x+a)^(3/2),x,method=_RETURNVERBOSE)

[Out]

e^2*(1/6*x*(c*x^2+b*x+a)^(5/2)/c-7/12*b/c*(1/5*(c*x^2+b*x+a)^(5/2)/c-1/2*b/c*(1/8*(2*c*x+b)*(c*x^2+b*x+a)^(3/2
)/c+3/16*(4*a*c-b^2)/c*(1/4*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/c+1/8*(4*a*c-b^2)/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*
x^2+b*x+a)^(1/2)))))-1/6*a/c*(1/8*(2*c*x+b)*(c*x^2+b*x+a)^(3/2)/c+3/16*(4*a*c-b^2)/c*(1/4*(2*c*x+b)*(c*x^2+b*x
+a)^(1/2)/c+1/8*(4*a*c-b^2)/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2)))))+2*d*e*(1/5*(c*x^2+b*x+a)^(5
/2)/c-1/2*b/c*(1/8*(2*c*x+b)*(c*x^2+b*x+a)^(3/2)/c+3/16*(4*a*c-b^2)/c*(1/4*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/c+1/8
*(4*a*c-b^2)/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2)))))+d^2*(1/8*(2*c*x+b)*(c*x^2+b*x+a)^(3/2)/c+3
/16*(4*a*c-b^2)/c*(1/4*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/c+1/8*(4*a*c-b^2)/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b
*x+a)^(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((e*x+d)^2*(c*x^2+b*x+a)^(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(4*a*c-b^2>0)', see `assume?` f
or more deta

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Fricas [A]
time = 3.13, size = 877, normalized size = 3.41 \begin {gather*} \left [-\frac {15 \, {\left (24 \, {\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} d^{2} - 24 \, {\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} d e + {\left (7 \, b^{6} - 60 \, a b^{4} c + 144 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} e^{2}\right )} \sqrt {c} \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 (1920 \, c^{6} d^{2} x^{3} + 2880 \, b c^{5} d^{2} x^{2} + 240 \, {\left (b^{2} c^{4} + 20 \, a c^{5}\right )} d^{2} x - 120 \, {\left (3 \, b^{3} c^{3} - 20 \, a b c^{4}\right )} d^{2} + {\left (1280 \, c^{6} x^{5} + 1664 \, b c^{5} x^{4} - 105 \, b^{5} c + 760 \, a b^{3} c^{2} - 1296 \, a^{2} b c^{3} + 16 \, {\left (3 \, b^{2} c^{4} + 140 \, a c^{5}\right )} x^{3} - 8 \, {\left (7 \, b^{3} c^{3} - 36 \, a b c^{4}\right )} x^{2} + 2 \, {\left (35 \, b^{4} c^{2} - 216 \, a b^{2} c^{3} + 240 \, a^{2} c^{4}\right )} x\right )} e^{2} + 24 \, {\left (128 \, c^{6} d x^{4} + 176 \, b c^{5} d x^{3} + 8 \, {\left (b^{2} c^{4} + 32 \, a c^{5}\right )} d x^{2} - 2 \, {\left (5 \, b^{3} c^{3} - 28 \, a b c^{4}\right )} d x + {\left (15 \, b^{4} c^{2} - 100 \, a b^{2} c^{3} + 128 \, a^{2} c^{4}\right )} d\right )} e\right )} \sqrt {c x^{2} + b x + a}}{30720 \, c^{5}}, -\frac {15 \, {\left (24 \, {\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} d^{2} - 24 \, {\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} d e + {\left (7 \, b^{6} - 60 \, a b^{4} c + 144 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} e^{2}\right )} \sqrt {-c} \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 (1920 \, c^{6} d^{2} x^{3} + 2880 \, b c^{5} d^{2} x^{2} + 240 \, {\left (b^{2} c^{4} + 20 \, a c^{5}\right )} d^{2} x - 120 \, {\left (3 \, b^{3} c^{3} - 20 \, a b c^{4}\right )} d^{2} + {\left (1280 \, c^{6} x^{5} + 1664 \, b c^{5} x^{4} - 105 \, b^{5} c + 760 \, a b^{3} c^{2} - 1296 \, a^{2} b c^{3} + 16 \, {\left (3 \, b^{2} c^{4} + 140 \, a c^{5}\right )} x^{3} - 8 \, {\left (7 \, b^{3} c^{3} - 36 \, a b c^{4}\right )} x^{2} + 2 \, {\left (35 \, b^{4} c^{2} - 216 \, a b^{2} c^{3} + 240 \, a^{2} c^{4}\right )} x\right )} e^{2} + 24 \, {\left (128 \, c^{6} d x^{4} + 176 \, b c^{5} d x^{3} + 8 \, {\left (b^{2} c^{4} + 32 \, a c^{5}\right )} d x^{2} - 2 \, {\left (5 \, b^{3} c^{3} - 28 \, a b c^{4}\right )} d x + {\left (15 \, b^{4} c^{2} - 100 \, a b^{2} c^{3} + 128 \, a^{2} c^{4}\right )} d\right )} e\right )} \sqrt {c x^{2} + b x + a}}{15360 \, c^{5}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/30720*(15*(24*(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d^2 - 24*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d*e + (7*
b^6 - 60*a*b^4*c + 144*a^2*b^2*c^2 - 64*a^3*c^3)*e^2)*sqrt(c)*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*(1920*c^6*d^2*x^3 + 2880*b*c^5*d^2*x^2 + 240*(b^2*c^4 + 20*a*c^5)*d^
2*x - 120*(3*b^3*c^3 - 20*a*b*c^4)*d^2 + (1280*c^6*x^5 + 1664*b*c^5*x^4 - 105*b^5*c + 760*a*b^3*c^2 - 1296*a^2
*b*c^3 + 16*(3*b^2*c^4 + 140*a*c^5)*x^3 - 8*(7*b^3*c^3 - 36*a*b*c^4)*x^2 + 2*(35*b^4*c^2 - 216*a*b^2*c^3 + 240
*a^2*c^4)*x)*e^2 + 24*(128*c^6*d*x^4 + 176*b*c^5*d*x^3 + 8*(b^2*c^4 + 32*a*c^5)*d*x^2 - 2*(5*b^3*c^3 - 28*a*b*
c^4)*d*x + (15*b^4*c^2 - 100*a*b^2*c^3 + 128*a^2*c^4)*d)*e)*sqrt(c*x^2 + b*x + a))/c^5, -1/15360*(15*(24*(b^4*
c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d^2 - 24*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d*e + (7*b^6 - 60*a*b^4*c + 144*
a^2*b^2*c^2 - 64*a^3*c^3)*e^2)*sqrt(-c)*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*(1920*c^6*d^2*x^3 + 2880*b*c^5*d^2*x^2 + 240*(b^2*c^4 + 20*a*c^5)*d^2*x - 120*(3*b^3*c^3 - 20*a*b
*c^4)*d^2 + (1280*c^6*x^5 + 1664*b*c^5*x^4 - 105*b^5*c + 760*a*b^3*c^2 - 1296*a^2*b*c^3 + 16*(3*b^2*c^4 + 140*
a*c^5)*x^3 - 8*(7*b^3*c^3 - 36*a*b*c^4)*x^2 + 2*(35*b^4*c^2 - 216*a*b^2*c^3 + 240*a^2*c^4)*x)*e^2 + 24*(128*c^
6*d*x^4 + 176*b*c^5*d*x^3 + 8*(b^2*c^4 + 32*a*c^5)*d*x^2 - 2*(5*b^3*c^3 - 28*a*b*c^4)*d*x + (15*b^4*c^2 - 100*
a*b^2*c^3 + 128*a^2*c^4)*d)*e)*sqrt(c*x^2 + b*x + a))/c^5]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 2.13, size = 463, normalized size = 1.80 \begin {gather*} \frac {1}{7680} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (10 \, c x e^{2} + \frac {24 \, c^{6} d e + 13 \, b c^{5} e^{2}}{c^{5}}\right )} x + \frac {120 \, c^{6} d^{2} + 264 \, b c^{5} d e + 3 \, b^{2} c^{4} e^{2} + 140 \, a c^{5} e^{2}}{c^{5}}\right )} x + \frac {360 \, b c^{5} d^{2} + 24 \, b^{2} c^{4} d e + 768 \, a c^{5} d e - 7 \, b^{3} c^{3} e^{2} + 36 \, a b c^{4} e^{2}}{c^{5}}\right )} x + \frac {120 \, b^{2} c^{4} d^{2} + 2400 \, a c^{5} d^{2} - 120 \, b^{3} c^{3} d e + 672 \, a b c^{4} d e + 35 \, b^{4} c^{2} e^{2} - 216 \, a b^{2} c^{3} e^{2} + 240 \, a^{2} c^{4} e^{2}}{c^{5}}\right )} x - \frac {360 \, b^{3} c^{3} d^{2} - 2400 \, a b c^{4} d^{2} - 360 \, b^{4} c^{2} d e + 2400 \, a b^{2} c^{3} d e - 3072 \, a^{2} c^{4} d e + 105 \, b^{5} c e^{2} - 760 \, a b^{3} c^{2} e^{2} + 1296 \, a^{2} b c^{3} e^{2}}{c^{5}}\right )} - \frac {{\left (24 \, b^{4} c^{2} d^{2} - 192 \, a b^{2} c^{3} d^{2} + 384 \, a^{2} c^{4} d^{2} - 24 \, b^{5} c d e + 192 \, a b^{3} c^{2} d e - 384 \, a^{2} b c^{3} d e + 7 \, b^{6} e^{2} - 60 \, a b^{4} c e^{2} + 144 \, a^{2} b^{2} c^{2} e^{2} - 64 \, a^{3} c^{3} e^{2}\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} - b \right |}\right )}{1024 \, c^{\frac {9}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7680*sqrt(c*x^2 + b*x + a)*(2*(4*(2*(8*(10*c*x*e^2 + (24*c^6*d*e + 13*b*c^5*e^2)/c^5)*x + (120*c^6*d^2 + 264
*b*c^5*d*e + 3*b^2*c^4*e^2 + 140*a*c^5*e^2)/c^5)*x + (360*b*c^5*d^2 + 24*b^2*c^4*d*e + 768*a*c^5*d*e - 7*b^3*c
^3*e^2 + 36*a*b*c^4*e^2)/c^5)*x + (120*b^2*c^4*d^2 + 2400*a*c^5*d^2 - 120*b^3*c^3*d*e + 672*a*b*c^4*d*e + 35*b
^4*c^2*e^2 - 216*a*b^2*c^3*e^2 + 240*a^2*c^4*e^2)/c^5)*x - (360*b^3*c^3*d^2 - 2400*a*b*c^4*d^2 - 360*b^4*c^2*d
*e + 2400*a*b^2*c^3*d*e - 3072*a^2*c^4*d*e + 105*b^5*c*e^2 - 760*a*b^3*c^2*e^2 + 1296*a^2*b*c^3*e^2)/c^5) - 1/
1024*(24*b^4*c^2*d^2 - 192*a*b^2*c^3*d^2 + 384*a^2*c^4*d^2 - 24*b^5*c*d*e + 192*a*b^3*c^2*d*e - 384*a^2*b*c^3*
d*e + 7*b^6*e^2 - 60*a*b^4*c*e^2 + 144*a^2*b^2*c^2*e^2 - 64*a^3*c^3*e^2)*log(abs(-2*(sqrt(c)*x - sqrt(c*x^2 +
b*x + a))*sqrt(c) - b))/c^(9/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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