3.2.5 \(\int (a+b x+c x^2)^{3/2} (d+e x+f x^2) \, dx\) [105]

Optimal. Leaf size=236 \[ -\frac {\left (b^2-4 a c\right ) \left (24 c^2 d+7 b^2 f-4 c (3 b e+a f)\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{512 c^4}+\frac {\left (24 c^2 d-12 b c e+7 b^2 f-4 a c f\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{192 c^3}+\frac {(12 c e-7 b f) \left (a+b x+c x^2\right )^{5/2}}{60 c^2}+\frac {f x \left (a+b x+c x^2\right )^{5/2}}{6 c}+\frac {\left (b^2-4 a c\right )^2 \left (24 c^2 d+7 b^2 f-4 c (3 b e+a f)\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{1024 c^{9/2}} \]

[Out]

1/192*(-4*a*c*f+7*b^2*f-12*b*c*e+24*c^2*d)*(2*c*x+b)*(c*x^2+b*x+a)^(3/2)/c^3+1/60*(-7*b*f+12*c*e)*(c*x^2+b*x+a
)^(5/2)/c^2+1/6*f*x*(c*x^2+b*x+a)^(5/2)/c+1/1024*(-4*a*c+b^2)^2*(24*c^2*d+7*b^2*f-4*c*(a*f+3*b*e))*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+7*b^2*f-4*c*(a*f+3*b*e))*(2*c*x+b
)*(c*x^2+b*x+a)^(1/2)/c^4

________________________________________________________________________________________

Rubi [A]
time = 0.14, antiderivative size = 236, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1675, 654, 626, 635, 212} \begin {gather*} \frac {\left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (-4 c (a f+3 b e)+7 b^2 f+24 c^2 d\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 (a f+3 b e)+7 b^2 f+24 c^2 d\right )}{512 c^4}+\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2} \left (-4 a c f+7 b^2 f-12 b c e+24 c^2 d\right )}{192 c^3}+\frac {\left (a+b x+c x^2\right )^{5/2} (12 c e-7 b f)}{60 c^2}+\frac {f x \left (a+b x+c x^2\right )^{5/2}}{6 c} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/512*((b^2 - 4*a*c)*(24*c^2*d + 7*b^2*f - 4*c*(3*b*e + a*f))*(b + 2*c*x)*Sqrt[a + b*x + c*x^2])/c^4 + ((24*c
^2*d - 12*b*c*e + 7*b^2*f - 4*a*c*f)*(b + 2*c*x)*(a + b*x + c*x^2)^(3/2))/(192*c^3) + ((12*c*e - 7*b*f)*(a + b
*x + c*x^2)^(5/2))/(60*c^2) + (f*x*(a + b*x + c*x^2)^(5/2))/(6*c) + ((b^2 - 4*a*c)^2*(24*c^2*d + 7*b^2*f - 4*c
*(3*b*e + a*f))*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 1675

Int[(Pq_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], e = Coeff[Pq, x, Expo
n[Pq, x]]}, Simp[e*x^(q - 1)*((a + b*x + c*x^2)^(p + 1)/(c*(q + 2*p + 1))), x] + Dist[1/(c*(q + 2*p + 1)), Int
[(a + b*x + c*x^2)^p*ExpandToSum[c*(q + 2*p + 1)*Pq - a*e*(q - 1)*x^(q - 2) - b*e*(q + p)*x^(q - 1) - c*e*(q +
2*p + 1)*x^q, x], x], x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] &&  !LeQ[p, -1]

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]
time = 1.22, size = 290, normalized size = 1.23 \begin {gather*} \frac {2 \sqrt {c} \sqrt {a+x (b+c x)} \left (-105 b^5 f+10 b^4 c (18 e+7 f x)-8 b^3 c (45 c d-95 a f+c x (15 e+7 f x))+48 b^2 c^2 (-a (25 e+9 f x)+c x (5 d+x (2 e+f x)))+16 b c^2 \left (-81 a^2 f+6 a c (25 d+x (7 e+3 f x))+4 c^2 x^2 (45 d+x (33 e+26 f x))\right )+32 c^3 \left (3 a^2 (16 e+5 f x)+4 c^2 x^3 (15 d+2 x (6 e+5 f x))+2 a c x (75 d+x (48 e+35 f x))\right )\right )-15 \left (b^2-4 a c\right )^2 \left (24 c^2 d+7 b^2 f-4 c (3 b e+a f)\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[(a + b*x + c*x^2)^(3/2)*(d + e*x + f*x^2),x]

[Out]

(2*Sqrt[c]*Sqrt[a + x*(b + c*x)]*(-105*b^5*f + 10*b^4*c*(18*e + 7*f*x) - 8*b^3*c*(45*c*d - 95*a*f + c*x*(15*e
+ 7*f*x)) + 48*b^2*c^2*(-(a*(25*e + 9*f*x)) + c*x*(5*d + x*(2*e + f*x))) + 16*b*c^2*(-81*a^2*f + 6*a*c*(25*d +
x*(7*e + 3*f*x)) + 4*c^2*x^2*(45*d + x*(33*e + 26*f*x))) + 32*c^3*(3*a^2*(16*e + 5*f*x) + 4*c^2*x^3*(15*d + 2
*x*(6*e + 5*f*x)) + 2*a*c*x*(75*d + x*(48*e + 35*f*x)))) - 15*(b^2 - 4*a*c)^2*(24*c^2*d + 7*b^2*f - 4*c*(3*b*e
+ a*f))*Log[b + 2*c*x - 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]])/(15360*c^(9/2))

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(498\) vs. \(2(210)=420\).
time = 0.14, size = 499, normalized size = 2.11

method	result	size
default	\(f \left (\frac {x \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}{6 c}-\frac {7 b \left (\frac {\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}{5 c}-\frac {b \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{8 c}+\frac {3 \left (4 a c -b^{2}\right ) \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{2 c}\right )}{12 c}-\frac {a \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{8 c}+\frac {3 \left (4 a c -b^{2}\right ) \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{6 c}\right )+e \left (\frac {\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}{5 c}-\frac {b \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{8 c}+\frac {3 \left (4 a c -b^{2}\right ) \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{2 c}\right )+d \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{8 c}+\frac {3 \left (4 a c -b^{2}\right ) \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )\)	\(499\)
risch	\(-\frac {\left (-1280 c^{5} f \,x^{5}-1664 b \,c^{4} f \,x^{4}-1536 c^{5} e \,x^{4}-2240 a \,c^{4} f \,x^{3}-48 b^{2} c^{3} f \,x^{3}-2112 b \,c^{4} e \,x^{3}-1920 c^{5} d \,x^{3}-288 a b \,c^{3} f \,x^{2}-3072 a \,c^{4} e \,x^{2}+56 b^{3} c^{2} f \,x^{2}-96 b^{2} c^{3} e \,x^{2}-2880 b \,c^{4} d \,x^{2}-480 a^{2} c^{3} f x +432 a \,b^{2} c^{2} f x -672 a b \,c^{3} e x -4800 a \,c^{4} d x -70 b^{4} c f x +120 b^{3} c^{2} e x -240 b^{2} c^{3} d x +1296 a^{2} b \,c^{2} f -1536 a^{2} c^{3} e -760 a \,b^{3} c f +1200 a \,b^{2} c^{2} e -2400 a b \,c^{3} d +105 b^{5} f -180 b^{4} c e +360 b^{3} c^{2} d \right ) \sqrt {c \,x^{2}+b x +a}}{7680 c^{4}}-\frac {\ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) a^{3} f}{16 c^{\frac {3}{2}}}+\frac {9 \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) a^{2} b^{2} f}{64 c^{\frac {5}{2}}}-\frac {3 \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) a^{2} b e}{16 c^{\frac {3}{2}}}+\frac {3 \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) a^{2} d}{8 \sqrt {c}}-\frac {15 \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) a \,b^{4} f}{256 c^{\frac {7}{2}}}+\frac {3 \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) a \,b^{3} e}{32 c^{\frac {5}{2}}}-\frac {3 \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) a \,b^{2} d}{16 c^{\frac {3}{2}}}+\frac {7 \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) b^{6} f}{1024 c^{\frac {9}{2}}}-\frac {3 \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) b^{5} e}{256 c^{\frac {7}{2}}}+\frac {3 \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) b^{4} d}{128 c^{\frac {5}{2}}}\)	\(624\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

f*(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*(c*x^2+b*x+a)^(3/2
)+3/16*(4*a*c-b^2)/c*(1/4*(2*c*x+b)/c*(c*x^2+b*x+a)^(1/2)+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*(c*x^2+b*x+a)^(3/2)+3/16*(4*a*c-b^2)/c*(1/4*(2*c*x+b)/c*(c*x^2+b*x
+a)^(1/2)+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)))))+e*(1/5*(c*x^2+b*x+a)^(5/2)/c-
1/2*b/c*(1/8*(2*c*x+b)/c*(c*x^2+b*x+a)^(3/2)+3/16*(4*a*c-b^2)/c*(1/4*(2*c*x+b)/c*(c*x^2+b*x+a)^(1/2)+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*(1/8*(2*c*x+b)/c*(c*x^2+b*x+a)^(3/2)+3/16*(4*a
*c-b^2)/c*(1/4*(2*c*x+b)/c*(c*x^2+b*x+a)^(1/2)+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))))

________________________________________________________________________________________

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

________________________________________________________________________________________

Fricas [A]
time = 5.34, size = 849, normalized size = 3.60 \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 + {\left (7 \, b^{6} - 60 \, a b^{4} c + 144 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} f - 12 \, {\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} e\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 (1280 \, c^{6} f x^{5} + 1664 \, b c^{5} f x^{4} + 16 \, {\left (120 \, c^{6} d + {\left (3 \, b^{2} c^{4} + 140 \, a c^{5}\right )} f\right )} x^{3} + 8 \, {\left (360 \, b c^{5} d - {\left (7 \, b^{3} c^{3} - 36 \, a b c^{4}\right )} f\right )} x^{2} - 120 \, {\left (3 \, b^{3} c^{3} - 20 \, a b c^{4}\right )} d - {\left (105 \, b^{5} c - 760 \, a b^{3} c^{2} + 1296 \, a^{2} b c^{3}\right )} f + 2 \, {\left (120 \, {\left (b^{2} c^{4} + 20 \, a c^{5}\right )} d + {\left (35 \, b^{4} c^{2} - 216 \, a b^{2} c^{3} + 240 \, a^{2} c^{4}\right )} f\right )} x + 12 \, {\left (128 \, c^{6} x^{4} + 176 \, b c^{5} x^{3} + 15 \, b^{4} c^{2} - 100 \, a b^{2} c^{3} + 128 \, a^{2} c^{4} + 8 \, {\left (b^{2} c^{4} + 32 \, a c^{5}\right )} x^{2} - 2 \, {\left (5 \, b^{3} c^{3} - 28 \, a b c^{4}\right )} x\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 + {\left (7 \, b^{6} - 60 \, a b^{4} c + 144 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} f - 12 \, {\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} e\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 (1280 \, c^{6} f x^{5} + 1664 \, b c^{5} f x^{4} + 16 \, {\left (120 \, c^{6} d + {\left (3 \, b^{2} c^{4} + 140 \, a c^{5}\right )} f\right )} x^{3} + 8 \, {\left (360 \, b c^{5} d - {\left (7 \, b^{3} c^{3} - 36 \, a b c^{4}\right )} f\right )} x^{2} - 120 \, {\left (3 \, b^{3} c^{3} - 20 \, a b c^{4}\right )} d - {\left (105 \, b^{5} c - 760 \, a b^{3} c^{2} + 1296 \, a^{2} b c^{3}\right )} f + 2 \, {\left (120 \, {\left (b^{2} c^{4} + 20 \, a c^{5}\right )} d + {\left (35 \, b^{4} c^{2} - 216 \, a b^{2} c^{3} + 240 \, a^{2} c^{4}\right )} f\right )} x + 12 \, {\left (128 \, c^{6} x^{4} + 176 \, b c^{5} x^{3} + 15 \, b^{4} c^{2} - 100 \, a b^{2} c^{3} + 128 \, a^{2} c^{4} + 8 \, {\left (b^{2} c^{4} + 32 \, a c^{5}\right )} x^{2} - 2 \, {\left (5 \, b^{3} c^{3} - 28 \, a b c^{4}\right )} x\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((c*x^2+b*x+a)^(3/2)*(f*x^2+e*x+d),x, algorithm="fricas")

[Out]

[-1/30720*(15*(24*(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d + (7*b^6 - 60*a*b^4*c + 144*a^2*b^2*c^2 - 64*a^3*c^3)
*f - 12*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*e)*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*(1280*c^6*f*x^5 + 1664*b*c^5*f*x^4 + 16*(120*c^6*d + (3*b^2*c^4 + 140*a*c^
5)*f)*x^3 + 8*(360*b*c^5*d - (7*b^3*c^3 - 36*a*b*c^4)*f)*x^2 - 120*(3*b^3*c^3 - 20*a*b*c^4)*d - (105*b^5*c - 7
60*a*b^3*c^2 + 1296*a^2*b*c^3)*f + 2*(120*(b^2*c^4 + 20*a*c^5)*d + (35*b^4*c^2 - 216*a*b^2*c^3 + 240*a^2*c^4)*
f)*x + 12*(128*c^6*x^4 + 176*b*c^5*x^3 + 15*b^4*c^2 - 100*a*b^2*c^3 + 128*a^2*c^4 + 8*(b^2*c^4 + 32*a*c^5)*x^2
- 2*(5*b^3*c^3 - 28*a*b*c^4)*x)*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 + (7*b^6 - 60*a*b^4*c + 144*a^2*b^2*c^2 - 64*a^3*c^3)*f - 12*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*e)
*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*(1280*c^6*f*x^5 +
1664*b*c^5*f*x^4 + 16*(120*c^6*d + (3*b^2*c^4 + 140*a*c^5)*f)*x^3 + 8*(360*b*c^5*d - (7*b^3*c^3 - 36*a*b*c^4)
*f)*x^2 - 120*(3*b^3*c^3 - 20*a*b*c^4)*d - (105*b^5*c - 760*a*b^3*c^2 + 1296*a^2*b*c^3)*f + 2*(120*(b^2*c^4 +
20*a*c^5)*d + (35*b^4*c^2 - 216*a*b^2*c^3 + 240*a^2*c^4)*f)*x + 12*(128*c^6*x^4 + 176*b*c^5*x^3 + 15*b^4*c^2 -
100*a*b^2*c^3 + 128*a^2*c^4 + 8*(b^2*c^4 + 32*a*c^5)*x^2 - 2*(5*b^3*c^3 - 28*a*b*c^4)*x)*e)*sqrt(c*x^2 + b*x
+ a))/c^5]

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [A]
time = 7.86, size = 417, normalized size = 1.77 \begin {gather*} \frac {1}{7680} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (10 \, c f x + \frac {13 \, b c^{5} f + 12 \, c^{6} e}{c^{5}}\right )} x + \frac {120 \, c^{6} d + 3 \, b^{2} c^{4} f + 140 \, a c^{5} f + 132 \, b c^{5} e}{c^{5}}\right )} x + \frac {360 \, b c^{5} d - 7 \, b^{3} c^{3} f + 36 \, a b c^{4} f + 12 \, b^{2} c^{4} e + 384 \, a c^{5} e}{c^{5}}\right )} x + \frac {120 \, b^{2} c^{4} d + 2400 \, a c^{5} d + 35 \, b^{4} c^{2} f - 216 \, a b^{2} c^{3} f + 240 \, a^{2} c^{4} f - 60 \, b^{3} c^{3} e + 336 \, a b c^{4} e}{c^{5}}\right )} x - \frac {360 \, b^{3} c^{3} d - 2400 \, a b c^{4} d + 105 \, b^{5} c f - 760 \, a b^{3} c^{2} f + 1296 \, a^{2} b c^{3} f - 180 \, b^{4} c^{2} e + 1200 \, a b^{2} c^{3} e - 1536 \, a^{2} c^{4} e}{c^{5}}\right )} - \frac {{\left (24 \, b^{4} c^{2} d - 192 \, a b^{2} c^{3} d + 384 \, a^{2} c^{4} d + 7 \, b^{6} f - 60 \, a b^{4} c f + 144 \, a^{2} b^{2} c^{2} f - 64 \, a^{3} c^{3} f - 12 \, b^{5} c e + 96 \, a b^{3} c^{2} e - 192 \, a^{2} b c^{3} e\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((c*x^2+b*x+a)^(3/2)*(f*x^2+e*x+d),x, algorithm="giac")

[Out]

1/7680*sqrt(c*x^2 + b*x + a)*(2*(4*(2*(8*(10*c*f*x + (13*b*c^5*f + 12*c^6*e)/c^5)*x + (120*c^6*d + 3*b^2*c^4*f
+ 140*a*c^5*f + 132*b*c^5*e)/c^5)*x + (360*b*c^5*d - 7*b^3*c^3*f + 36*a*b*c^4*f + 12*b^2*c^4*e + 384*a*c^5*e)
/c^5)*x + (120*b^2*c^4*d + 2400*a*c^5*d + 35*b^4*c^2*f - 216*a*b^2*c^3*f + 240*a^2*c^4*f - 60*b^3*c^3*e + 336*
a*b*c^4*e)/c^5)*x - (360*b^3*c^3*d - 2400*a*b*c^4*d + 105*b^5*c*f - 760*a*b^3*c^2*f + 1296*a^2*b*c^3*f - 180*b
^4*c^2*e + 1200*a*b^2*c^3*e - 1536*a^2*c^4*e)/c^5) - 1/1024*(24*b^4*c^2*d - 192*a*b^2*c^3*d + 384*a^2*c^4*d +
7*b^6*f - 60*a*b^4*c*f + 144*a^2*b^2*c^2*f - 64*a^3*c^3*f - 12*b^5*c*e + 96*a*b^3*c^2*e - 192*a^2*b*c^3*e)*log
(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) - b))/c^(9/2)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________