\(\int \frac {\sqrt {a+b x+c x^2}}{(b d+2 c d x)^7} \, dx\) [1201]

Optimal result

Integrand size = 26, antiderivative size = 175 \[ \int \frac {\sqrt {a+b x+c x^2}}{(b d+2 c d x)^7} \, dx=-\frac {\sqrt {a+b x+c x^2}}{12 c d^7 (b+2 c x)^6}+\frac {\sqrt {a+b x+c x^2}}{48 c \left (b^2-4 a c\right ) d^7 (b+2 c x)^4}+\frac {\sqrt {a+b x+c x^2}}{32 c \left (b^2-4 a c\right )^2 d^7 (b+2 c x)^2}+\frac {\arctan \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )}{64 c^{3/2} \left (b^2-4 a c\right )^{5/2} d^7} \]

[Out]

1/64*arctan(2*c^(1/2)*(c*x^2+b*x+a)^(1/2)/(-4*a*c+b^2)^(1/2))/c^(3/2)/(-4*a*c+b^2)^(5/2)/d^7-1/12*(c*x^2+b*x+a
)^(1/2)/c/d^7/(2*c*x+b)^6+1/48*(c*x^2+b*x+a)^(1/2)/c/(-4*a*c+b^2)/d^7/(2*c*x+b)^4+1/32*(c*x^2+b*x+a)^(1/2)/c/(
-4*a*c+b^2)^2/d^7/(2*c*x+b)^2

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {698, 707, 702, 211} \[ \int \frac {\sqrt {a+b x+c x^2}}{(b d+2 c d x)^7} \, dx=\frac {\arctan \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )}{64 c^{3/2} d^7 \left (b^2-4 a c\right )^{5/2}}+\frac {\sqrt {a+b x+c x^2}}{32 c d^7 \left (b^2-4 a c\right )^2 (b+2 c x)^2}+\frac {\sqrt {a+b x+c x^2}}{48 c d^7 \left (b^2-4 a c\right ) (b+2 c x)^4}-\frac {\sqrt {a+b x+c x^2}}{12 c d^7 (b+2 c x)^6} \]

[In]

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

[Out]

-1/12*Sqrt[a + b*x + c*x^2]/(c*d^7*(b + 2*c*x)^6) + Sqrt[a + b*x + c*x^2]/(48*c*(b^2 - 4*a*c)*d^7*(b + 2*c*x)^
4) + Sqrt[a + b*x + c*x^2]/(32*c*(b^2 - 4*a*c)^2*d^7*(b + 2*c*x)^2) + ArcTan[(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])
/Sqrt[b^2 - 4*a*c]]/(64*c^(3/2)*(b^2 - 4*a*c)^(5/2)*d^7)

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

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 702

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

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*} \text {integral}& = -\frac {\sqrt {a+b x+c x^2}}{12 c d^7 (b+2 c x)^6}+\frac {\int \frac {1}{(b d+2 c d x)^5 \sqrt {a+b x+c x^2}} \, dx}{24 c d^2} \\ & = -\frac {\sqrt {a+b x+c x^2}}{12 c d^7 (b+2 c x)^6}+\frac {\sqrt {a+b x+c x^2}}{48 c \left (b^2-4 a c\right ) d^7 (b+2 c x)^4}+\frac {\int \frac {1}{(b d+2 c d x)^3 \sqrt {a+b x+c x^2}} \, dx}{32 c \left (b^2-4 a c\right ) d^4} \\ & = -\frac {\sqrt {a+b x+c x^2}}{12 c d^7 (b+2 c x)^6}+\frac {\sqrt {a+b x+c x^2}}{48 c \left (b^2-4 a c\right ) d^7 (b+2 c x)^4}+\frac {\sqrt {a+b x+c x^2}}{32 c \left (b^2-4 a c\right )^2 d^7 (b+2 c x)^2}+\frac {\int \frac {1}{(b d+2 c d x) \sqrt {a+b x+c x^2}} \, dx}{64 c \left (b^2-4 a c\right )^2 d^6} \\ & = -\frac {\sqrt {a+b x+c x^2}}{12 c d^7 (b+2 c x)^6}+\frac {\sqrt {a+b x+c x^2}}{48 c \left (b^2-4 a c\right ) d^7 (b+2 c x)^4}+\frac {\sqrt {a+b x+c x^2}}{32 c \left (b^2-4 a c\right )^2 d^7 (b+2 c x)^2}+\frac {\text {Subst}\left (\int \frac {1}{2 b^2 c d-8 a c^2 d+8 c^2 d x^2} \, dx,x,\sqrt {a+b x+c x^2}\right )}{16 \left (b^2-4 a c\right )^2 d^6} \\ & = -\frac {\sqrt {a+b x+c x^2}}{12 c d^7 (b+2 c x)^6}+\frac {\sqrt {a+b x+c x^2}}{48 c \left (b^2-4 a c\right ) d^7 (b+2 c x)^4}+\frac {\sqrt {a+b x+c x^2}}{32 c \left (b^2-4 a c\right )^2 d^7 (b+2 c x)^2}+\frac {\tan ^{-1}\left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )}{64 c^{3/2} \left (b^2-4 a c\right )^{5/2} d^7} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 10.05 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.35 \[ \int \frac {\sqrt {a+b x+c x^2}}{(b d+2 c d x)^7} \, dx=\frac {2 (a+x (b+c x))^{3/2} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},4,\frac {5}{2},\frac {4 c (a+x (b+c x))}{-b^2+4 a c}\right )}{3 \left (b^2-4 a c\right )^4 d^7} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 2.78 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.90

method	result	size
pseudoelliptic	\(-\frac {\frac {3 \left (2 c x +b \right )^{6} \operatorname {arctanh}\left (\frac {2 c \sqrt {c \,x^{2}+b x +a}}{\sqrt {4 c^{2} a -b^{2} c}}\right )}{256}+\left (\frac {3 c^{2} x^{2}}{4}+\left (\frac {3 b x}{4}+a \right ) c -\frac {b^{2}}{16}\right ) \sqrt {4 c^{2} a -b^{2} c}\, \left (-\frac {c^{2} x^{2}}{2}+\left (-\frac {b x}{2}+a \right ) c -\frac {3 b^{2}}{8}\right ) \sqrt {c \,x^{2}+b x +a}}{12 \sqrt {4 c^{2} a -b^{2} c}\, d^{7} \left (2 c x +b \right )^{6} c \left (-\frac {b^{2}}{4}+a c \right )^{2}}\)	\(157\)
default	\(\frac {-\frac {2 c \left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {3}{2}}}{3 \left (4 a c -b^{2}\right ) \left (x +\frac {b}{2 c}\right )^{6}}-\frac {2 c^{2} \left (-\frac {c \left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {3}{2}}}{\left (4 a c -b^{2}\right ) \left (x +\frac {b}{2 c}\right )^{4}}-\frac {c^{2} \left (-\frac {2 c \left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {3}{2}}}{\left (4 a c -b^{2}\right ) \left (x +\frac {b}{2 c}\right )^{2}}+\frac {2 c^{2} \left (\frac {\sqrt {4 \left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{c}}}{2}-\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {4 a c -b^{2}}{2 c}+\frac {\sqrt {\frac {4 a c -b^{2}}{c}}\, \sqrt {4 \left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{c}}}{2}}{x +\frac {b}{2 c}}\right )}{2 c \sqrt {\frac {4 a c -b^{2}}{c}}}\right )}{4 a c -b^{2}}\right )}{4 a c -b^{2}}\right )}{4 a c -b^{2}}}{128 d^{7} c^{7}}\)	\(368\)

[In]

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

[Out]

-1/12*(3/256*(2*c*x+b)^6*arctanh(2*c*(c*x^2+b*x+a)^(1/2)/(4*a*c^2-b^2*c)^(1/2))+(3/4*c^2*x^2+(3/4*b*x+a)*c-1/1
6*b^2)*(4*a*c^2-b^2*c)^(1/2)*(-1/2*c^2*x^2+(-1/2*b*x+a)*c-3/8*b^2)*(c*x^2+b*x+a)^(1/2))/(4*a*c^2-b^2*c)^(1/2)/
d^7/(2*c*x+b)^6/c/(-1/4*b^2+a*c)^2

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 595 vs. \(2 (151) = 302\).

Time = 3.04 (sec) , antiderivative size = 1220, normalized size of antiderivative = 6.97 \[ \int \frac {\sqrt {a+b x+c x^2}}{(b d+2 c d x)^7} \, dx=\text {Too large to display} \]

[In]

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

[Out]

[-1/384*(3*(64*c^6*x^6 + 192*b*c^5*x^5 + 240*b^2*c^4*x^4 + 160*b^3*c^3*x^3 + 60*b^4*c^2*x^2 + 12*b^5*c*x + b^6
)*sqrt(-b^2*c + 4*a*c^2)*log(-(4*c^2*x^2 + 4*b*c*x - b^2 + 8*a*c - 4*sqrt(-b^2*c + 4*a*c^2)*sqrt(c*x^2 + b*x +
a))/(4*c^2*x^2 + 4*b*c*x + b^2)) + 4*(3*b^6*c - 68*a*b^4*c^2 + 352*a^2*b^2*c^3 - 512*a^3*c^4 - 48*(b^2*c^5 -
4*a*c^6)*x^4 - 96*(b^3*c^4 - 4*a*b*c^5)*x^3 - 16*(5*b^4*c^3 - 22*a*b^2*c^4 + 8*a^2*c^5)*x^2 - 32*(b^5*c^2 - 5*
a*b^3*c^3 + 4*a^2*b*c^4)*x)*sqrt(c*x^2 + b*x + a))/(64*(b^6*c^8 - 12*a*b^4*c^9 + 48*a^2*b^2*c^10 - 64*a^3*c^11
)*d^7*x^6 + 192*(b^7*c^7 - 12*a*b^5*c^8 + 48*a^2*b^3*c^9 - 64*a^3*b*c^10)*d^7*x^5 + 240*(b^8*c^6 - 12*a*b^6*c^
7 + 48*a^2*b^4*c^8 - 64*a^3*b^2*c^9)*d^7*x^4 + 160*(b^9*c^5 - 12*a*b^7*c^6 + 48*a^2*b^5*c^7 - 64*a^3*b^3*c^8)*
d^7*x^3 + 60*(b^10*c^4 - 12*a*b^8*c^5 + 48*a^2*b^6*c^6 - 64*a^3*b^4*c^7)*d^7*x^2 + 12*(b^11*c^3 - 12*a*b^9*c^4
+ 48*a^2*b^7*c^5 - 64*a^3*b^5*c^6)*d^7*x + (b^12*c^2 - 12*a*b^10*c^3 + 48*a^2*b^8*c^4 - 64*a^3*b^6*c^5)*d^7),
-1/192*(3*(64*c^6*x^6 + 192*b*c^5*x^5 + 240*b^2*c^4*x^4 + 160*b^3*c^3*x^3 + 60*b^4*c^2*x^2 + 12*b^5*c*x + b^6
)*sqrt(b^2*c - 4*a*c^2)*arctan(1/2*sqrt(b^2*c - 4*a*c^2)*sqrt(c*x^2 + b*x + a)/(c^2*x^2 + b*c*x + a*c)) + 2*(3
*b^6*c - 68*a*b^4*c^2 + 352*a^2*b^2*c^3 - 512*a^3*c^4 - 48*(b^2*c^5 - 4*a*c^6)*x^4 - 96*(b^3*c^4 - 4*a*b*c^5)*
x^3 - 16*(5*b^4*c^3 - 22*a*b^2*c^4 + 8*a^2*c^5)*x^2 - 32*(b^5*c^2 - 5*a*b^3*c^3 + 4*a^2*b*c^4)*x)*sqrt(c*x^2 +
b*x + a))/(64*(b^6*c^8 - 12*a*b^4*c^9 + 48*a^2*b^2*c^10 - 64*a^3*c^11)*d^7*x^6 + 192*(b^7*c^7 - 12*a*b^5*c^8
+ 48*a^2*b^3*c^9 - 64*a^3*b*c^10)*d^7*x^5 + 240*(b^8*c^6 - 12*a*b^6*c^7 + 48*a^2*b^4*c^8 - 64*a^3*b^2*c^9)*d^7
*x^4 + 160*(b^9*c^5 - 12*a*b^7*c^6 + 48*a^2*b^5*c^7 - 64*a^3*b^3*c^8)*d^7*x^3 + 60*(b^10*c^4 - 12*a*b^8*c^5 +
48*a^2*b^6*c^6 - 64*a^3*b^4*c^7)*d^7*x^2 + 12*(b^11*c^3 - 12*a*b^9*c^4 + 48*a^2*b^7*c^5 - 64*a^3*b^5*c^6)*d^7*
x + (b^12*c^2 - 12*a*b^10*c^3 + 48*a^2*b^8*c^4 - 64*a^3*b^6*c^5)*d^7)]

Sympy [F]

\[ \int \frac {\sqrt {a+b x+c x^2}}{(b d+2 c d x)^7} \, dx=\frac {\int \frac {\sqrt {a + b x + c x^{2}}}{b^{7} + 14 b^{6} c x + 84 b^{5} c^{2} x^{2} + 280 b^{4} c^{3} x^{3} + 560 b^{3} c^{4} x^{4} + 672 b^{2} c^{5} x^{5} + 448 b c^{6} x^{6} + 128 c^{7} x^{7}}\, dx}{d^{7}} \]

[In]

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

[Out]

Integral(sqrt(a + b*x + c*x**2)/(b**7 + 14*b**6*c*x + 84*b**5*c**2*x**2 + 280*b**4*c**3*x**3 + 560*b**3*c**4*x
**4 + 672*b**2*c**5*x**5 + 448*b*c**6*x**6 + 128*c**7*x**7), x)/d**7

Maxima [F(-2)]

Exception generated. \[ \int \frac {\sqrt {a+b x+c x^2}}{(b d+2 c d x)^7} \, dx=\text {Exception raised: ValueError} \]

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` f
or more deta

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1336 vs. \(2 (151) = 302\).

Time = 0.36 (sec) , antiderivative size = 1336, normalized size of antiderivative = 7.63 \[ \int \frac {\sqrt {a+b x+c x^2}}{(b d+2 c d x)^7} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/32*arctan(-(2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*c + b*sqrt(c))/sqrt(b^2*c - 4*a*c^2))/((b^4*c*d^7 - 8*a*b^
2*c^2*d^7 + 16*a^2*c^3*d^7)*sqrt(b^2*c - 4*a*c^2)) - 1/96*(96*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^11*c^5 + 528
*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^10*b*c^(9/2) + 1456*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*b^2*c^4 - 544*(
sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*a*c^5 + 2592*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*b^3*c^(7/2) - 2448*(sq
rt(c)*x - sqrt(c*x^2 + b*x + a))^8*a*b*c^(9/2) + 2976*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*b^4*c^3 - 3072*(sq
rt(c)*x - sqrt(c*x^2 + b*x + a))^7*a*b^2*c^4 - 3648*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*a^2*c^5 + 2016*(sqrt
(c)*x - sqrt(c*x^2 + b*x + a))^6*b^5*c^(5/2) + 672*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*a*b^3*c^(7/2) - 12768
*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*a^2*b*c^(9/2) + 624*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*b^6*c^2 + 460
8*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a*b^4*c^3 - 16416*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a^2*b^2*c^4 -
3648*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a^3*c^5 - 72*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*b^7*c^(3/2) + 41
28*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a*b^5*c^(5/2) - 9120*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a^2*b^3*c^
(7/2) - 9120*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a^3*b*c^(9/2) - 118*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*b
^8*c + 1600*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*b^6*c^2 - 1344*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^2*b
^4*c^3 - 8576*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^3*b^2*c^4 - 544*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^
4*c^5 - 33*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b^9*sqrt(c) + 240*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a*b^7
*c^(3/2) + 720*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^2*b^5*c^(5/2) - 3744*(sqrt(c)*x - sqrt(c*x^2 + b*x + a)
)^2*a^3*b^3*c^(7/2) - 816*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^4*b*c^(9/2) - 3*(sqrt(c)*x - sqrt(c*x^2 + b*
x + a))*b^10 - 6*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a*b^8*c + 288*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^2*b^6
*c^2 - 672*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^3*b^4*c^3 - 528*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^4*b^2*c
^4 + 96*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^5*c^5 - 3*a*b^9*sqrt(c) + 24*a^2*b^7*c^(3/2) - 16*a^3*b^5*c^(5/2
) - 128*a^4*b^3*c^(7/2) + 48*a^5*b*c^(9/2))/((b^4*c*d^7 - 8*a*b^2*c^2*d^7 + 16*a^2*c^3*d^7)*(2*(sqrt(c)*x - sq
rt(c*x^2 + b*x + a))^2*c + 2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b*sqrt(c) + b^2 - 2*a*c)^6)

Mupad [F(-1)]

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

[In]

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

[Out]

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