\(\int \frac {(a+b x+c x^2)^{5/2}}{(b d+2 c d x)^{15/2}} \, dx\) [1362]

Optimal result

Integrand size = 28, antiderivative size = 357 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^{15/2}} \, dx=-\frac {\sqrt {a+b x+c x^2}}{156 c^3 d^5 (b d+2 c d x)^{5/2}}+\frac {\sqrt {a+b x+c x^2}}{78 c^3 \left (b^2-4 a c\right ) d^7 \sqrt {b d+2 c d x}}-\frac {5 \left (a+b x+c x^2\right )^{3/2}}{234 c^2 d^3 (b d+2 c d x)^{9/2}}-\frac {\left (a+b x+c x^2\right )^{5/2}}{13 c d (b d+2 c d x)^{13/2}}-\frac {\sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\arcsin \left (\frac {\sqrt {b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\right |-1\right )}{156 c^4 \sqrt [4]{b^2-4 a c} d^{15/2} \sqrt {a+b x+c x^2}}+\frac {\sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right ),-1\right )}{156 c^4 \sqrt [4]{b^2-4 a c} d^{15/2} \sqrt {a+b x+c x^2}} \]

[Out]

-5/234*(c*x^2+b*x+a)^(3/2)/c^2/d^3/(2*c*d*x+b*d)^(9/2)-1/13*(c*x^2+b*x+a)^(5/2)/c/d/(2*c*d*x+b*d)^(13/2)-1/156
*(c*x^2+b*x+a)^(1/2)/c^3/d^5/(2*c*d*x+b*d)^(5/2)+1/78*(c*x^2+b*x+a)^(1/2)/c^3/(-4*a*c+b^2)/d^7/(2*c*d*x+b*d)^(
1/2)-1/156*EllipticE((2*c*d*x+b*d)^(1/2)/(-4*a*c+b^2)^(1/4)/d^(1/2),I)*(-c*(c*x^2+b*x+a)/(-4*a*c+b^2))^(1/2)/c
^4/(-4*a*c+b^2)^(1/4)/d^(15/2)/(c*x^2+b*x+a)^(1/2)+1/156*EllipticF((2*c*d*x+b*d)^(1/2)/(-4*a*c+b^2)^(1/4)/d^(1
/2),I)*(-c*(c*x^2+b*x+a)/(-4*a*c+b^2))^(1/2)/c^4/(-4*a*c+b^2)^(1/4)/d^(15/2)/(c*x^2+b*x+a)^(1/2)

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 357, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {698, 707, 705, 704, 313, 227, 1213, 435} \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^{15/2}} \, dx=\frac {\sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right ),-1\right )}{156 c^4 d^{15/2} \sqrt [4]{b^2-4 a c} \sqrt {a+b x+c x^2}}-\frac {\sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\arcsin \left (\frac {\sqrt {b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\right |-1\right )}{156 c^4 d^{15/2} \sqrt [4]{b^2-4 a c} \sqrt {a+b x+c x^2}}+\frac {\sqrt {a+b x+c x^2}}{78 c^3 d^7 \left (b^2-4 a c\right ) \sqrt {b d+2 c d x}}-\frac {\sqrt {a+b x+c x^2}}{156 c^3 d^5 (b d+2 c d x)^{5/2}}-\frac {5 \left (a+b x+c x^2\right )^{3/2}}{234 c^2 d^3 (b d+2 c d x)^{9/2}}-\frac {\left (a+b x+c x^2\right )^{5/2}}{13 c d (b d+2 c d x)^{13/2}} \]

[In]

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

[Out]

-1/156*Sqrt[a + b*x + c*x^2]/(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
*Sqrt[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 227

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 313

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 435

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

Rule 698

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 704

Int[Sqrt[(d_) + (e_.)*(x_)]/Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[(4/e)*Sqrt[-c/(b^2 - 4*
a*c)], 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 705

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 707

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 1213

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]

Rubi steps \begin{align*} \text {integral}& = -\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}} \text {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}} \text {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}} \text {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}} \text {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] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 10.07 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.31 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^{15/2}} \, dx=-\frac {\left (b^2-4 a c\right )^2 \sqrt {d (b+2 c x)} \sqrt {a+x (b+c x)} \operatorname {Hypergeometric2F1}\left (-\frac {13}{4},-\frac {5}{2},-\frac {9}{4},\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{416 c^3 d^8 (b+2 c x)^7 \sqrt {\frac {c (a+x (b+c x))}{-b^2+4 a c}}} \]

[In]

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

[Out]

-1/416*((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)])/(c^3*d^8*(b + 2*c*x)^7*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1233\) vs. \(2(301)=602\).

Time = 5.69 (sec) , antiderivative size = 1234, normalized size of antiderivative = 3.46

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1234\)
default	\(\text {Expression too large to display}\)	\(2125\)

[In]

int((c*x^2+b*x+a)^(5/2)/(2*c*d*x+b*d)^(15/2),x,method=_RETURNVERBOSE)

[Out]

(d*(2*c*x+b)*(c*x^2+b*x+a))^(1/2)/(d*(2*c*x+b))^(1/2)/(c*x^2+b*x+a)^(1/2)*(-1/26624*(16*a^2*c^2-8*a*b^2*c+b^4)
/c^10/d^8*(2*c^2*d*x^3+3*b*c*d*x^2+2*a*c*d*x+b^2*d*x+a*b*d)^(1/2)/(x+1/2/c*b)^7-7/14976/c^8*(4*a*c-b^2)/d^8*(2
*c^2*d*x^3+3*b*c*d*x^2+2*a*c*d*x+b^2*d*x+a*b*d)^(1/2)/(x+1/2/c*b)^5-31/14976/c^6/d^8*(2*c^2*d*x^3+3*b*c*d*x^2+
2*a*c*d*x+b^2*d*x+a*b*d)^(1/2)/(x+1/2/c*b)^3-1/156*(2*c^2*d*x^2+2*b*c*d*x+2*a*c*d)/c^4/(4*a*c-b^2)/d^8/((x+1/2
/c*b)*(2*c^2*d*x^2+2*b*c*d*x+2*a*c*d))^(1/2)+1/156/c^3*b/(4*a*c-b^2)/d^7*(1/2/c*(-b+(-4*a*c+b^2)^(1/2))+1/2*(b
+(-4*a*c+b^2)^(1/2))/c)*((x+1/2*(b+(-4*a*c+b^2)^(1/2))/c)/(1/2/c*(-b+(-4*a*c+b^2)^(1/2))+1/2*(b+(-4*a*c+b^2)^(
1/2))/c))^(1/2)*((x+1/2/c*b)/(-1/2*(b+(-4*a*c+b^2)^(1/2))/c+1/2/c*b))^(1/2)*((x-1/2/c*(-b+(-4*a*c+b^2)^(1/2)))
/(-1/2*(b+(-4*a*c+b^2)^(1/2))/c-1/2/c*(-b+(-4*a*c+b^2)^(1/2))))^(1/2)/(2*c^2*d*x^3+3*b*c*d*x^2+2*a*c*d*x+b^2*d
*x+a*b*d)^(1/2)*EllipticF(((x+1/2*(b+(-4*a*c+b^2)^(1/2))/c)/(1/2/c*(-b+(-4*a*c+b^2)^(1/2))+1/2*(b+(-4*a*c+b^2)
^(1/2))/c))^(1/2),((-1/2*(b+(-4*a*c+b^2)^(1/2))/c-1/2/c*(-b+(-4*a*c+b^2)^(1/2)))/(-1/2*(b+(-4*a*c+b^2)^(1/2))/
c+1/2/c*b))^(1/2))+1/78/c^2/(4*a*c-b^2)/d^7*(1/2/c*(-b+(-4*a*c+b^2)^(1/2))+1/2*(b+(-4*a*c+b^2)^(1/2))/c)*((x+1
/2*(b+(-4*a*c+b^2)^(1/2))/c)/(1/2/c*(-b+(-4*a*c+b^2)^(1/2))+1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2)*((x+1/2/c*b)/
(-1/2*(b+(-4*a*c+b^2)^(1/2))/c+1/2/c*b))^(1/2)*((x-1/2/c*(-b+(-4*a*c+b^2)^(1/2)))/(-1/2*(b+(-4*a*c+b^2)^(1/2))
/c-1/2/c*(-b+(-4*a*c+b^2)^(1/2))))^(1/2)/(2*c^2*d*x^3+3*b*c*d*x^2+2*a*c*d*x+b^2*d*x+a*b*d)^(1/2)*((-1/2*(b+(-4
*a*c+b^2)^(1/2))/c+1/2/c*b)*EllipticE(((x+1/2*(b+(-4*a*c+b^2)^(1/2))/c)/(1/2/c*(-b+(-4*a*c+b^2)^(1/2))+1/2*(b+
(-4*a*c+b^2)^(1/2))/c))^(1/2),((-1/2*(b+(-4*a*c+b^2)^(1/2))/c-1/2/c*(-b+(-4*a*c+b^2)^(1/2)))/(-1/2*(b+(-4*a*c+
b^2)^(1/2))/c+1/2/c*b))^(1/2))-1/2/c*b*EllipticF(((x+1/2*(b+(-4*a*c+b^2)^(1/2))/c)/(1/2/c*(-b+(-4*a*c+b^2)^(1/
2))+1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2),((-1/2*(b+(-4*a*c+b^2)^(1/2))/c-1/2/c*(-b+(-4*a*c+b^2)^(1/2)))/(-1/2*
(b+(-4*a*c+b^2)^(1/2))/c+1/2/c*b))^(1/2))))

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.18 (sec) , antiderivative size = 496, normalized size of antiderivative = 1.39 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^{15/2}} \, dx=\frac {3 \, \sqrt {2} {\left (128 \, c^{7} x^{7} + 448 \, b c^{6} x^{6} + 672 \, b^{2} c^{5} x^{5} + 560 \, b^{3} c^{4} x^{4} + 280 \, b^{4} c^{3} x^{3} + 84 \, b^{5} c^{2} x^{2} + 14 \, b^{6} c x + b^{7}\right )} \sqrt {c^{2} d} {\rm weierstrassZeta}\left (\frac {b^{2} - 4 \, a c}{c^{2}}, 0, {\rm weierstrassPInverse}\left (\frac {b^{2} - 4 \, a c}{c^{2}}, 0, \frac {2 \, c x + b}{2 \, c}\right )\right ) + {\left (384 \, c^{7} x^{6} + 1152 \, b c^{6} x^{5} + 3 \, b^{6} c + 2 \, a b^{4} c^{2} + 4 \, a^{2} b^{2} c^{3} + 144 \, a^{3} c^{4} + 4 \, {\left (329 \, b^{2} c^{5} + 124 \, a c^{6}\right )} x^{4} + 8 \, {\left (89 \, b^{3} c^{4} + 124 \, a b c^{5}\right )} x^{3} + 2 \, {\left (101 \, b^{4} c^{3} + 260 \, a b^{2} c^{4} + 224 \, a^{2} c^{5}\right )} x^{2} + 2 \, {\left (19 \, b^{5} c^{2} + 12 \, a b^{3} c^{3} + 224 \, a^{2} b c^{4}\right )} x\right )} \sqrt {2 \, c d x + b d} \sqrt {c x^{2} + b x + a}}{468 \, {\left (128 \, {\left (b^{2} c^{11} - 4 \, a c^{12}\right )} d^{8} x^{7} + 448 \, {\left (b^{3} c^{10} - 4 \, a b c^{11}\right )} d^{8} x^{6} + 672 \, {\left (b^{4} c^{9} - 4 \, a b^{2} c^{10}\right )} d^{8} x^{5} + 560 \, {\left (b^{5} c^{8} - 4 \, a b^{3} c^{9}\right )} d^{8} x^{4} + 280 \, {\left (b^{6} c^{7} - 4 \, a b^{4} c^{8}\right )} d^{8} x^{3} + 84 \, {\left (b^{7} c^{6} - 4 \, a b^{5} c^{7}\right )} d^{8} x^{2} + 14 \, {\left (b^{8} c^{5} - 4 \, a b^{6} c^{6}\right )} d^{8} x + {\left (b^{9} c^{4} - 4 \, a b^{7} c^{5}\right )} d^{8}\right )}} \]

[In]

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

[Out]

1/468*(3*sqrt(2)*(128*c^7*x^7 + 448*b*c^6*x^6 + 672*b^2*c^5*x^5 + 560*b^3*c^4*x^4 + 280*b^4*c^3*x^3 + 84*b^5*c
^2*x^2 + 14*b^6*c*x + b^7)*sqrt(c^2*d)*weierstrassZeta((b^2 - 4*a*c)/c^2, 0, weierstrassPInverse((b^2 - 4*a*c)
/c^2, 0, 1/2*(2*c*x + b)/c)) + (384*c^7*x^6 + 1152*b*c^6*x^5 + 3*b^6*c + 2*a*b^4*c^2 + 4*a^2*b^2*c^3 + 144*a^3
*c^4 + 4*(329*b^2*c^5 + 124*a*c^6)*x^4 + 8*(89*b^3*c^4 + 124*a*b*c^5)*x^3 + 2*(101*b^4*c^3 + 260*a*b^2*c^4 + 2
24*a^2*c^5)*x^2 + 2*(19*b^5*c^2 + 12*a*b^3*c^3 + 224*a^2*b*c^4)*x)*sqrt(2*c*d*x + b*d)*sqrt(c*x^2 + b*x + a))/
(128*(b^2*c^11 - 4*a*c^12)*d^8*x^7 + 448*(b^3*c^10 - 4*a*b*c^11)*d^8*x^6 + 672*(b^4*c^9 - 4*a*b^2*c^10)*d^8*x^
5 + 560*(b^5*c^8 - 4*a*b^3*c^9)*d^8*x^4 + 280*(b^6*c^7 - 4*a*b^4*c^8)*d^8*x^3 + 84*(b^7*c^6 - 4*a*b^5*c^7)*d^8
*x^2 + 14*(b^8*c^5 - 4*a*b^6*c^6)*d^8*x + (b^9*c^4 - 4*a*b^7*c^5)*d^8)

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^{15/2}} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

\[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^{15/2}} \, dx=\int { \frac {{\left (c x^{2} + b x + a\right )}^{\frac {5}{2}}}{{\left (2 \, c d x + b d\right )}^{\frac {15}{2}}} \,d x } \]

[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)

Giac [F]

\[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^{15/2}} \, dx=\int { \frac {{\left (c x^{2} + b x + a\right )}^{\frac {5}{2}}}{{\left (2 \, c d x + b d\right )}^{\frac {15}{2}}} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^{15/2}} \, dx=\int \frac {{\left (c\,x^2+b\,x+a\right )}^{5/2}}{{\left (b\,d+2\,c\,d\,x\right )}^{15/2}} \,d x \]

[In]

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

[Out]

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