### 3.1362 $$\int \frac{(a+b x+c x^2)^{5/2}}{(b d+2 c d x)^{15/2}} \, dx$$

Optimal. Leaf size=357 $\frac{\sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{b d+2 c d x}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right ),-1\right )}{156 c^4 d^{15/2} \sqrt [4]{b^2-4 a c} \sqrt{a+b x+c x^2}}+\frac{\sqrt{a+b x+c x^2}}{78 c^3 d^7 \left (b^2-4 a c\right ) \sqrt{b d+2 c d x}}-\frac{\sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{156 c^4 d^{15/2} \sqrt [4]{b^2-4 a c} \sqrt{a+b x+c x^2}}-\frac{5 \left (a+b x+c x^2\right )^{3/2}}{234 c^2 d^3 (b d+2 c d x)^{9/2}}-\frac{\sqrt{a+b x+c x^2}}{156 c^3 d^5 (b d+2 c d x)^{5/2}}-\frac{\left (a+b x+c x^2\right )^{5/2}}{13 c d (b d+2 c d x)^{13/2}}$

[Out]

-Sqrt[a + b*x + c*x^2]/(156*c^3*d^5*(b*d + 2*c*d*x)^(5/2)) + Sqrt[a + b*x + c*x^2]/(78*c^3*(b^2 - 4*a*c)*d^7*S
qrt[b*d + 2*c*d*x]) - (5*(a + b*x + c*x^2)^(3/2))/(234*c^2*d^3*(b*d + 2*c*d*x)^(9/2)) - (a + b*x + c*x^2)^(5/2
)/(13*c*d*(b*d + 2*c*d*x)^(13/2)) - (Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticE[ArcSin[Sqrt[b*d +
2*c*d*x]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])], -1])/(156*c^4*(b^2 - 4*a*c)^(1/4)*d^(15/2)*Sqrt[a + b*x + c*x^2]) + (
Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticF[ArcSin[Sqrt[b*d + 2*c*d*x]/((b^2 - 4*a*c)^(1/4)*Sqrt[d]
)], -1])/(156*c^4*(b^2 - 4*a*c)^(1/4)*d^(15/2)*Sqrt[a + b*x + c*x^2])

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Rubi [A]  time = 0.337042, antiderivative size = 357, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 28, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.286, Rules used = {684, 693, 691, 690, 307, 221, 1199, 424} $\frac{\sqrt{a+b x+c x^2}}{78 c^3 d^7 \left (b^2-4 a c\right ) \sqrt{b d+2 c d x}}+\frac{\sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{156 c^4 d^{15/2} \sqrt [4]{b^2-4 a c} \sqrt{a+b x+c x^2}}-\frac{\sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{156 c^4 d^{15/2} \sqrt [4]{b^2-4 a c} \sqrt{a+b x+c x^2}}-\frac{5 \left (a+b x+c x^2\right )^{3/2}}{234 c^2 d^3 (b d+2 c d x)^{9/2}}-\frac{\sqrt{a+b x+c x^2}}{156 c^3 d^5 (b d+2 c d x)^{5/2}}-\frac{\left (a+b x+c x^2\right )^{5/2}}{13 c d (b d+2 c d x)^{13/2}}$

Antiderivative was successfully veriﬁed.

[In]

Int[(a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^(15/2),x]

[Out]

-Sqrt[a + b*x + c*x^2]/(156*c^3*d^5*(b*d + 2*c*d*x)^(5/2)) + Sqrt[a + b*x + c*x^2]/(78*c^3*(b^2 - 4*a*c)*d^7*S
qrt[b*d + 2*c*d*x]) - (5*(a + b*x + c*x^2)^(3/2))/(234*c^2*d^3*(b*d + 2*c*d*x)^(9/2)) - (a + b*x + c*x^2)^(5/2
)/(13*c*d*(b*d + 2*c*d*x)^(13/2)) - (Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticE[ArcSin[Sqrt[b*d +
2*c*d*x]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])], -1])/(156*c^4*(b^2 - 4*a*c)^(1/4)*d^(15/2)*Sqrt[a + b*x + c*x^2]) + (
Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticF[ArcSin[Sqrt[b*d + 2*c*d*x]/((b^2 - 4*a*c)^(1/4)*Sqrt[d]
)], -1])/(156*c^4*(b^2 - 4*a*c)^(1/4)*d^(15/2)*Sqrt[a + b*x + c*x^2])

Rule 684

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 + 1)), x] - Dist[(b*p)/(d*e*(m + 1)), Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2)^(p - 1
), x], x] /; FreeQ[{a, b, c, d, e}, 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] &&  !(IntegerQ[m/2] && LtQ[m + 2*p + 3, 0]) && IntegerQ[2*p]

Rule 693

Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-2*b*d*(d + e*x)^(m
+ 1)*(a + b*x + c*x^2)^(p + 1))/(d^2*(m + 1)*(b^2 - 4*a*c)), x] + Dist[(b^2*(m + 2*p + 3))/(d^2*(m + 1)*(b^2
- 4*a*c)), Int[(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] && LtQ[m, -1] && (IntegerQ[2*p] || (IntegerQ[m] && Rationa
lQ[p]) || IntegerQ[(m + 2*p + 3)/2])

Rule 691

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

Rule 690

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

Rule 307

Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[-(b/a), 2]}, -Dist[q^(-1), Int[1/Sqrt[a + b*x^
4], x], x] + Dist[1/q, Int[(1 + q*x^2)/Sqrt[a + b*x^4], x], x]] /; FreeQ[{a, b}, x] && NegQ[b/a]

Rule 221

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[EllipticF[ArcSin[(Rt[-b, 4]*x)/Rt[a, 4]], -1]/(Rt[a, 4]*Rt[
-b, 4]), x] /; FreeQ[{a, b}, x] && NegQ[b/a] && GtQ[a, 0]

Rule 1199

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> Dist[d/Sqrt[a], Int[Sqrt[1 + (e*x^2)/d]/Sqrt
[1 - (e*x^2)/d], x], x] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] && EqQ[c*d^2 + a*e^2, 0] && GtQ[a, 0]

Rule 424

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]*EllipticE[ArcSin[Rt[-(d/c)
, 2]*x], (b*c)/(a*d)])/(Sqrt[c]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[
a, 0]

Rubi steps

\begin{align*} \int \frac{\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^{15/2}} \, dx &=-\frac{\left (a+b x+c x^2\right )^{5/2}}{13 c d (b d+2 c d x)^{13/2}}+\frac{5 \int \frac{\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^{11/2}} \, dx}{26 c d^2}\\ &=-\frac{5 \left (a+b x+c x^2\right )^{3/2}}{234 c^2 d^3 (b d+2 c d x)^{9/2}}-\frac{\left (a+b x+c x^2\right )^{5/2}}{13 c d (b d+2 c d x)^{13/2}}+\frac{5 \int \frac{\sqrt{a+b x+c x^2}}{(b d+2 c d x)^{7/2}} \, dx}{156 c^2 d^4}\\ &=-\frac{\sqrt{a+b x+c x^2}}{156 c^3 d^5 (b d+2 c d x)^{5/2}}-\frac{5 \left (a+b x+c x^2\right )^{3/2}}{234 c^2 d^3 (b d+2 c d x)^{9/2}}-\frac{\left (a+b x+c x^2\right )^{5/2}}{13 c d (b d+2 c d x)^{13/2}}+\frac{\int \frac{1}{(b d+2 c d x)^{3/2} \sqrt{a+b x+c x^2}} \, dx}{312 c^3 d^6}\\ &=-\frac{\sqrt{a+b x+c x^2}}{156 c^3 d^5 (b d+2 c d x)^{5/2}}+\frac{\sqrt{a+b x+c x^2}}{78 c^3 \left (b^2-4 a c\right ) d^7 \sqrt{b d+2 c d x}}-\frac{5 \left (a+b x+c x^2\right )^{3/2}}{234 c^2 d^3 (b d+2 c d x)^{9/2}}-\frac{\left (a+b x+c x^2\right )^{5/2}}{13 c d (b d+2 c d x)^{13/2}}-\frac{\int \frac{\sqrt{b d+2 c d x}}{\sqrt{a+b x+c x^2}} \, dx}{312 c^3 \left (b^2-4 a c\right ) d^8}\\ &=-\frac{\sqrt{a+b x+c x^2}}{156 c^3 d^5 (b d+2 c d x)^{5/2}}+\frac{\sqrt{a+b x+c x^2}}{78 c^3 \left (b^2-4 a c\right ) d^7 \sqrt{b d+2 c d x}}-\frac{5 \left (a+b x+c x^2\right )^{3/2}}{234 c^2 d^3 (b d+2 c d x)^{9/2}}-\frac{\left (a+b x+c x^2\right )^{5/2}}{13 c d (b d+2 c d x)^{13/2}}-\frac{\sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \int \frac{\sqrt{b d+2 c d x}}{\sqrt{-\frac{a c}{b^2-4 a c}-\frac{b c x}{b^2-4 a c}-\frac{c^2 x^2}{b^2-4 a c}}} \, dx}{312 c^3 \left (b^2-4 a c\right ) d^8 \sqrt{a+b x+c x^2}}\\ &=-\frac{\sqrt{a+b x+c x^2}}{156 c^3 d^5 (b d+2 c d x)^{5/2}}+\frac{\sqrt{a+b x+c x^2}}{78 c^3 \left (b^2-4 a c\right ) d^7 \sqrt{b d+2 c d x}}-\frac{5 \left (a+b x+c x^2\right )^{3/2}}{234 c^2 d^3 (b d+2 c d x)^{9/2}}-\frac{\left (a+b x+c x^2\right )^{5/2}}{13 c d (b d+2 c d x)^{13/2}}-\frac{\sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{1-\frac{x^4}{\left (b^2-4 a c\right ) d^2}}} \, dx,x,\sqrt{b d+2 c d x}\right )}{156 c^4 \left (b^2-4 a c\right ) d^9 \sqrt{a+b x+c x^2}}\\ &=-\frac{\sqrt{a+b x+c x^2}}{156 c^3 d^5 (b d+2 c d x)^{5/2}}+\frac{\sqrt{a+b x+c x^2}}{78 c^3 \left (b^2-4 a c\right ) d^7 \sqrt{b d+2 c d x}}-\frac{5 \left (a+b x+c x^2\right )^{3/2}}{234 c^2 d^3 (b d+2 c d x)^{9/2}}-\frac{\left (a+b x+c x^2\right )^{5/2}}{13 c d (b d+2 c d x)^{13/2}}+\frac{\sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x^4}{\left (b^2-4 a c\right ) d^2}}} \, dx,x,\sqrt{b d+2 c d x}\right )}{156 c^4 \sqrt{b^2-4 a c} d^8 \sqrt{a+b x+c x^2}}-\frac{\sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname{Subst}\left (\int \frac{1+\frac{x^2}{\sqrt{b^2-4 a c} d}}{\sqrt{1-\frac{x^4}{\left (b^2-4 a c\right ) d^2}}} \, dx,x,\sqrt{b d+2 c d x}\right )}{156 c^4 \sqrt{b^2-4 a c} d^8 \sqrt{a+b x+c x^2}}\\ &=-\frac{\sqrt{a+b x+c x^2}}{156 c^3 d^5 (b d+2 c d x)^{5/2}}+\frac{\sqrt{a+b x+c x^2}}{78 c^3 \left (b^2-4 a c\right ) d^7 \sqrt{b d+2 c d x}}-\frac{5 \left (a+b x+c x^2\right )^{3/2}}{234 c^2 d^3 (b d+2 c d x)^{9/2}}-\frac{\left (a+b x+c x^2\right )^{5/2}}{13 c d (b d+2 c d x)^{13/2}}+\frac{\sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{156 c^4 \sqrt [4]{b^2-4 a c} d^{15/2} \sqrt{a+b x+c x^2}}-\frac{\sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{x^2}{\sqrt{b^2-4 a c} d}}}{\sqrt{1-\frac{x^2}{\sqrt{b^2-4 a c} d}}} \, dx,x,\sqrt{b d+2 c d x}\right )}{156 c^4 \sqrt{b^2-4 a c} d^8 \sqrt{a+b x+c x^2}}\\ &=-\frac{\sqrt{a+b x+c x^2}}{156 c^3 d^5 (b d+2 c d x)^{5/2}}+\frac{\sqrt{a+b x+c x^2}}{78 c^3 \left (b^2-4 a c\right ) d^7 \sqrt{b d+2 c d x}}-\frac{5 \left (a+b x+c x^2\right )^{3/2}}{234 c^2 d^3 (b d+2 c d x)^{9/2}}-\frac{\left (a+b x+c x^2\right )^{5/2}}{13 c d (b d+2 c d x)^{13/2}}-\frac{\sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{156 c^4 \sqrt [4]{b^2-4 a c} d^{15/2} \sqrt{a+b x+c x^2}}+\frac{\sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{156 c^4 \sqrt [4]{b^2-4 a c} d^{15/2} \sqrt{a+b x+c x^2}}\\ \end{align*}

Mathematica [C]  time = 0.0876682, size = 109, normalized size = 0.31 $-\frac{\left (b^2-4 a c\right )^2 \sqrt{a+x (b+c x)} \sqrt{d (b+2 c x)} \, _2F_1\left (-\frac{13}{4},-\frac{5}{2};-\frac{9}{4};\frac{(b+2 c x)^2}{b^2-4 a c}\right )}{416 c^3 d^8 (b+2 c x)^7 \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}}}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^(15/2),x]

[Out]

-((b^2 - 4*a*c)^2*Sqrt[d*(b + 2*c*x)]*Sqrt[a + x*(b + c*x)]*Hypergeometric2F1[-13/4, -5/2, -9/4, (b + 2*c*x)^2
/(b^2 - 4*a*c)])/(416*c^3*d^8*(b + 2*c*x)^7*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)])

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Maple [B]  time = 0.33, size = 2125, normalized size = 6. \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)/(2*c*d*x+b*d)^(15/2),x)

[Out]

1/936*(c*x^2+b*x+a)^(1/2)*(d*(2*c*x+b))^(1/2)*(-768*x^8*c^8-6*x*b^7*c+720*EllipticE(1/2*((b+2*c*x+(-4*a*c+b^2)
^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),2^(1/2))*x^2*a*b^4*c^3*((-b-2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(
1/2))^(1/2)*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*(-(2*c*x+b)/(-4*a*c+b^2)^(1/2))^(1/2)+144*
EllipticE(1/2*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),2^(1/2))*x*a*b^5*c^2*((-b-2*c*x+
(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*(-(2*c*x
+b)/(-4*a*c+b^2)^(1/2))^(1/2)+2304*EllipticE(1/2*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/
2),2^(1/2))*x^5*a*b*c^6*((-b-2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*((b+2*c*x+(-4*a*c+b^2)^(1/2))
/(-4*a*c+b^2)^(1/2))^(1/2)*(-(2*c*x+b)/(-4*a*c+b^2)^(1/2))^(1/2)+2880*EllipticE(1/2*((b+2*c*x+(-4*a*c+b^2)^(1/
2))/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),2^(1/2))*x^4*a*b^2*c^5*((-b-2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2)
)^(1/2)*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*(-(2*c*x+b)/(-4*a*c+b^2)^(1/2))^(1/2)+1920*Ell
ipticE(1/2*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),2^(1/2))*x^3*a*b^3*c^4*((-b-2*c*x+(
-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*(-(2*c*x+
b)/(-4*a*c+b^2)^(1/2))^(1/2)-3*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*(-(2*c*x+b)/(-4*a*c+b^2
)^(1/2))^(1/2)*((-b-2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*EllipticE(1/2*((b+2*c*x+(-4*a*c+b^2)^(
1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),2^(1/2))*b^8+768*EllipticE(1/2*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b
^2)^(1/2))^(1/2)*2^(1/2),2^(1/2))*x^6*a*c^7*((-b-2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*((b+2*c*x
+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*(-(2*c*x+b)/(-4*a*c+b^2)^(1/2))^(1/2)-192*EllipticE(1/2*((b+2*c
*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),2^(1/2))*x^6*b^2*c^6*((-b-2*c*x+(-4*a*c+b^2)^(1/2))/(
-4*a*c+b^2)^(1/2))^(1/2)*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*(-(2*c*x+b)/(-4*a*c+b^2)^(1/2
))^(1/2)-576*EllipticE(1/2*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),2^(1/2))*x^5*b^3*c^
5*((-b-2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(
1/2)*(-(2*c*x+b)/(-4*a*c+b^2)^(1/2))^(1/2)-720*EllipticE(1/2*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))
^(1/2)*2^(1/2),2^(1/2))*x^4*b^4*c^4*((-b-2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*((b+2*c*x+(-4*a*c
+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*(-(2*c*x+b)/(-4*a*c+b^2)^(1/2))^(1/2)-480*EllipticE(1/2*((b+2*c*x+(-4*a
*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),2^(1/2))*x^3*b^5*c^3*((-b-2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b
^2)^(1/2))^(1/2)*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*(-(2*c*x+b)/(-4*a*c+b^2)^(1/2))^(1/2)
-180*EllipticE(1/2*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),2^(1/2))*x^2*b^6*c^2*((-b-2
*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*(-(
2*c*x+b)/(-4*a*c+b^2)^(1/2))^(1/2)-36*EllipticE(1/2*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*2^
(1/2),2^(1/2))*x*b^7*c*((-b-2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*((b+2*c*x+(-4*a*c+b^2)^(1/2))/
(-4*a*c+b^2)^(1/2))^(1/2)*(-(2*c*x+b)/(-4*a*c+b^2)^(1/2))^(1/2)-1828*x^4*b^4*c^4-4056*x^5*b^3*c^5-1760*x^6*a*c
^7-1888*x^4*a^2*c^6-1184*x^2*a^3*c^5-3072*x^7*b*c^7-480*x^3*b^5*c^3-82*x^2*b^6*c^2-4936*x^6*b^2*c^6-288*a^4*c^
4-8*a^3*b^2*c^3-4*a^2*b^4*c^2-6*a*b^6*c-80*x*a*b^5*c^2-5280*x^5*a*b*c^6-5656*x^4*a*b^2*c^5-3776*x^3*a^2*b*c^5-
2512*x^3*a*b^3*c^4-1944*x^2*a^2*b^2*c^4-456*x^2*a*b^4*c^3-1184*x*a^3*b*c^4-56*x*a^2*b^3*c^3+12*((b+2*c*x+(-4*a
*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*(-(2*c*x+b)/(-4*a*c+b^2)^(1/2))^(1/2)*((-b-2*c*x+(-4*a*c+b^2)^(1/2))/
(-4*a*c+b^2)^(1/2))^(1/2)*EllipticE(1/2*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),2^(1/2
))*a*b^6*c)/d^8/(2*c^2*x^3+3*b*c*x^2+2*a*c*x+b^2*x+a*b)/(2*c*x+b)^6/(4*a*c-b^2)/c^4

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (c x^{2} + b x + a\right )}^{\frac{5}{2}}}{{\left (2 \, c d x + b d\right )}^{\frac{15}{2}}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^(5/2)/(2*c*d*x+b*d)^(15/2),x, algorithm="maxima")

[Out]

integrate((c*x^2 + b*x + a)^(5/2)/(2*c*d*x + b*d)^(15/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x +{\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )} \sqrt{2 \, c d x + b d} \sqrt{c x^{2} + b x + a}}{256 \, c^{8} d^{8} x^{8} + 1024 \, b c^{7} d^{8} x^{7} + 1792 \, b^{2} c^{6} d^{8} x^{6} + 1792 \, b^{3} c^{5} d^{8} x^{5} + 1120 \, b^{4} c^{4} d^{8} x^{4} + 448 \, b^{5} c^{3} d^{8} x^{3} + 112 \, b^{6} c^{2} d^{8} x^{2} + 16 \, b^{7} c d^{8} x + b^{8} d^{8}}, x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^(5/2)/(2*c*d*x+b*d)^(15/2),x, algorithm="fricas")

[Out]

integral((c^2*x^4 + 2*b*c*x^3 + 2*a*b*x + (b^2 + 2*a*c)*x^2 + a^2)*sqrt(2*c*d*x + b*d)*sqrt(c*x^2 + b*x + a)/(
256*c^8*d^8*x^8 + 1024*b*c^7*d^8*x^7 + 1792*b^2*c^6*d^8*x^6 + 1792*b^3*c^5*d^8*x^5 + 1120*b^4*c^4*d^8*x^4 + 44
8*b^5*c^3*d^8*x^3 + 112*b^6*c^2*d^8*x^2 + 16*b^7*c*d^8*x + b^8*d^8), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+b*x+a)**(5/2)/(2*c*d*x+b*d)**(15/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (c x^{2} + b x + a\right )}^{\frac{5}{2}}}{{\left (2 \, c d x + b d\right )}^{\frac{15}{2}}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^(5/2)/(2*c*d*x+b*d)^(15/2),x, algorithm="giac")

[Out]

integrate((c*x^2 + b*x + a)^(5/2)/(2*c*d*x + b*d)^(15/2), x)