3.14.94 \(\int \frac {1}{(b d+2 c d x)^{5/2} (a+b x+c x^2)^{5/2}} \, dx\) [1394]

Optimal. Leaf size=228 \[ -\frac {2}{3 \left (b^2-4 a c\right ) d (b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )^{3/2}}+\frac {12 c}{\left (b^2-4 a c\right )^2 d (b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2}}+\frac {80 c^2 \sqrt {a+b x+c x^2}}{\left (b^2-4 a c\right )^3 d (b d+2 c d x)^{3/2}}+\frac {40 c \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\right |-1\right )}{\left (b^2-4 a c\right )^{11/4} d^{5/2} \sqrt {a+b x+c x^2}} \]

[Out]

-2/3/(-4*a*c+b^2)/d/(2*c*d*x+b*d)^(3/2)/(c*x^2+b*x+a)^(3/2)+12*c/(-4*a*c+b^2)^2/d/(2*c*d*x+b*d)^(3/2)/(c*x^2+b
*x+a)^(1/2)+80*c^2*(c*x^2+b*x+a)^(1/2)/(-4*a*c+b^2)^3/d/(2*c*d*x+b*d)^(3/2)+40*c*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)/(-4*a*c+b^2)^(11/4)/d^(5/2)/(c*x^2+b*x+a)
^(1/2)

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Rubi [A]
time = 0.12, antiderivative size = 228, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {701, 707, 705, 703, 227} \begin {gather*} \frac {40 c \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\text {ArcSin}\left (\frac {\sqrt {b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\right |-1\right )}{d^{5/2} \left (b^2-4 a c\right )^{11/4} \sqrt {a+b x+c x^2}}+\frac {80 c^2 \sqrt {a+b x+c x^2}}{d \left (b^2-4 a c\right )^3 (b d+2 c d x)^{3/2}}+\frac {12 c}{d \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2} (b d+2 c d x)^{3/2}}-\frac {2}{3 d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2} (b d+2 c d x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((b*d + 2*c*d*x)^(5/2)*(a + b*x + c*x^2)^(5/2)),x]

[Out]

-2/(3*(b^2 - 4*a*c)*d*(b*d + 2*c*d*x)^(3/2)*(a + b*x + c*x^2)^(3/2)) + (12*c)/((b^2 - 4*a*c)^2*d*(b*d + 2*c*d*
x)^(3/2)*Sqrt[a + b*x + c*x^2]) + (80*c^2*Sqrt[a + b*x + c*x^2])/((b^2 - 4*a*c)^3*d*(b*d + 2*c*d*x)^(3/2)) + (
40*c*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)*Sq
rt[d])], -1])/((b^2 - 4*a*c)^(11/4)*d^(5/2)*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 701

Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[2*c*(d + e*x)^(m + 1
)*((a + b*x + c*x^2)^(p + 1)/(e*(p + 1)*(b^2 - 4*a*c))), x] - Dist[2*c*e*((m + 2*p + 3)/(e*(p + 1)*(b^2 - 4*a*
c))), Int[(d + e*x)^m*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0]
&& EqQ[2*c*d - b*e, 0] && NeQ[m + 2*p + 3, 0] && LtQ[p, -1] &&  !GtQ[m, 1] && RationalQ[m] && IntegerQ[2*p]

Rule 703

Int[1/(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[1/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])

Rubi steps

\begin {align*} \int \frac {1}{(b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )^{5/2}} \, dx &=-\frac {2}{3 \left (b^2-4 a c\right ) d (b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )^{3/2}}-\frac {(6 c) \int \frac {1}{(b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )^{3/2}} \, dx}{b^2-4 a c}\\ &=-\frac {2}{3 \left (b^2-4 a c\right ) d (b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )^{3/2}}+\frac {12 c}{\left (b^2-4 a c\right )^2 d (b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2}}+\frac {\left (60 c^2\right ) \int \frac {1}{(b d+2 c d x)^{5/2} \sqrt {a+b x+c x^2}} \, dx}{\left (b^2-4 a c\right )^2}\\ &=-\frac {2}{3 \left (b^2-4 a c\right ) d (b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )^{3/2}}+\frac {12 c}{\left (b^2-4 a c\right )^2 d (b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2}}+\frac {80 c^2 \sqrt {a+b x+c x^2}}{\left (b^2-4 a c\right )^3 d (b d+2 c d x)^{3/2}}+\frac {\left (20 c^2\right ) \int \frac {1}{\sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}} \, dx}{\left (b^2-4 a c\right )^3 d^2}\\ &=-\frac {2}{3 \left (b^2-4 a c\right ) d (b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )^{3/2}}+\frac {12 c}{\left (b^2-4 a c\right )^2 d (b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2}}+\frac {80 c^2 \sqrt {a+b x+c x^2}}{\left (b^2-4 a c\right )^3 d (b d+2 c d x)^{3/2}}+\frac {\left (20 c^2 \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \int \frac {1}{\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}{\left (b^2-4 a c\right )^3 d^2 \sqrt {a+b x+c x^2}}\\ &=-\frac {2}{3 \left (b^2-4 a c\right ) d (b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )^{3/2}}+\frac {12 c}{\left (b^2-4 a c\right )^2 d (b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2}}+\frac {80 c^2 \sqrt {a+b x+c x^2}}{\left (b^2-4 a c\right )^3 d (b d+2 c d x)^{3/2}}+\frac {\left (40 c \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \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 )}{\left (b^2-4 a c\right )^3 d^3 \sqrt {a+b x+c x^2}}\\ &=-\frac {2}{3 \left (b^2-4 a c\right ) d (b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )^{3/2}}+\frac {12 c}{\left (b^2-4 a c\right )^2 d (b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2}}+\frac {80 c^2 \sqrt {a+b x+c x^2}}{\left (b^2-4 a c\right )^3 d (b d+2 c d x)^{3/2}}+\frac {40 c \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 )}{\left (b^2-4 a c\right )^{11/4} d^{5/2} \sqrt {a+b x+c x^2}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 10.07, size = 99, normalized size = 0.43 \begin {gather*} -\frac {32 c \sqrt {\frac {c (a+x (b+c x))}{-b^2+4 a c}} \, _2F_1\left (-\frac {3}{4},\frac {5}{2};\frac {1}{4};\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{3 \left (b^2-4 a c\right )^2 d (d (b+2 c x))^{3/2} \sqrt {a+x (b+c x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((b*d + 2*c*d*x)^(5/2)*(a + b*x + c*x^2)^(5/2)),x]

[Out]

(-32*c*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)]*Hypergeometric2F1[-3/4, 5/2, 1/4, (b + 2*c*x)^2/(b^2 - 4*a*c
)])/(3*(b^2 - 4*a*c)^2*d*(d*(b + 2*c*x))^(3/2)*Sqrt[a + x*(b + c*x)])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(796\) vs. \(2(200)=400\).
time = 0.78, size = 797, normalized size = 3.50

method	result	size
elliptic	\(\frac {\sqrt {d \left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )}\, \left (\frac {\left (-\frac {2 x^{2}}{c^{2} \left (4 a c -b^{2}\right )^{2} d^{3}}-\frac {2 b x}{c^{3} \left (4 a c -b^{2}\right )^{2} d^{3}}-\frac {8 a c +b^{2}}{6 \left (4 a c -b^{2}\right )^{2} d^{3} c^{4}}\right ) \sqrt {2 c^{2} d \,x^{3}+3 b c d \,x^{2}+2 a c d x +b^{2} d x +a b d}}{\left (x^{3}+\frac {3 b \,x^{2}}{2 c}+\frac {\left (2 a c +b^{2}\right ) x}{2 c^{2}}+\frac {a b}{2 c^{2}}\right )^{2}}-\frac {2 c^{2} d \left (\frac {10 x}{d^{3} \left (4 a c -b^{2}\right )^{3}}+\frac {5 b}{c \,d^{3} \left (4 a c -b^{2}\right )^{3}}\right ) \sqrt {2}}{\sqrt {\left (x^{3}+\frac {3 b \,x^{2}}{2 c}+\frac {\left (2 a c +b^{2}\right ) x}{2 c^{2}}+\frac {a b}{2 c^{2}}\right ) c^{2} d}}-\frac {40 c^{2} \left (\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \sqrt {\frac {x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {\frac {x +\frac {b}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b}{2 c}}}\, \sqrt {\frac {x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \EllipticF \left (\sqrt {\frac {x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}, \sqrt {\frac {-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b}{2 c}}}\right )}{d^{2} \left (4 a c -b^{2}\right )^{3} \sqrt {2 c^{2} d \,x^{3}+3 b c d \,x^{2}+2 a c d x +b^{2} d x +a b d}}\right )}{\sqrt {d \left (2 c x +b \right )}\, \sqrt {c \,x^{2}+b x +a}}\)	\(664\)
default	\(-\frac {2 \left (60 \sqrt {-4 a c +b^{2}}\, \sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-\frac {2 c x +b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {-b -2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \EllipticF \left (\frac {\sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {2}\right ) c^{3} x^{3}+90 \sqrt {-4 a c +b^{2}}\, \sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-\frac {2 c x +b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {-b -2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \EllipticF \left (\frac {\sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {2}\right ) b \,c^{2} x^{2}+60 \sqrt {-4 a c +b^{2}}\, \sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-\frac {2 c x +b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {-b -2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \EllipticF \left (\frac {\sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {2}\right ) a \,c^{2} x +30 \sqrt {-4 a c +b^{2}}\, \sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-\frac {2 c x +b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {-b -2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \EllipticF \left (\frac {\sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {2}\right ) b^{2} c x +30 \sqrt {-4 a c +b^{2}}\, \sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-\frac {2 c x +b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {-b -2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \EllipticF \left (\frac {\sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {2}\right ) a b c +120 c^{4} x^{4}+240 b \,c^{3} x^{3}+168 x^{2} c^{3} a +138 b^{2} c^{2} x^{2}+168 x a b \,c^{2}+18 b^{3} c x +32 a^{2} c^{2}+26 a c \,b^{2}-b^{4}\right ) \sqrt {d \left (2 c x +b \right )}}{3 d^{3} \left (2 c x +b \right )^{2} \left (4 a c -b^{2}\right )^{3} \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}\)	\(797\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(2*c*d*x+b*d)^(5/2)/(c*x^2+b*x+a)^(5/2),x,method=_RETURNVERBOSE)

[Out]

-2/3*(60*(-4*a*c+b^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)*((-b-2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*EllipticF(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))*c^3*x^3+90*(-4*a*c+b^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)*((-b-2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))
^(1/2)*EllipticF(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*c^2*x^2+60*(-4
*a*c+b^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)*
((-b-2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*EllipticF(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*c^2*x+30*(-4*a*c+b^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)*((-b-2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*Ellip
ticF(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^2*c*x+30*(-4*a*c+b^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)*((-b-2*c*x+(-4
*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*EllipticF(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*c+120*c^4*x^4+240*b*c^3*x^3+168*x^2*c^3*a+138*b^2*c^2*x^2+168*x*a*b*c^2+18*b^3*c*x+32*
a^2*c^2+26*a*c*b^2-b^4)*(d*(2*c*x+b))^(1/2)/d^3/(2*c*x+b)^2/(4*a*c-b^2)^3/(c*x^2+b*x+a)^(3/2)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2*c*d*x+b*d)^(5/2)/(c*x^2+b*x+a)^(5/2),x, algorithm="maxima")

[Out]

integrate(1/((2*c*d*x + b*d)^(5/2)*(c*x^2 + b*x + a)^(5/2)), x)

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.88, size = 597, normalized size = 2.62 \begin {gather*} \frac {2 \, {\left (30 \, \sqrt {2} {\left (4 \, c^{4} x^{6} + 12 \, b c^{3} x^{5} + {\left (13 \, b^{2} c^{2} + 8 \, a c^{3}\right )} x^{4} + a^{2} b^{2} + 2 \, {\left (3 \, b^{3} c + 8 \, a b c^{2}\right )} x^{3} + {\left (b^{4} + 10 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} x^{2} + 2 \, {\left (a b^{3} + 2 \, a^{2} b c\right )} x\right )} \sqrt {c^{2} d} {\rm weierstrassPInverse}\left (\frac {b^{2} - 4 \, a c}{c^{2}}, 0, \frac {2 \, c x + b}{2 \, c}\right ) + {\left (120 \, c^{4} x^{4} + 240 \, b c^{3} x^{3} - b^{4} + 26 \, a b^{2} c + 32 \, a^{2} c^{2} + 6 \, {\left (23 \, b^{2} c^{2} + 28 \, a c^{3}\right )} x^{2} + 6 \, {\left (3 \, b^{3} c + 28 \, a b c^{2}\right )} x\right )} \sqrt {2 \, c d x + b d} \sqrt {c x^{2} + b x + a}\right )}}{3 \, {\left (4 \, {\left (b^{6} c^{4} - 12 \, a b^{4} c^{5} + 48 \, a^{2} b^{2} c^{6} - 64 \, a^{3} c^{7}\right )} d^{3} x^{6} + 12 \, {\left (b^{7} c^{3} - 12 \, a b^{5} c^{4} + 48 \, a^{2} b^{3} c^{5} - 64 \, a^{3} b c^{6}\right )} d^{3} x^{5} + {\left (13 \, b^{8} c^{2} - 148 \, a b^{6} c^{3} + 528 \, a^{2} b^{4} c^{4} - 448 \, a^{3} b^{2} c^{5} - 512 \, a^{4} c^{6}\right )} d^{3} x^{4} + 2 \, {\left (3 \, b^{9} c - 28 \, a b^{7} c^{2} + 48 \, a^{2} b^{5} c^{3} + 192 \, a^{3} b^{3} c^{4} - 512 \, a^{4} b c^{5}\right )} d^{3} x^{3} + {\left (b^{10} - 2 \, a b^{8} c - 68 \, a^{2} b^{6} c^{2} + 368 \, a^{3} b^{4} c^{3} - 448 \, a^{4} b^{2} c^{4} - 256 \, a^{5} c^{5}\right )} d^{3} x^{2} + 2 \, {\left (a b^{9} - 10 \, a^{2} b^{7} c + 24 \, a^{3} b^{5} c^{2} + 32 \, a^{4} b^{3} c^{3} - 128 \, a^{5} b c^{4}\right )} d^{3} x + {\left (a^{2} b^{8} - 12 \, a^{3} b^{6} c + 48 \, a^{4} b^{4} c^{2} - 64 \, a^{5} b^{2} c^{3}\right )} d^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2*c*d*x+b*d)^(5/2)/(c*x^2+b*x+a)^(5/2),x, algorithm="fricas")

[Out]

2/3*(30*sqrt(2)*(4*c^4*x^6 + 12*b*c^3*x^5 + (13*b^2*c^2 + 8*a*c^3)*x^4 + a^2*b^2 + 2*(3*b^3*c + 8*a*b*c^2)*x^3
+ (b^4 + 10*a*b^2*c + 4*a^2*c^2)*x^2 + 2*(a*b^3 + 2*a^2*b*c)*x)*sqrt(c^2*d)*weierstrassPInverse((b^2 - 4*a*c)
/c^2, 0, 1/2*(2*c*x + b)/c) + (120*c^4*x^4 + 240*b*c^3*x^3 - b^4 + 26*a*b^2*c + 32*a^2*c^2 + 6*(23*b^2*c^2 + 2
8*a*c^3)*x^2 + 6*(3*b^3*c + 28*a*b*c^2)*x)*sqrt(2*c*d*x + b*d)*sqrt(c*x^2 + b*x + a))/(4*(b^6*c^4 - 12*a*b^4*c
^5 + 48*a^2*b^2*c^6 - 64*a^3*c^7)*d^3*x^6 + 12*(b^7*c^3 - 12*a*b^5*c^4 + 48*a^2*b^3*c^5 - 64*a^3*b*c^6)*d^3*x^
5 + (13*b^8*c^2 - 148*a*b^6*c^3 + 528*a^2*b^4*c^4 - 448*a^3*b^2*c^5 - 512*a^4*c^6)*d^3*x^4 + 2*(3*b^9*c - 28*a
*b^7*c^2 + 48*a^2*b^5*c^3 + 192*a^3*b^3*c^4 - 512*a^4*b*c^5)*d^3*x^3 + (b^10 - 2*a*b^8*c - 68*a^2*b^6*c^2 + 36
8*a^3*b^4*c^3 - 448*a^4*b^2*c^4 - 256*a^5*c^5)*d^3*x^2 + 2*(a*b^9 - 10*a^2*b^7*c + 24*a^3*b^5*c^2 + 32*a^4*b^3
*c^3 - 128*a^5*b*c^4)*d^3*x + (a^2*b^8 - 12*a^3*b^6*c + 48*a^4*b^4*c^2 - 64*a^5*b^2*c^3)*d^3)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (d \left (b + 2 c x\right )\right )^{\frac {5}{2}} \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(1/(2*c*d*x+b*d)**(5/2)/(c*x**2+b*x+a)**(5/2),x)

[Out]

Integral(1/((d*(b + 2*c*x))**(5/2)*(a + b*x + c*x**2)**(5/2)), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2*c*d*x+b*d)^(5/2)/(c*x^2+b*x+a)^(5/2),x, algorithm="giac")

[Out]

integrate(1/((2*c*d*x + b*d)^(5/2)*(c*x^2 + b*x + a)^(5/2)), x)

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

[Out]

int(1/((b*d + 2*c*d*x)^(5/2)*(a + b*x + c*x^2)^(5/2)), x)

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