### 3.178 $$\int (a+b x+c x^2)^{5/2} (A+C x^2) \, dx$$

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

[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))

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Rubi [A]  time = 0.23808, antiderivative size = 267, normalized size of antiderivative = 1., 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 = {1661, 640, 612, 621, 206} $\frac{(b+2 c x) \left (a+b x+c x^2\right )^{5/2} \left (-4 a c C+32 A c^2+9 b^2 C\right )}{384 c^3}-\frac{5 \left (b^2-4 a c\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2} \left (-4 a c C+32 A c^2+9 b^2 C\right )}{6144 c^4}+\frac{5 \left (b^2-4 a c\right )^2 (b+2 c x) \sqrt{a+b x+c x^2} \left (-4 a c C+32 A c^2+9 b^2 C\right )}{16384 c^5}-\frac{5 \left (b^2-4 a c\right )^3 \left (-4 a c C+32 A c^2+9 b^2 C\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{32768 c^{11/2}}-\frac{9 b C \left (a+b x+c x^2\right )^{7/2}}{112 c^2}+\frac{C x \left (a+b x+c x^2\right )^{7/2}}{8 c}$

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 1661

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]

Rule 640

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 612

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 621

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 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

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 ) \operatorname{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*}

Mathematica [A]  time = 0.851162, size = 344, normalized size = 1.29 $\frac{-\frac{1120 A \left (b^2-4 a c\right ) \left (16 c^{3/2} (b+2 c x) (a+x (b+c x))^{3/2}-3 \left (b^2-4 a c\right ) \left (2 \sqrt{c} (b+2 c x) \sqrt{a+x (b+c x)}-\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right )\right )\right )}{c^{5/2}}+57344 A (b+2 c x) (a+x (b+c x))^{5/2}+\frac{7 C \left (9 b^2-4 a c\right ) \left (256 c^{5/2} (b+2 c x) (a+x (b+c x))^{5/2}-5 \left (b^2-4 a c\right ) \left (16 c^{3/2} (b+2 c x) (a+x (b+c x))^{3/2}-3 \left (b^2-4 a c\right ) \left (2 \sqrt{c} (b+2 c x) \sqrt{a+x (b+c x)}-\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right )\right )\right )\right )}{c^{9/2}}-\frac{55296 b C (a+x (b+c x))^{7/2}}{c}+86016 C x (a+x (b+c x))^{7/2}}{688128 c}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(57344*A*(b + 2*c*x)*(a + x*(b + c*x))^(5/2) - (55296*b*C*(a + x*(b + c*x))^(7/2))/c + 86016*C*x*(a + x*(b + c
*x))^(7/2) - (1120*A*(b^2 - 4*a*c)*(16*c^(3/2)*(b + 2*c*x)*(a + x*(b + c*x))^(3/2) - 3*(b^2 - 4*a*c)*(2*Sqrt[c
]*(b + 2*c*x)*Sqrt[a + x*(b + c*x)] - (b^2 - 4*a*c)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])])))/
c^(5/2) + (7*(9*b^2 - 4*a*c)*C*(256*c^(5/2)*(b + 2*c*x)*(a + x*(b + c*x))^(5/2) - 5*(b^2 - 4*a*c)*(16*c^(3/2)*
(b + 2*c*x)*(a + x*(b + c*x))^(3/2) - 3*(b^2 - 4*a*c)*(2*Sqrt[c]*(b + 2*c*x)*Sqrt[a + x*(b + c*x)] - (b^2 - 4*
a*c)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])]))))/c^(9/2))/(688128*c)

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Maple [B]  time = 0.055, size = 997, normalized size = 3.7 \begin{align*} \text{result too large to display} \end{align*}

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)

[Out]

-15/64*A/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*b^2*a^2-5/96*A/c*(c*x^2+b*x+a)^(3/2)*x*b^2+15/256
*A/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*b^4*a+5/256*A/c^2*(c*x^2+b*x+a)^(1/2)*x*b^4+5/32*A/c*(c
*x^2+b*x+a)^(1/2)*b*a^2-5/64*A/c^2*(c*x^2+b*x+a)^(1/2)*b^3*a+15/128*C*b^2/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^
2+b*x+a)^(1/2))*a^3-15/1024*C*b^4/c^3*(c*x^2+b*x+a)^(3/2)*x-5/192*C*a^2/c*(c*x^2+b*x+a)^(3/2)*x+3/64*C*b^2/c^2
*(c*x^2+b*x+a)^(5/2)*x+35/2048*C*b^6/c^(9/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a+1/6*A*(c*x^2+b*x+a)
^(5/2)*x-1/48*C*a/c*(c*x^2+b*x+a)^(5/2)*x+5/48*A/c*(c*x^2+b*x+a)^(3/2)*b*a-1/96*C*a/c^2*(c*x^2+b*x+a)^(5/2)*b-
75/1024*C*b^4/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a^2-5/256*C*a^3/c^2*(c*x^2+b*x+a)^(1/2)*b-5/
128*C*a^3/c*(c*x^2+b*x+a)^(1/2)*x+3/128*C*b^3/c^3*(c*x^2+b*x+a)^(5/2)+45/8192*C*b^6/c^4*(c*x^2+b*x+a)^(1/2)*x+
55/1024*C*b^3/c^3*(c*x^2+b*x+a)^(1/2)*a^2-95/4096*C*b^5/c^4*(c*x^2+b*x+a)^(1/2)*a-5/384*C*a^2/c^2*(c*x^2+b*x+a
)^(3/2)*b-5/1024*A/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*b^6-5/32*A/c*(c*x^2+b*x+a)^(1/2)*x*a*b^
2+1/12*A/c*(c*x^2+b*x+a)^(5/2)*b+5/24*A*(c*x^2+b*x+a)^(3/2)*x*a-5/192*A/c^2*(c*x^2+b*x+a)^(3/2)*b^3+5/16*A*(c*
x^2+b*x+a)^(1/2)*x*a^2+5/512*A/c^3*(c*x^2+b*x+a)^(1/2)*b^5+5/16*A/c^(1/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)
^(1/2))*a^3-15/2048*C*b^5/c^4*(c*x^2+b*x+a)^(3/2)-45/32768*C*b^8/c^(11/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)
^(1/2))+45/16384*C*b^7/c^5*(c*x^2+b*x+a)^(1/2)-5/128*C*a^4/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))
-95/2048*C*b^4/c^3*(c*x^2+b*x+a)^(1/2)*x*a-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+25/
384*C*b^2/c^2*(c*x^2+b*x+a)^(3/2)*x*a+55/512*C*b^2/c^2*(c*x^2+b*x+a)^(1/2)*x*a^2+25/768*C*b^3/c^3*(c*x^2+b*x+a
)^(3/2)*a

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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

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Fricas [B]  time = 2.88837, size = 2342, normalized size = 8.77 \begin{align*} \text{result too large to display} \end{align*}

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., size = 0, normalized size = 0. \begin{align*} \int \left (A + C x^{2}\right ) \left (a + b x + c x^{2}\right )^{\frac{5}{2}}\, dx \end{align*}

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]  time = 1.25004, size = 651, normalized size = 2.44 \begin{align*} \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{align*}

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)