3.2.78 \(\int (a+b x+c x^2)^{5/2} (A+C x^2) \, dx\) [178]

Optimal. Leaf size=267 \[ \frac {5 \left (b^2-4 a c\right )^2 \left (32 A c^2+9 b^2 C-4 a c C\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{16384 c^5}-\frac {5 \left (b^2-4 a c\right ) \left (32 A c^2+9 b^2 C-4 a c C\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{6144 c^4}+\frac {\left (32 A c^2+9 b^2 C-4 a c C\right ) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{384 c^3}-\frac {9 b C \left (a+b x+c x^2\right )^{7/2}}{112 c^2}+\frac {C x \left (a+b x+c x^2\right )^{7/2}}{8 c}-\frac {5 \left (b^2-4 a c\right )^3 \left (32 A c^2+9 b^2 C-4 a c C\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{32768 c^{11/2}} \]

[Out]

-5/6144*(-4*a*c+b^2)*(32*A*c^2-4*C*a*c+9*C*b^2)*(2*c*x+b)*(c*x^2+b*x+a)^(3/2)/c^4+1/384*(32*A*c^2-4*C*a*c+9*C*
b^2)*(2*c*x+b)*(c*x^2+b*x+a)^(5/2)/c^3-9/112*b*C*(c*x^2+b*x+a)^(7/2)/c^2+1/8*C*x*(c*x^2+b*x+a)^(7/2)/c-5/32768
*(-4*a*c+b^2)^3*(32*A*c^2-4*C*a*c+9*C*b^2)*arctanh(1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b*x+a)^(1/2))/c^(11/2)+5/16384
*(-4*a*c+b^2)^2*(32*A*c^2-4*C*a*c+9*C*b^2)*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/c^5

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Rubi [A]
time = 0.15, antiderivative size = 267, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {1675, 654, 626, 635, 212} \begin {gather*} -\frac {5 \left (b^2-4 a c\right )^3 \left (-4 a c C+32 A c^2+9 b^2 C\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{32768 c^{11/2}}+\frac {5 \left (b^2-4 a c\right )^2 (b+2 c x) \sqrt {a+b x+c x^2} \left (-4 a c C+32 A c^2+9 b^2 C\right )}{16384 c^5}-\frac {5 \left (b^2-4 a c\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2} \left (-4 a c C+32 A c^2+9 b^2 C\right )}{6144 c^4}+\frac {(b+2 c x) \left (a+b x+c x^2\right )^{5/2} \left (-4 a c C+32 A c^2+9 b^2 C\right )}{384 c^3}-\frac {9 b C \left (a+b x+c x^2\right )^{7/2}}{112 c^2}+\frac {C x \left (a+b x+c x^2\right )^{7/2}}{8 c} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(5*(b^2 - 4*a*c)^2*(32*A*c^2 + 9*b^2*C - 4*a*c*C)*(b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(16384*c^5) - (5*(b^2 - 4
*a*c)*(32*A*c^2 + 9*b^2*C - 4*a*c*C)*(b + 2*c*x)*(a + b*x + c*x^2)^(3/2))/(6144*c^4) + ((32*A*c^2 + 9*b^2*C -
4*a*c*C)*(b + 2*c*x)*(a + b*x + c*x^2)^(5/2))/(384*c^3) - (9*b*C*(a + b*x + c*x^2)^(7/2))/(112*c^2) + (C*x*(a
+ b*x + c*x^2)^(7/2))/(8*c) - (5*(b^2 - 4*a*c)^3*(32*A*c^2 + 9*b^2*C - 4*a*c*C)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]
*Sqrt[a + b*x + c*x^2])])/(32768*c^(11/2))

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 )^{5/2} \left (A+C x^2\right ) \, dx &=\frac {C x \left (a+b x+c x^2\right )^{7/2}}{8 c}+\frac {\int \left (8 A c-a C-\frac {9 b C x}{2}\right ) \left (a+b x+c x^2\right )^{5/2} \, dx}{8 c}\\ &=-\frac {9 b C \left (a+b x+c x^2\right )^{7/2}}{112 c^2}+\frac {C x \left (a+b x+c x^2\right )^{7/2}}{8 c}+\frac {\left (\frac {9 b^2 C}{2}+2 c (8 A c-a C)\right ) \int \left (a+b x+c x^2\right )^{5/2} \, dx}{16 c^2}\\ &=\frac {\left (32 A c^2+9 b^2 C-4 a c C\right ) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{384 c^3}-\frac {9 b C \left (a+b x+c x^2\right )^{7/2}}{112 c^2}+\frac {C x \left (a+b x+c x^2\right )^{7/2}}{8 c}-\frac {\left (5 \left (b^2-4 a c\right ) \left (32 A c^2+9 b^2 C-4 a c C\right )\right ) \int \left (a+b x+c x^2\right )^{3/2} \, dx}{768 c^3}\\ &=-\frac {5 \left (b^2-4 a c\right ) \left (32 A c^2+9 b^2 C-4 a c C\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{6144 c^4}+\frac {\left (32 A c^2+9 b^2 C-4 a c C\right ) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{384 c^3}-\frac {9 b C \left (a+b x+c x^2\right )^{7/2}}{112 c^2}+\frac {C x \left (a+b x+c x^2\right )^{7/2}}{8 c}+\frac {\left (5 \left (b^2-4 a c\right )^2 \left (32 A c^2+9 b^2 C-4 a c C\right )\right ) \int \sqrt {a+b x+c x^2} \, dx}{4096 c^4}\\ &=\frac {5 \left (b^2-4 a c\right )^2 \left (32 A c^2+9 b^2 C-4 a c C\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{16384 c^5}-\frac {5 \left (b^2-4 a c\right ) \left (32 A c^2+9 b^2 C-4 a c C\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{6144 c^4}+\frac {\left (32 A c^2+9 b^2 C-4 a c C\right ) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{384 c^3}-\frac {9 b C \left (a+b x+c x^2\right )^{7/2}}{112 c^2}+\frac {C x \left (a+b x+c x^2\right )^{7/2}}{8 c}-\frac {\left (5 \left (b^2-4 a c\right )^3 \left (32 A c^2+9 b^2 C-4 a c C\right )\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{32768 c^5}\\ &=\frac {5 \left (b^2-4 a c\right )^2 \left (32 A c^2+9 b^2 C-4 a c C\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{16384 c^5}-\frac {5 \left (b^2-4 a c\right ) \left (32 A c^2+9 b^2 C-4 a c C\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{6144 c^4}+\frac {\left (32 A c^2+9 b^2 C-4 a c C\right ) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{384 c^3}-\frac {9 b C \left (a+b x+c x^2\right )^{7/2}}{112 c^2}+\frac {C x \left (a+b x+c x^2\right )^{7/2}}{8 c}-\frac {\left (5 \left (b^2-4 a c\right )^3 \left (32 A c^2+9 b^2 C-4 a c C\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 )}{16384 c^5}\\ &=\frac {5 \left (b^2-4 a c\right )^2 \left (32 A c^2+9 b^2 C-4 a c C\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{16384 c^5}-\frac {5 \left (b^2-4 a c\right ) \left (32 A c^2+9 b^2 C-4 a c C\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{6144 c^4}+\frac {\left (32 A c^2+9 b^2 C-4 a c C\right ) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{384 c^3}-\frac {9 b C \left (a+b x+c x^2\right )^{7/2}}{112 c^2}+\frac {C x \left (a+b x+c x^2\right )^{7/2}}{8 c}-\frac {5 \left (b^2-4 a c\right )^3 \left (32 A c^2+9 b^2 C-4 a c C\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{32768 c^{11/2}}\\ \end {align*}

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Mathematica [A]
time = 1.50, size = 357, normalized size = 1.34 \begin {gather*} \frac {2 \sqrt {c} \sqrt {a+x (b+c x)} \left (224 A c^2 (b+2 c x) \left (15 b^4-40 b^3 c x+32 b c^2 x \left (13 a+8 c x^2\right )+8 b^2 c \left (-20 a+11 c x^2\right )+16 c^2 \left (33 a^2+26 a c x^2+8 c^2 x^4\right )\right )+C \left (945 b^7-630 b^6 c x+8 b^4 c^2 x \left (791 a-54 c x^2\right )+84 b^5 c \left (-125 a+6 c x^2\right )+16 b^3 c^2 \left (2359 a^2-284 a c x^2+24 c^2 x^4\right )+96 b^2 c^3 x \left (-199 a^2+36 a c x^2+648 c^2 x^4\right )+896 c^4 x \left (15 a^3+118 a^2 c x^2+136 a c^2 x^4+48 c^3 x^6\right )+64 b c^3 \left (-663 a^3+174 a^2 c x^2+2456 a c^2 x^4+1584 c^3 x^6\right )\right )\right )+105 \left (b^2-4 a c\right )^3 \left (32 A c^2+9 b^2 C-4 a c C\right ) \log \left (b+2 c x-2 \sqrt {c} \sqrt {a+x (b+c x)}\right )}{688128 c^{11/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[c]*Sqrt[a + x*(b + c*x)]*(224*A*c^2*(b + 2*c*x)*(15*b^4 - 40*b^3*c*x + 32*b*c^2*x*(13*a + 8*c*x^2) + 8
*b^2*c*(-20*a + 11*c*x^2) + 16*c^2*(33*a^2 + 26*a*c*x^2 + 8*c^2*x^4)) + C*(945*b^7 - 630*b^6*c*x + 8*b^4*c^2*x
*(791*a - 54*c*x^2) + 84*b^5*c*(-125*a + 6*c*x^2) + 16*b^3*c^2*(2359*a^2 - 284*a*c*x^2 + 24*c^2*x^4) + 96*b^2*
c^3*x*(-199*a^2 + 36*a*c*x^2 + 648*c^2*x^4) + 896*c^4*x*(15*a^3 + 118*a^2*c*x^2 + 136*a*c^2*x^4 + 48*c^3*x^6)
+ 64*b*c^3*(-663*a^3 + 174*a^2*c*x^2 + 2456*a*c^2*x^4 + 1584*c^3*x^6))) + 105*(b^2 - 4*a*c)^3*(32*A*c^2 + 9*b^
2*C - 4*a*c*C)*Log[b + 2*c*x - 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]])/(688128*c^(11/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(486\) vs. \(2(237)=474\).
time = 0.15, size = 487, normalized size = 1.82

method	result	size
default	\(C \left (\frac {x \left (c \,x^{2}+b x +a \right )^{\frac {7}{2}}}{8 c}-\frac {9 b \left (\frac {\left (c \,x^{2}+b x +a \right )^{\frac {7}{2}}}{7 c}-\frac {b \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}{12 c}+\frac {5 \left (4 a c -b^{2}\right ) \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{8 c}+\frac {3 \left (4 a c -b^{2}\right ) \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{24 c}\right )}{2 c}\right )}{16 c}-\frac {a \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}{12 c}+\frac {5 \left (4 a c -b^{2}\right ) \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{8 c}+\frac {3 \left (4 a c -b^{2}\right ) \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{24 c}\right )}{8 c}\right )+A \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}{12 c}+\frac {5 \left (4 a c -b^{2}\right ) \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{8 c}+\frac {3 \left (4 a c -b^{2}\right ) \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{24 c}\right )\)	\(487\)
risch	\(\frac {\left (43008 c^{7} C \,x^{7}+101376 b \,c^{6} C \,x^{6}+57344 A \,c^{7} x^{5}+121856 C a \,c^{6} x^{5}+62208 C \,b^{2} c^{5} x^{5}+143360 A b \,c^{6} x^{4}+157184 C a b \,c^{5} x^{4}+384 C \,b^{3} c^{4} x^{4}+186368 A a \,c^{6} x^{3}+96768 A \,b^{2} c^{5} x^{3}+105728 C \,a^{2} c^{5} x^{3}+3456 C a \,b^{2} c^{4} x^{3}-432 C \,b^{4} c^{3} x^{3}+279552 A a b \,c^{5} x^{2}+1792 A \,b^{3} c^{4} x^{2}+11136 C \,a^{2} b \,c^{4} x^{2}-4544 C a \,b^{3} c^{3} x^{2}+504 C \,b^{5} c^{2} x^{2}+236544 A \,a^{2} c^{5} x +21504 A a \,b^{2} c^{4} x -2240 A \,b^{4} c^{3} x +13440 C \,a^{3} c^{4} x -19104 C \,a^{2} b^{2} c^{3} x +6328 C a \,b^{4} c^{2} x -630 C \,b^{6} c x +118272 A \,a^{2} b \,c^{4}-35840 A a \,b^{3} c^{3}+3360 A \,b^{5} c^{2}-42432 C \,a^{3} b \,c^{3}+37744 C \,a^{2} b^{3} c^{2}-10500 C a \,b^{5} c +945 C \,b^{7}\right ) \sqrt {c \,x^{2}+b x +a}}{344064 c^{5}}+\frac {5 \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) A \,a^{3}}{16 \sqrt {c}}-\frac {15 \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) A \,a^{2} b^{2}}{64 c^{\frac {3}{2}}}+\frac {15 \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) A a \,b^{4}}{256 c^{\frac {5}{2}}}-\frac {5 \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) A \,b^{6}}{1024 c^{\frac {7}{2}}}-\frac {5 \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) C \,a^{4}}{128 c^{\frac {3}{2}}}+\frac {15 \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) C \,a^{3} b^{2}}{128 c^{\frac {5}{2}}}-\frac {75 \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) C \,a^{2} b^{4}}{1024 c^{\frac {7}{2}}}+\frac {35 \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) C a \,b^{6}}{2048 c^{\frac {9}{2}}}-\frac {45 \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) C \,b^{8}}{32768 c^{\frac {11}{2}}}\)	\(678\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

C*(1/8*x*(c*x^2+b*x+a)^(7/2)/c-9/16*b/c*(1/7*(c*x^2+b*x+a)^(7/2)/c-1/2*b/c*(1/12*(2*c*x+b)/c*(c*x^2+b*x+a)^(5/
2)+5/24*(4*a*c-b^2)/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))))))-1/8*a/c*(1/12*(2*c*x+b)/c*(c*x^2+
b*x+a)^(5/2)+5/24*(4*a*c-b^2)/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))))))+A*(1/12*(2*c*x+b)/c*(c*
x^2+b*x+a)^(5/2)+5/24*(4*a*c-b^2)/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)))))

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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((c*x^2+b*x+a)^(5/2)*(C*x^2+A),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 [B] Leaf count of result is larger than twice the leaf count of optimal. 475 vs. \(2 (237) = 474\).
time = 0.52, size = 953, normalized size = 3.57 \begin {gather*} \left [\frac {105 \, {\left (9 \, C b^{8} - 112 \, C a b^{6} c - 2048 \, A a^{3} c^{5} + 256 \, {\left (C a^{4} + 6 \, A a^{2} b^{2}\right )} c^{4} - 384 \, {\left (2 \, C a^{3} b^{2} + A a b^{4}\right )} c^{3} + 32 \, {\left (15 \, C a^{2} b^{4} + A b^{6}\right )} c^{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 (43008 \, C c^{8} x^{7} + 101376 \, C b c^{7} x^{6} + 945 \, C b^{7} c - 10500 \, C a b^{5} c^{2} + 118272 \, A a^{2} b c^{5} + 256 \, {\left (243 \, C b^{2} c^{6} + 476 \, C a c^{7} + 224 \, A c^{8}\right )} x^{5} - 64 \, {\left (663 \, C a^{3} b + 560 \, A a b^{3}\right )} c^{4} + 128 \, {\left (3 \, C b^{3} c^{5} + 1228 \, C a b c^{6} + 1120 \, A b c^{7}\right )} x^{4} + 112 \, {\left (337 \, C a^{2} b^{3} + 30 \, A b^{5}\right )} c^{3} - 16 \, {\left (27 \, C b^{4} c^{4} - 216 \, C a b^{2} c^{5} - 11648 \, A a c^{7} - 112 \, {\left (59 \, C a^{2} + 54 \, A b^{2}\right )} c^{6}\right )} x^{3} + 8 \, {\left (63 \, C b^{5} c^{3} - 568 \, C a b^{3} c^{4} + 34944 \, A a b c^{6} + 16 \, {\left (87 \, C a^{2} b + 14 \, A b^{3}\right )} c^{5}\right )} x^{2} - 2 \, {\left (315 \, C b^{6} c^{2} - 3164 \, C a b^{4} c^{3} - 118272 \, A a^{2} c^{6} - 1344 \, {\left (5 \, C a^{3} + 8 \, A a b^{2}\right )} c^{5} + 16 \, {\left (597 \, C a^{2} b^{2} + 70 \, A b^{4}\right )} c^{4}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{1376256 \, c^{6}}, \frac {105 \, {\left (9 \, C b^{8} - 112 \, C a b^{6} c - 2048 \, A a^{3} c^{5} + 256 \, {\left (C a^{4} + 6 \, A a^{2} b^{2}\right )} c^{4} - 384 \, {\left (2 \, C a^{3} b^{2} + A a b^{4}\right )} c^{3} + 32 \, {\left (15 \, C a^{2} b^{4} + A b^{6}\right )} c^{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 (43008 \, C c^{8} x^{7} + 101376 \, C b c^{7} x^{6} + 945 \, C b^{7} c - 10500 \, C a b^{5} c^{2} + 118272 \, A a^{2} b c^{5} + 256 \, {\left (243 \, C b^{2} c^{6} + 476 \, C a c^{7} + 224 \, A c^{8}\right )} x^{5} - 64 \, {\left (663 \, C a^{3} b + 560 \, A a b^{3}\right )} c^{4} + 128 \, {\left (3 \, C b^{3} c^{5} + 1228 \, C a b c^{6} + 1120 \, A b c^{7}\right )} x^{4} + 112 \, {\left (337 \, C a^{2} b^{3} + 30 \, A b^{5}\right )} c^{3} - 16 \, {\left (27 \, C b^{4} c^{4} - 216 \, C a b^{2} c^{5} - 11648 \, A a c^{7} - 112 \, {\left (59 \, C a^{2} + 54 \, A b^{2}\right )} c^{6}\right )} x^{3} + 8 \, {\left (63 \, C b^{5} c^{3} - 568 \, C a b^{3} c^{4} + 34944 \, A a b c^{6} + 16 \, {\left (87 \, C a^{2} b + 14 \, A b^{3}\right )} c^{5}\right )} x^{2} - 2 \, {\left (315 \, C b^{6} c^{2} - 3164 \, C a b^{4} c^{3} - 118272 \, A a^{2} c^{6} - 1344 \, {\left (5 \, C a^{3} + 8 \, A a b^{2}\right )} c^{5} + 16 \, {\left (597 \, C a^{2} b^{2} + 70 \, A b^{4}\right )} c^{4}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{688128 \, c^{6}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/1376256*(105*(9*C*b^8 - 112*C*a*b^6*c - 2048*A*a^3*c^5 + 256*(C*a^4 + 6*A*a^2*b^2)*c^4 - 384*(2*C*a^3*b^2 +
A*a*b^4)*c^3 + 32*(15*C*a^2*b^4 + A*b^6)*c^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*(43008*C*c^8*x^7 + 101376*C*b*c^7*x^6 + 945*C*b^7*c - 10500*C*a*b^5*c^2 + 1
18272*A*a^2*b*c^5 + 256*(243*C*b^2*c^6 + 476*C*a*c^7 + 224*A*c^8)*x^5 - 64*(663*C*a^3*b + 560*A*a*b^3)*c^4 + 1
28*(3*C*b^3*c^5 + 1228*C*a*b*c^6 + 1120*A*b*c^7)*x^4 + 112*(337*C*a^2*b^3 + 30*A*b^5)*c^3 - 16*(27*C*b^4*c^4 -
216*C*a*b^2*c^5 - 11648*A*a*c^7 - 112*(59*C*a^2 + 54*A*b^2)*c^6)*x^3 + 8*(63*C*b^5*c^3 - 568*C*a*b^3*c^4 + 34
944*A*a*b*c^6 + 16*(87*C*a^2*b + 14*A*b^3)*c^5)*x^2 - 2*(315*C*b^6*c^2 - 3164*C*a*b^4*c^3 - 118272*A*a^2*c^6 -
1344*(5*C*a^3 + 8*A*a*b^2)*c^5 + 16*(597*C*a^2*b^2 + 70*A*b^4)*c^4)*x)*sqrt(c*x^2 + b*x + a))/c^6, 1/688128*(
105*(9*C*b^8 - 112*C*a*b^6*c - 2048*A*a^3*c^5 + 256*(C*a^4 + 6*A*a^2*b^2)*c^4 - 384*(2*C*a^3*b^2 + A*a*b^4)*c^
3 + 32*(15*C*a^2*b^4 + A*b^6)*c^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*(43008*C*c^8*x^7 + 101376*C*b*c^7*x^6 + 945*C*b^7*c - 10500*C*a*b^5*c^2 + 118272*A*a^2*b*c^5
+ 256*(243*C*b^2*c^6 + 476*C*a*c^7 + 224*A*c^8)*x^5 - 64*(663*C*a^3*b + 560*A*a*b^3)*c^4 + 128*(3*C*b^3*c^5 +
1228*C*a*b*c^6 + 1120*A*b*c^7)*x^4 + 112*(337*C*a^2*b^3 + 30*A*b^5)*c^3 - 16*(27*C*b^4*c^4 - 216*C*a*b^2*c^5 -
11648*A*a*c^7 - 112*(59*C*a^2 + 54*A*b^2)*c^6)*x^3 + 8*(63*C*b^5*c^3 - 568*C*a*b^3*c^4 + 34944*A*a*b*c^6 + 16
*(87*C*a^2*b + 14*A*b^3)*c^5)*x^2 - 2*(315*C*b^6*c^2 - 3164*C*a*b^4*c^3 - 118272*A*a^2*c^6 - 1344*(5*C*a^3 + 8
*A*a*b^2)*c^5 + 16*(597*C*a^2*b^2 + 70*A*b^4)*c^4)*x)*sqrt(c*x^2 + b*x + a))/c^6]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((A + C*x**2)*(a + b*x + c*x**2)**(5/2), x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 482 vs. \(2 (237) = 474\).
time = 3.76, size = 482, normalized size = 1.81 \begin {gather*} \frac {1}{344064} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (2 \, {\left (12 \, {\left (14 \, C c^{2} x + 33 \, C b c\right )} x + \frac {243 \, C b^{2} c^{7} + 476 \, C a c^{8} + 224 \, A c^{9}}{c^{7}}\right )} x + \frac {3 \, C b^{3} c^{6} + 1228 \, C a b c^{7} + 1120 \, A b c^{8}}{c^{7}}\right )} x - \frac {27 \, C b^{4} c^{5} - 216 \, C a b^{2} c^{6} - 6608 \, C a^{2} c^{7} - 6048 \, A b^{2} c^{7} - 11648 \, A a c^{8}}{c^{7}}\right )} x + \frac {63 \, C b^{5} c^{4} - 568 \, C a b^{3} c^{5} + 1392 \, C a^{2} b c^{6} + 224 \, A b^{3} c^{6} + 34944 \, A a b c^{7}}{c^{7}}\right )} x - \frac {315 \, C b^{6} c^{3} - 3164 \, C a b^{4} c^{4} + 9552 \, C a^{2} b^{2} c^{5} + 1120 \, A b^{4} c^{5} - 6720 \, C a^{3} c^{6} - 10752 \, A a b^{2} c^{6} - 118272 \, A a^{2} c^{7}}{c^{7}}\right )} x + \frac {945 \, C b^{7} c^{2} - 10500 \, C a b^{5} c^{3} + 37744 \, C a^{2} b^{3} c^{4} + 3360 \, A b^{5} c^{4} - 42432 \, C a^{3} b c^{5} - 35840 \, A a b^{3} c^{5} + 118272 \, A a^{2} b c^{6}}{c^{7}}\right )} + \frac {5 \, {\left (9 \, C b^{8} - 112 \, C a b^{6} c + 480 \, C a^{2} b^{4} c^{2} + 32 \, A b^{6} c^{2} - 768 \, C a^{3} b^{2} c^{3} - 384 \, A a b^{4} c^{3} + 256 \, C a^{4} c^{4} + 1536 \, A a^{2} b^{2} c^{4} - 2048 \, A a^{3} c^{5}\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} - b \right |}\right )}{32768 \, c^{\frac {11}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/344064*sqrt(c*x^2 + b*x + a)*(2*(4*(2*(8*(2*(12*(14*C*c^2*x + 33*C*b*c)*x + (243*C*b^2*c^7 + 476*C*a*c^8 + 2
24*A*c^9)/c^7)*x + (3*C*b^3*c^6 + 1228*C*a*b*c^7 + 1120*A*b*c^8)/c^7)*x - (27*C*b^4*c^5 - 216*C*a*b^2*c^6 - 66
08*C*a^2*c^7 - 6048*A*b^2*c^7 - 11648*A*a*c^8)/c^7)*x + (63*C*b^5*c^4 - 568*C*a*b^3*c^5 + 1392*C*a^2*b*c^6 + 2
24*A*b^3*c^6 + 34944*A*a*b*c^7)/c^7)*x - (315*C*b^6*c^3 - 3164*C*a*b^4*c^4 + 9552*C*a^2*b^2*c^5 + 1120*A*b^4*c
^5 - 6720*C*a^3*c^6 - 10752*A*a*b^2*c^6 - 118272*A*a^2*c^7)/c^7)*x + (945*C*b^7*c^2 - 10500*C*a*b^5*c^3 + 3774
4*C*a^2*b^3*c^4 + 3360*A*b^5*c^4 - 42432*C*a^3*b*c^5 - 35840*A*a*b^3*c^5 + 118272*A*a^2*b*c^6)/c^7) + 5/32768*
(9*C*b^8 - 112*C*a*b^6*c + 480*C*a^2*b^4*c^2 + 32*A*b^6*c^2 - 768*C*a^3*b^2*c^3 - 384*A*a*b^4*c^3 + 256*C*a^4*
c^4 + 1536*A*a^2*b^2*c^4 - 2048*A*a^3*c^5)*log(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) - b))/c^(11/
2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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