\(\int \frac {1}{(b d+2 c d x)^{7/2} \sqrt {a+b x+c x^2}} \, dx\) [1372]

Optimal result

Integrand size = 28, antiderivative size = 287 \[ \int \frac {1}{(b d+2 c d x)^{7/2} \sqrt {a+b x+c x^2}} \, dx=\frac {4 \sqrt {a+b x+c x^2}}{5 \left (b^2-4 a c\right ) d (b d+2 c d x)^{5/2}}+\frac {12 \sqrt {a+b x+c x^2}}{5 \left (b^2-4 a c\right )^2 d^3 \sqrt {b d+2 c d x}}-\frac {6 \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\arcsin \left (\frac {\sqrt {b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\right |-1\right )}{5 c \left (b^2-4 a c\right )^{5/4} d^{7/2} \sqrt {a+b x+c x^2}}+\frac {6 \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right ),-1\right )}{5 c \left (b^2-4 a c\right )^{5/4} d^{7/2} \sqrt {a+b x+c x^2}} \]

[Out]

4/5*(c*x^2+b*x+a)^(1/2)/(-4*a*c+b^2)/d/(2*c*d*x+b*d)^(5/2)+12/5*(c*x^2+b*x+a)^(1/2)/(-4*a*c+b^2)^2/d^3/(2*c*d*
x+b*d)^(1/2)-6/5*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*a*c+b^2)^(5/4)/d^(7/2)/(c*x^2+b*x+a)^(1/2)+6/5*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*a*c+b^2)^(5/4)/d^(7/2)/(c*x^2+b*x+a)^(1/2)

Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {707, 705, 704, 313, 227, 1213, 435} \[ \int \frac {1}{(b d+2 c d x)^{7/2} \sqrt {a+b x+c x^2}} \, dx=\frac {6 \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right ),-1\right )}{5 c d^{7/2} \left (b^2-4 a c\right )^{5/4} \sqrt {a+b x+c x^2}}-\frac {6 \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\arcsin \left (\frac {\sqrt {b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\right |-1\right )}{5 c d^{7/2} \left (b^2-4 a c\right )^{5/4} \sqrt {a+b x+c x^2}}+\frac {12 \sqrt {a+b x+c x^2}}{5 d^3 \left (b^2-4 a c\right )^2 \sqrt {b d+2 c d x}}+\frac {4 \sqrt {a+b x+c x^2}}{5 d \left (b^2-4 a c\right ) (b d+2 c d x)^{5/2}} \]

[In]

Int[1/((b*d + 2*c*d*x)^(7/2)*Sqrt[a + b*x + c*x^2]),x]

[Out]

(4*Sqrt[a + b*x + c*x^2])/(5*(b^2 - 4*a*c)*d*(b*d + 2*c*d*x)^(5/2)) + (12*Sqrt[a + b*x + c*x^2])/(5*(b^2 - 4*a
*c)^2*d^3*Sqrt[b*d + 2*c*d*x]) - (6*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])/(5*c*(b^2 - 4*a*c)^(5/4)*d^(7/2)*Sqrt[a + b*x + c*x^2]) + (6*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])/(5*c*(b^2 - 4*a*c)^(5/4)*d^(7/2)*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 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 {4 \sqrt {a+b x+c x^2}}{5 \left (b^2-4 a c\right ) d (b d+2 c d x)^{5/2}}+\frac {3 \int \frac {1}{(b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2}} \, dx}{5 \left (b^2-4 a c\right ) d^2} \\ & = \frac {4 \sqrt {a+b x+c x^2}}{5 \left (b^2-4 a c\right ) d (b d+2 c d x)^{5/2}}+\frac {12 \sqrt {a+b x+c x^2}}{5 \left (b^2-4 a c\right )^2 d^3 \sqrt {b d+2 c d x}}-\frac {3 \int \frac {\sqrt {b d+2 c d x}}{\sqrt {a+b x+c x^2}} \, dx}{5 \left (b^2-4 a c\right )^2 d^4} \\ & = \frac {4 \sqrt {a+b x+c x^2}}{5 \left (b^2-4 a c\right ) d (b d+2 c d x)^{5/2}}+\frac {12 \sqrt {a+b x+c x^2}}{5 \left (b^2-4 a c\right )^2 d^3 \sqrt {b d+2 c d x}}-\frac {\left (3 \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \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}{5 \left (b^2-4 a c\right )^2 d^4 \sqrt {a+b x+c x^2}} \\ & = \frac {4 \sqrt {a+b x+c x^2}}{5 \left (b^2-4 a c\right ) d (b d+2 c d x)^{5/2}}+\frac {12 \sqrt {a+b x+c x^2}}{5 \left (b^2-4 a c\right )^2 d^3 \sqrt {b d+2 c d x}}-\frac {\left (6 \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \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 )}{5 c \left (b^2-4 a c\right )^2 d^5 \sqrt {a+b x+c x^2}} \\ & = \frac {4 \sqrt {a+b x+c x^2}}{5 \left (b^2-4 a c\right ) d (b d+2 c d x)^{5/2}}+\frac {12 \sqrt {a+b x+c x^2}}{5 \left (b^2-4 a c\right )^2 d^3 \sqrt {b d+2 c d x}}+\frac {\left (6 \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 )}{5 c \left (b^2-4 a c\right )^{3/2} d^4 \sqrt {a+b x+c x^2}}-\frac {\left (6 \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \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 )}{5 c \left (b^2-4 a c\right )^{3/2} d^4 \sqrt {a+b x+c x^2}} \\ & = \frac {4 \sqrt {a+b x+c x^2}}{5 \left (b^2-4 a c\right ) d (b d+2 c d x)^{5/2}}+\frac {12 \sqrt {a+b x+c x^2}}{5 \left (b^2-4 a c\right )^2 d^3 \sqrt {b d+2 c d x}}+\frac {6 \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 )}{5 c \left (b^2-4 a c\right )^{5/4} d^{7/2} \sqrt {a+b x+c x^2}}-\frac {\left (6 \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \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 )}{5 c \left (b^2-4 a c\right )^{3/2} d^4 \sqrt {a+b x+c x^2}} \\ & = \frac {4 \sqrt {a+b x+c x^2}}{5 \left (b^2-4 a c\right ) d (b d+2 c d x)^{5/2}}+\frac {12 \sqrt {a+b x+c x^2}}{5 \left (b^2-4 a c\right )^2 d^3 \sqrt {b d+2 c d x}}-\frac {6 \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 )}{5 c \left (b^2-4 a c\right )^{5/4} d^{7/2} \sqrt {a+b x+c x^2}}+\frac {6 \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 )}{5 c \left (b^2-4 a c\right )^{5/4} d^{7/2} \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 = 10.05 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.32 \[ \int \frac {1}{(b d+2 c d x)^{7/2} \sqrt {a+b x+c x^2}} \, dx=-\frac {2 \sqrt {\frac {c (a+x (b+c x))}{-b^2+4 a c}} \operatorname {Hypergeometric2F1}\left (-\frac {5}{4},\frac {1}{2},-\frac {1}{4},\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{5 c d (d (b+2 c x))^{5/2} \sqrt {a+x (b+c x)}} \]

[In]

Integrate[1/((b*d + 2*c*d*x)^(7/2)*Sqrt[a + b*x + c*x^2]),x]

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(873\) vs. \(2(243)=486\).

Time = 4.32 (sec) , antiderivative size = 874, normalized size of antiderivative = 3.05

method	result	size
default	\(-\frac {\left (48 \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}}}}\, E\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^{3} x^{2}-12 \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}}}}\, E\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^{2} x^{2}+48 \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}}}}\, E\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^{2} x -12 \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}}}}\, E\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^{3} c x +12 \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}}}}\, E\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^{2} c -3 \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}}}}\, E\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^{4}-48 c^{4} x^{4}-96 b \,c^{3} x^{3}-32 x^{2} c^{3} a -64 b^{2} c^{2} x^{2}-32 a b \,c^{2} x -16 b^{3} c x +16 a^{2} c^{2}-16 a \,b^{2} c \right ) \sqrt {d \left (2 c x +b \right )}}{5 d^{4} \sqrt {c \,x^{2}+b x +a}\, \left (2 c x +b \right )^{3} \left (4 a c -b^{2}\right )^{2} c}\)	\(874\)
elliptic	\(\text {Expression too large to display}\)	\(1104\)

[In]

int(1/(2*c*d*x+b*d)^(7/2)/(c*x^2+b*x+a)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/5*(48*((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*c^3*x^2-12*((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))*b^2*c^2*x^2+48*((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*c^2*x-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))*b^3*c*x+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^2*c-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^4-48*c^4*x^4-96*b*c^3*x^3-32*x^2*c^3*a-
64*b^2*c^2*x^2-32*a*b*c^2*x-16*b^3*c*x+16*a^2*c^2-16*a*b^2*c)*(d*(2*c*x+b))^(1/2)/d^4/(c*x^2+b*x+a)^(1/2)/(2*c
*x+b)^3/(4*a*c-b^2)^2/c

Fricas [C] (verification not implemented)

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

Time = 0.10 (sec) , antiderivative size = 268, normalized size of antiderivative = 0.93 \[ \int \frac {1}{(b d+2 c d x)^{7/2} \sqrt {a+b x+c x^2}} \, dx=\frac {2 \, {\left (3 \, \sqrt {2} {\left (8 \, c^{3} x^{3} + 12 \, b c^{2} x^{2} + 6 \, b^{2} c x + b^{3}\right )} \sqrt {c^{2} d} {\rm weierstrassZeta}\left (\frac {b^{2} - 4 \, a c}{c^{2}}, 0, {\rm weierstrassPInverse}\left (\frac {b^{2} - 4 \, a c}{c^{2}}, 0, \frac {2 \, c x + b}{2 \, c}\right )\right ) + 8 \, {\left (3 \, c^{3} x^{2} + 3 \, b c^{2} x + b^{2} c - a c^{2}\right )} \sqrt {2 \, c d x + b d} \sqrt {c x^{2} + b x + a}\right )}}{5 \, {\left (8 \, {\left (b^{4} c^{4} - 8 \, a b^{2} c^{5} + 16 \, a^{2} c^{6}\right )} d^{4} x^{3} + 12 \, {\left (b^{5} c^{3} - 8 \, a b^{3} c^{4} + 16 \, a^{2} b c^{5}\right )} d^{4} x^{2} + 6 \, {\left (b^{6} c^{2} - 8 \, a b^{4} c^{3} + 16 \, a^{2} b^{2} c^{4}\right )} d^{4} x + {\left (b^{7} c - 8 \, a b^{5} c^{2} + 16 \, a^{2} b^{3} c^{3}\right )} d^{4}\right )}} \]

[In]

integrate(1/(2*c*d*x+b*d)^(7/2)/(c*x^2+b*x+a)^(1/2),x, algorithm="fricas")

[Out]

2/5*(3*sqrt(2)*(8*c^3*x^3 + 12*b*c^2*x^2 + 6*b^2*c*x + b^3)*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)) + 8*(3*c^3*x^2 + 3*b*c^2*x + b^2*c - a*c^2)*sqrt
(2*c*d*x + b*d)*sqrt(c*x^2 + b*x + a))/(8*(b^4*c^4 - 8*a*b^2*c^5 + 16*a^2*c^6)*d^4*x^3 + 12*(b^5*c^3 - 8*a*b^3
*c^4 + 16*a^2*b*c^5)*d^4*x^2 + 6*(b^6*c^2 - 8*a*b^4*c^3 + 16*a^2*b^2*c^4)*d^4*x + (b^7*c - 8*a*b^5*c^2 + 16*a^
2*b^3*c^3)*d^4)

Sympy [F]

\[ \int \frac {1}{(b d+2 c d x)^{7/2} \sqrt {a+b x+c x^2}} \, dx=\int \frac {1}{\left (d \left (b + 2 c x\right )\right )^{\frac {7}{2}} \sqrt {a + b x + c x^{2}}}\, dx \]

[In]

integrate(1/(2*c*d*x+b*d)**(7/2)/(c*x**2+b*x+a)**(1/2),x)

[Out]

Integral(1/((d*(b + 2*c*x))**(7/2)*sqrt(a + b*x + c*x**2)), x)

Maxima [F]

\[ \int \frac {1}{(b d+2 c d x)^{7/2} \sqrt {a+b x+c x^2}} \, dx=\int { \frac {1}{{\left (2 \, c d x + b d\right )}^{\frac {7}{2}} \sqrt {c x^{2} + b x + a}} \,d x } \]

[In]

integrate(1/(2*c*d*x+b*d)^(7/2)/(c*x^2+b*x+a)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/((2*c*d*x + b*d)^(7/2)*sqrt(c*x^2 + b*x + a)), x)

Giac [F]

\[ \int \frac {1}{(b d+2 c d x)^{7/2} \sqrt {a+b x+c x^2}} \, dx=\int { \frac {1}{{\left (2 \, c d x + b d\right )}^{\frac {7}{2}} \sqrt {c x^{2} + b x + a}} \,d x } \]

[In]

integrate(1/(2*c*d*x+b*d)^(7/2)/(c*x^2+b*x+a)^(1/2),x, algorithm="giac")

[Out]

integrate(1/((2*c*d*x + b*d)^(7/2)*sqrt(c*x^2 + b*x + a)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(b d+2 c d x)^{7/2} \sqrt {a+b x+c x^2}} \, dx=\int \frac {1}{{\left (b\,d+2\,c\,d\,x\right )}^{7/2}\,\sqrt {c\,x^2+b\,x+a}} \,d x \]

[In]

int(1/((b*d + 2*c*d*x)^(7/2)*(a + b*x + c*x^2)^(1/2)),x)

[Out]

int(1/((b*d + 2*c*d*x)^(7/2)*(a + b*x + c*x^2)^(1/2)), x)