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

Optimal. Leaf size=175 \[ \frac {\tan ^{-1}\left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )}{64 c^{3/2} d^7 \left (b^2-4 a c\right )^{5/2}}+\frac {\sqrt {a+b x+c x^2}}{32 c d^7 \left (b^2-4 a c\right )^2 (b+2 c x)^2}+\frac {\sqrt {a+b x+c x^2}}{48 c d^7 \left (b^2-4 a c\right ) (b+2 c x)^4}-\frac {\sqrt {a+b x+c x^2}}{12 c d^7 (b+2 c x)^6} \]

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Rubi [A]  time = 0.13, antiderivative size = 175, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {684, 693, 688, 205} \begin {gather*} \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} 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} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-Sqrt[a + b*x + c*x^2]/(12*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])/S
qrt[b^2 - 4*a*c]]/(64*c^(3/2)*(b^2 - 4*a*c)^(5/2)*d^7)

Rule 205

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

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 688

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 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])

Rubi steps

\begin {align*} \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 {\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 {\operatorname {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*}

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Mathematica [C]  time = 0.02, size = 62, normalized size = 0.35 \begin {gather*} \frac {2 (a+x (b+c x))^{3/2} \, _2F_1\left (\frac {3}{2},4;\frac {5}{2};\frac {4 c (a+x (b+c x))}{4 a c-b^2}\right )}{3 d^7 \left (b^2-4 a c\right )^4} \end {gather*}

Antiderivative was successfully verified.

[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)

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IntegrateAlgebraic [F]  time = 180.04, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

$Aborted

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fricas [B]  time = 3.75, size = 1220, normalized size = 6.97

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[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)]

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Unab
le to divide, perhaps due to rounding error%%%{%%%{128,[7]%%%},[14,7,0,0]%%%}+%%%{%%{[%%%{-896,[6]%%%},0]:[1,0
,%%%{-1,[1]%%%}]%%},[13,7,1,0]%%%}+%%%{%%%{3136,[6]%%%},[12,7,2,0]%%%}+%%%{%%%{-896,[7]%%%},[12,7,0,1]%%%}+%%%
{%%{[%%%{-7168,[5]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[11,7,3,0]%%%}+%%%{%%{[%%%{5376,[6]%%%},0]:[1,0,%%%{-1,[1]%%
%}]%%},[11,7,1,1]%%%}+%%%{%%%{11872,[5]%%%},[10,7,4,0]%%%}+%%%{%%%{-16128,[6]%%%},[10,7,2,1]%%%}+%%%{%%%{2688,
[7]%%%},[10,7,0,2]%%%}+%%%{%%{[%%%{-15008,[4]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[9,7,5,0]%%%}+%%%{%%{[%%%{31360,[
5]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[9,7,3,1]%%%}+%%%{%%{[%%%{-13440,[6]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[9,7,1,2
]%%%}+%%%{%%%{14896,[4]%%%},[8,7,6,0]%%%}+%%%{%%%{-43680,[5]%%%},[8,7,4,1]%%%}+%%%{%%%{33600,[6]%%%},[8,7,2,2]
%%%}+%%%{%%%{-4480,[7]%%%},[8,7,0,3]%%%}+%%%{%%{[%%%{-11776,[3]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[7,7,7,0]%%%}+%
%%{%%{[%%%{45696,[4]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[7,7,5,1]%%%}+%%%{%%{[%%%{-53760,[5]%%%},0]:[1,0,%%%{-1,[1
]%%%}]%%},[7,7,3,2]%%%}+%%%{%%{[%%%{17920,[6]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[7,7,1,3]%%%}+%%%{%%%{7448,[3]%%%
},[6,7,8,0]%%%}+%%%{%%%{-36736,[4]%%%},[6,7,6,1]%%%}+%%%{%%%{60480,[5]%%%},[6,7,4,2]%%%}+%%%{%%%{-35840,[6]%%%
},[6,7,2,3]%%%}+%%%{%%%{4480,[7]%%%},[6,7,0,4]%%%}+%%%{%%{[%%%{-3752,[2]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[5,7,9
,0]%%%}+%%%{%%{[%%%{22848,[3]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[5,7,7,1]%%%}+%%%{%%{[%%%{-49728,[4]%%%},0]:[1,0,
%%%{-1,[1]%%%}]%%},[5,7,5,2]%%%}+%%%{%%{[%%%{44800,[5]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[5,7,3,3]%%%}+%%%{%%{[%%
%{-13440,[6]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[5,7,1,4]%%%}+%%%{%%%{1484,[2]%%%},[4,7,10,0]%%%}+%%%{%%%{-10920,[
3]%%%},[4,7,8,1]%%%}+%%%{%%%{30240,[4]%%%},[4,7,6,2]%%%}+%%%{%%%{-38080,[5]%%%},[4,7,4,3]%%%}+%%%{%%%{20160,[6
]%%%},[4,7,2,4]%%%}+%%%{%%%{-2688,[7]%%%},[4,7,0,5]%%%}+%%%{%%{[%%%{-448,[1]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3
,7,11,0]%%%}+%%%{%%{[%%%{3920,[2]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3,7,9,1]%%%}+%%%{%%{[%%%{-13440,[3]%%%},0]:[
1,0,%%%{-1,[1]%%%}]%%},[3,7,7,2]%%%}+%%%{%%{[%%%{22400,[4]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3,7,5,3]%%%}+%%%{%%
{[%%%{-17920,[5]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3,7,3,4]%%%}+%%%{%%{[%%%{5376,[6]%%%},0]:[1,0,%%%{-1,[1]%%%}]
%%},[3,7,1,5]%%%}+%%%{%%%{98,[1]%%%},[2,7,12,0]%%%}+%%%{%%%{-1008,[2]%%%},[2,7,10,1]%%%}+%%%{%%%{4200,[3]%%%},
[2,7,8,2]%%%}+%%%{%%%{-8960,[4]%%%},[2,7,6,3]%%%}+%%%{%%%{10080,[5]%%%},[2,7,4,4]%%%}+%%%{%%%{-5376,[6]%%%},[2
,7,2,5]%%%}+%%%{%%%{896,[7]%%%},[2,7,0,6]%%%}+%%%{%%{[-14,0]:[1,0,%%%{-1,[1]%%%}]%%},[1,7,13,0]%%%}+%%%{%%{[%%
%{168,[1]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,7,11,1]%%%}+%%%{%%{[%%%{-840,[2]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1
,7,9,2]%%%}+%%%{%%{[%%%{2240,[3]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,7,7,3]%%%}+%%%{%%{[%%%{-3360,[4]%%%},0]:[1,
0,%%%{-1,[1]%%%}]%%},[1,7,5,4]%%%}+%%%{%%{[%%%{2688,[5]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,7,3,5]%%%}+%%%{%%{[%
%%{-896,[6]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,7,1,6]%%%}+%%%{1,[0,7,14,0]%%%}+%%%{%%%{-14,[1]%%%},[0,7,12,1]%%
%}+%%%{%%%{84,[2]%%%},[0,7,10,2]%%%}+%%%{%%%{-280,[3]%%%},[0,7,8,3]%%%}+%%%{%%%{560,[4]%%%},[0,7,6,4]%%%}+%%%{
%%%{-672,[5]%%%},[0,7,4,5]%%%}+%%%{%%%{448,[6]%%%},[0,7,2,6]%%%}+%%%{%%%{-128,[7]%%%},[0,7,0,7]%%%} / %%%{%%{p
oly1[%%%{-128,[10]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[14,0,0,0]%%%}+%%%{%%%{896,[10]%%%},[13,0,1,0]%%%}+%%%{%%{po
ly1[%%%{-3136,[9]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[12,0,2,0]%%%}+%%%{%%{[%%%{896,[10]%%%},0]:[1,0,%%%{-1,[1]%%%
}]%%},[12,0,0,1]%%%}+%%%{%%%{7168,[9]%%%},[11,0,3,0]%%%}+%%%{%%%{-5376,[10]%%%},[11,0,1,1]%%%}+%%%{%%{poly1[%%
%{-11872,[8]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[10,0,4,0]%%%}+%%%{%%{[%%%{16128,[9]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%
},[10,0,2,1]%%%}+%%%{%%{poly1[%%%{-2688,[10]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[10,0,0,2]%%%}+%%%{%%%{15008,[8]%%
%},[9,0,5,0]%%%}+%%%{%%%{-31360,[9]%%%},[9,0,3,1]%%%}+%%%{%%%{13440,[10]%%%},[9,0,1,2]%%%}+%%%{%%{poly1[%%%{-1
4896,[7]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[8,0,6,0]%%%}+%%%{%%{[%%%{43680,[8]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[8,
0,4,1]%%%}+%%%{%%{poly1[%%%{-33600,[9]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[8,0,2,2]%%%}+%%%{%%{[%%%{4480,[10]%%%},
0]:[1,0,%%%{-1,[1]%%%}]%%},[8,0,0,3]%%%}+%%%{%%%{11776,[7]%%%},[7,0,7,0]%%%}+%%%{%%%{-45696,[8]%%%},[7,0,5,1]%
%%}+%%%{%%%{53760,[9]%%%},[7,0,3,2]%%%}+%%%{%%%{-17920,[10]%%%},[7,0,1,3]%%%}+%%%{%%{poly1[%%%{-7448,[6]%%%},0
]:[1,0,%%%{-1,[1]%%%}]%%},[6,0,8,0]%%%}+%%%{%%{[%%%{36736,[7]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[6,0,6,1]%%%}+%%%
{%%{poly1[%%%{-60480,[8]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[6,0,4,2]%%%}+%%%{%%{[%%%{35840,[9]%%%},0]:[1,0,%%%{-1
,[1]%%%}]%%},[6,0,2,3]%%%}+%%%{%%{poly1[%%%{-4480,[10]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[6,0,0,4]%%%}+%%%{%%%{37
52,[6]%%%},[5,0,9,0]%%%}+%%%{%%%{-22848,[7]%%%},[5,0,7,1]%%%}+%%%{%%%{49728,[8]%%%},[5,0,5,2]%%%}+%%%{%%%{-448
00,[9]%%%},[5,0,3,3]%%%}+%%%{%%%{13440,[10]%%%},[5,0,1,4]%%%}+%%%{%%{poly1[%%%{-1484,[5]%%%},0]:[1,0,%%%{-1,[1
]%%%}]%%},[4,0,10,0]%%%}+%%%{%%{[%%%{10920,[6]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[4,0,8,1]%%%}+%%%{%%{poly1[%%%{-
30240,[7]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[4,0,6,2]%%%}+%%%{%%{[%%%{38080,[8]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[4
,0,4,3]%%%}+%%%{%%{poly1[%%%{-20160,[9]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[4,0,2,4]%%%}+%%%{%%{[%%%{2688,[10]%%%}
,0]:[1,0,%%%{-1,[1]%%%}]%%},[4,0,0,5]%%%}+%%%{%%%{448,[5]%%%},[3,0,11,0]%%%}+%%%{%%%{-3920,[6]%%%},[3,0,9,1]%%
%}+%%%{%%%{13440,[7]%%%},[3,0,7,2]%%%}+%%%{%%%{-22400,[8]%%%},[3,0,5,3]%%%}+%%%{%%%{17920,[9]%%%},[3,0,3,4]%%%
}+%%%{%%%{-5376,[10]%%%},[3,0,1,5]%%%}+%%%{%%{poly1[%%%{-98,[4]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[2,0,12,0]%%%}+
%%%{%%{[%%%{1008,[5]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[2,0,10,1]%%%}+%%%{%%{poly1[%%%{-4200,[6]%%%},0]:[1,0,%%%{
-1,[1]%%%}]%%},[2,0,8,2]%%%}+%%%{%%{[%%%{8960,[7]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[2,0,6,3]%%%}+%%%{%%{poly1[%%
%{-10080,[8]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[2,0,4,4]%%%}+%%%{%%{[%%%{5376,[9]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},
[2,0,2,5]%%%}+%%%{%%{[%%%{-896,[10]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[2,0,0,6]%%%}+%%%{%%%{14,[4]%%%},[1,0,13,0]
%%%}+%%%{%%%{-168,[5]%%%},[1,0,11,1]%%%}+%%%{%%%{840,[6]%%%},[1,0,9,2]%%%}+%%%{%%%{-2240,[7]%%%},[1,0,7,3]%%%}
+%%%{%%%{3360,[8]%%%},[1,0,5,4]%%%}+%%%{%%%{-2688,[9]%%%},[1,0,3,5]%%%}+%%%{%%%{896,[10]%%%},[1,0,1,6]%%%}+%%%
{%%{poly1[%%%{-1,[3]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[0,0,14,0]%%%}+%%%{%%{[%%%{14,[4]%%%},0]:[1,0,%%%{-1,[1]%%
%}]%%},[0,0,12,1]%%%}+%%%{%%{poly1[%%%{-84,[5]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[0,0,10,2]%%%}+%%%{%%{[%%%{280,[
6]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[0,0,8,3]%%%}+%%%{%%{poly1[%%%{-560,[7]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[0,0,
6,4]%%%}+%%%{%%{[%%%{672,[8]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[0,0,4,5]%%%}+%%%{%%{[%%%{-448,[9]%%%},0]:[1,0,%%%
{-1,[1]%%%}]%%},[0,0,2,6]%%%}+%%%{%%{[%%%{128,[10]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[0,0,0,7]%%%} Error: Bad Arg
ument Value

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maple [B]  time = 0.06, size = 460, normalized size = 2.63 \begin {gather*} -\frac {a \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 )}{16 \left (4 a c -b^{2}\right )^{3} \sqrt {\frac {4 a c -b^{2}}{c}}\, c \,d^{7}}+\frac {b^{2} \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 )}{64 \left (4 a c -b^{2}\right )^{3} \sqrt {\frac {4 a c -b^{2}}{c}}\, c^{2} d^{7}}+\frac {\sqrt {4 \left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{c}}}{64 \left (4 a c -b^{2}\right )^{3} c \,d^{7}}-\frac {\left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {3}{2}}}{32 \left (4 a c -b^{2}\right )^{3} \left (x +\frac {b}{2 c}\right )^{2} c^{2} d^{7}}+\frac {\left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {3}{2}}}{64 \left (4 a c -b^{2}\right )^{2} \left (x +\frac {b}{2 c}\right )^{4} c^{4} d^{7}}-\frac {\left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {3}{2}}}{192 \left (4 a c -b^{2}\right ) \left (x +\frac {b}{2 c}\right )^{6} c^{6} d^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/192/d^7/c^6/(4*a*c-b^2)/(x+1/2*b/c)^6*((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(3/2)+1/64/d^7/c^4/(4*a*c-b^2)^2/
(x+1/2*b/c)^4*((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(3/2)-1/32/d^7/c^2/(4*a*c-b^2)^3/(x+1/2*b/c)^2*((x+1/2*b/c)^
2*c+1/4*(4*a*c-b^2)/c)^(3/2)+1/64/d^7/c/(4*a*c-b^2)^3*(4*(x+1/2*b/c)^2*c+(4*a*c-b^2)/c)^(1/2)-1/16/d^7/c/(4*a*
c-b^2)^3/((4*a*c-b^2)/c)^(1/2)*ln((1/2*(4*a*c-b^2)/c+1/2*((4*a*c-b^2)/c)^(1/2)*(4*(x+1/2*b/c)^2*c+(4*a*c-b^2)/
c)^(1/2))/(x+1/2*b/c))*a+1/64/d^7/c^2/(4*a*c-b^2)^3/((4*a*c-b^2)/c)^(1/2)*ln((1/2*(4*a*c-b^2)/c+1/2*((4*a*c-b^
2)/c)^(1/2)*(4*(x+1/2*b/c)^2*c+(4*a*c-b^2)/c)^(1/2))/(x+1/2*b/c))*b^2

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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 details)Is 4*a*c-b^2 zero or nonzero?

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\sqrt {c\,x^2+b\,x+a}}{{\left (b\,d+2\,c\,d\,x\right )}^7} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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