3.11.39 \(\int \frac {(a+b x+c x^2)^{5/2}}{(b d+2 c d x)^6} \, dx\)

Optimal. Leaf size=139 \[ \frac {\tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{64 c^{7/2} d^6}-\frac {\sqrt {a+b x+c x^2}}{32 c^3 d^6 (b+2 c x)}-\frac {\left (a+b x+c x^2\right )^{3/2}}{24 c^2 d^6 (b+2 c x)^3}-\frac {\left (a+b x+c x^2\right )^{5/2}}{10 c d^6 (b+2 c x)^5} \]

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Rubi [A]  time = 0.07, antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {684, 621, 206} \begin {gather*} -\frac {\left (a+b x+c x^2\right )^{3/2}}{24 c^2 d^6 (b+2 c x)^3}-\frac {\sqrt {a+b x+c x^2}}{32 c^3 d^6 (b+2 c x)}+\frac {\tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{64 c^{7/2} d^6}-\frac {\left (a+b x+c x^2\right )^{5/2}}{10 c d^6 (b+2 c x)^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^6,x]

[Out]

-Sqrt[a + b*x + c*x^2]/(32*c^3*d^6*(b + 2*c*x)) - (a + b*x + c*x^2)^(3/2)/(24*c^2*d^6*(b + 2*c*x)^3) - (a + b*
x + c*x^2)^(5/2)/(10*c*d^6*(b + 2*c*x)^5) + ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])]/(64*c^(7/2)
*d^6)

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 621

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

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]

Rubi steps

\begin {align*} \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^6} \, dx &=-\frac {\left (a+b x+c x^2\right )^{5/2}}{10 c d^6 (b+2 c x)^5}+\frac {\int \frac {\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^4} \, dx}{4 c d^2}\\ &=-\frac {\left (a+b x+c x^2\right )^{3/2}}{24 c^2 d^6 (b+2 c x)^3}-\frac {\left (a+b x+c x^2\right )^{5/2}}{10 c d^6 (b+2 c x)^5}+\frac {\int \frac {\sqrt {a+b x+c x^2}}{(b d+2 c d x)^2} \, dx}{16 c^2 d^4}\\ &=-\frac {\sqrt {a+b x+c x^2}}{32 c^3 d^6 (b+2 c x)}-\frac {\left (a+b x+c x^2\right )^{3/2}}{24 c^2 d^6 (b+2 c x)^3}-\frac {\left (a+b x+c x^2\right )^{5/2}}{10 c d^6 (b+2 c x)^5}+\frac {\int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{64 c^3 d^6}\\ &=-\frac {\sqrt {a+b x+c x^2}}{32 c^3 d^6 (b+2 c x)}-\frac {\left (a+b x+c x^2\right )^{3/2}}{24 c^2 d^6 (b+2 c x)^3}-\frac {\left (a+b x+c x^2\right )^{5/2}}{10 c d^6 (b+2 c x)^5}+\frac {\operatorname {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c x}{\sqrt {a+b x+c x^2}}\right )}{32 c^3 d^6}\\ &=-\frac {\sqrt {a+b x+c x^2}}{32 c^3 d^6 (b+2 c x)}-\frac {\left (a+b x+c x^2\right )^{3/2}}{24 c^2 d^6 (b+2 c x)^3}-\frac {\left (a+b x+c x^2\right )^{5/2}}{10 c d^6 (b+2 c x)^5}+\frac {\tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{64 c^{7/2} d^6}\\ \end {align*}

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Mathematica [C]  time = 0.05, size = 97, normalized size = 0.70 \begin {gather*} -\frac {\left (b^2-4 a c\right )^2 \sqrt {a+x (b+c x)} \, _2F_1\left (-\frac {5}{2},-\frac {5}{2};-\frac {3}{2};\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{320 c^3 d^6 (b+2 c x)^5 \sqrt {\frac {c (a+x (b+c x))}{4 a c-b^2}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^6,x]

[Out]

-1/320*((b^2 - 4*a*c)^2*Sqrt[a + x*(b + c*x)]*Hypergeometric2F1[-5/2, -5/2, -3/2, (b + 2*c*x)^2/(b^2 - 4*a*c)]
)/(c^3*d^6*(b + 2*c*x)^5*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)])

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IntegrateAlgebraic [A]  time = 1.02, size = 146, normalized size = 1.05 \begin {gather*} \frac {\sqrt {a+b x+c x^2} \left (-48 a^2 c^2-20 a b^2 c-176 a b c^2 x-176 a c^3 x^2-15 b^4-140 b^3 c x-508 b^2 c^2 x^2-736 b c^3 x^3-368 c^4 x^4\right )}{480 c^3 d^6 (b+2 c x)^5}-\frac {\log \left (-2 \sqrt {c} \sqrt {a+b x+c x^2}+b+2 c x\right )}{64 c^{7/2} d^6} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^6,x]

[Out]

(Sqrt[a + b*x + c*x^2]*(-15*b^4 - 20*a*b^2*c - 48*a^2*c^2 - 140*b^3*c*x - 176*a*b*c^2*x - 508*b^2*c^2*x^2 - 17
6*a*c^3*x^2 - 736*b*c^3*x^3 - 368*c^4*x^4))/(480*c^3*d^6*(b + 2*c*x)^5) - Log[b + 2*c*x - 2*Sqrt[c]*Sqrt[a + b
*x + c*x^2]]/(64*c^(7/2)*d^6)

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fricas [B]  time = 2.52, size = 549, normalized size = 3.95 \begin {gather*} \left [\frac {15 \, {\left (32 \, c^{5} x^{5} + 80 \, b c^{4} x^{4} + 80 \, b^{2} c^{3} x^{3} + 40 \, b^{3} c^{2} x^{2} + 10 \, b^{4} c x + b^{5}\right )} \sqrt {c} \log \left (-8 \, c^{2} x^{2} - 8 \, b c x - b^{2} - 4 \, \sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {c} - 4 \, a c\right ) - 4 \, {\left (368 \, c^{5} x^{4} + 736 \, b c^{4} x^{3} + 15 \, b^{4} c + 20 \, a b^{2} c^{2} + 48 \, a^{2} c^{3} + 4 \, {\left (127 \, b^{2} c^{3} + 44 \, a c^{4}\right )} x^{2} + 4 \, {\left (35 \, b^{3} c^{2} + 44 \, a b c^{3}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{1920 \, {\left (32 \, c^{9} d^{6} x^{5} + 80 \, b c^{8} d^{6} x^{4} + 80 \, b^{2} c^{7} d^{6} x^{3} + 40 \, b^{3} c^{6} d^{6} x^{2} + 10 \, b^{4} c^{5} d^{6} x + b^{5} c^{4} d^{6}\right )}}, -\frac {15 \, {\left (32 \, c^{5} x^{5} + 80 \, b c^{4} x^{4} + 80 \, b^{2} c^{3} x^{3} + 40 \, b^{3} c^{2} x^{2} + 10 \, b^{4} c x + b^{5}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{2} + b c x + a c\right )}}\right ) + 2 \, {\left (368 \, c^{5} x^{4} + 736 \, b c^{4} x^{3} + 15 \, b^{4} c + 20 \, a b^{2} c^{2} + 48 \, a^{2} c^{3} + 4 \, {\left (127 \, b^{2} c^{3} + 44 \, a c^{4}\right )} x^{2} + 4 \, {\left (35 \, b^{3} c^{2} + 44 \, a b c^{3}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{960 \, {\left (32 \, c^{9} d^{6} x^{5} + 80 \, b c^{8} d^{6} x^{4} + 80 \, b^{2} c^{7} d^{6} x^{3} + 40 \, b^{3} c^{6} d^{6} x^{2} + 10 \, b^{4} c^{5} d^{6} x + b^{5} c^{4} d^{6}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^(5/2)/(2*c*d*x+b*d)^6,x, algorithm="fricas")

[Out]

[1/1920*(15*(32*c^5*x^5 + 80*b*c^4*x^4 + 80*b^2*c^3*x^3 + 40*b^3*c^2*x^2 + 10*b^4*c*x + b^5)*sqrt(c)*log(-8*c^
2*x^2 - 8*b*c*x - b^2 - 4*sqrt(c*x^2 + b*x + a)*(2*c*x + b)*sqrt(c) - 4*a*c) - 4*(368*c^5*x^4 + 736*b*c^4*x^3
+ 15*b^4*c + 20*a*b^2*c^2 + 48*a^2*c^3 + 4*(127*b^2*c^3 + 44*a*c^4)*x^2 + 4*(35*b^3*c^2 + 44*a*b*c^3)*x)*sqrt(
c*x^2 + b*x + a))/(32*c^9*d^6*x^5 + 80*b*c^8*d^6*x^4 + 80*b^2*c^7*d^6*x^3 + 40*b^3*c^6*d^6*x^2 + 10*b^4*c^5*d^
6*x + b^5*c^4*d^6), -1/960*(15*(32*c^5*x^5 + 80*b*c^4*x^4 + 80*b^2*c^3*x^3 + 40*b^3*c^2*x^2 + 10*b^4*c*x + b^5
)*sqrt(-c)*arctan(1/2*sqrt(c*x^2 + b*x + a)*(2*c*x + b)*sqrt(-c)/(c^2*x^2 + b*c*x + a*c)) + 2*(368*c^5*x^4 + 7
36*b*c^4*x^3 + 15*b^4*c + 20*a*b^2*c^2 + 48*a^2*c^3 + 4*(127*b^2*c^3 + 44*a*c^4)*x^2 + 4*(35*b^3*c^2 + 44*a*b*
c^3)*x)*sqrt(c*x^2 + b*x + a))/(32*c^9*d^6*x^5 + 80*b*c^8*d^6*x^4 + 80*b^2*c^7*d^6*x^3 + 40*b^3*c^6*d^6*x^2 +
10*b^4*c^5*d^6*x + b^5*c^4*d^6)]

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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)^(5/2)/(2*c*d*x+b*d)^6,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%%%{%%%{64,[6]%%%},[12,6,12,0]%%%}+%%%{%%%{-1536,[7]%%%},[12,6,10,1
]%%%}+%%%{%%%{15360,[8]%%%},[12,6,8,2]%%%}+%%%{%%%{-81920,[9]%%%},[12,6,6,3]%%%}+%%%{%%%{245760,[10]%%%},[12,6
,4,4]%%%}+%%%{%%%{-393216,[11]%%%},[12,6,2,5]%%%}+%%%{%%%{262144,[12]%%%},[12,6,0,6]%%%}+%%%{%%{[%%%{-384,[5]%
%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[11,6,13,0]%%%}+%%%{%%{[%%%{9216,[6]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[11,6,11,1]
%%%}+%%%{%%{[%%%{-92160,[7]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[11,6,9,2]%%%}+%%%{%%{[%%%{491520,[8]%%%},0]:[1,0,%
%%{-1,[1]%%%}]%%},[11,6,7,3]%%%}+%%%{%%{[%%%{-1474560,[9]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[11,6,5,4]%%%}+%%%{%%
{[%%%{2359296,[10]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[11,6,3,5]%%%}+%%%{%%{[%%%{-1572864,[11]%%%},0]:[1,0,%%%{-1,
[1]%%%}]%%},[11,6,1,6]%%%}+%%%{%%%{1152,[5]%%%},[10,6,14,0]%%%}+%%%{%%%{-28032,[6]%%%},[10,6,12,1]%%%}+%%%{%%%
{285696,[7]%%%},[10,6,10,2]%%%}+%%%{%%%{-1566720,[8]%%%},[10,6,8,3]%%%}+%%%{%%%{4915200,[9]%%%},[10,6,6,4]%%%}
+%%%{%%%{-8552448,[10]%%%},[10,6,4,5]%%%}+%%%{%%%{7077888,[11]%%%},[10,6,2,6]%%%}+%%%{%%%{-1572864,[12]%%%},[1
0,6,0,7]%%%}+%%%{%%{[%%%{-2240,[4]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[9,6,15,0]%%%}+%%%{%%{[%%%{55680,[5]%%%},0]:
[1,0,%%%{-1,[1]%%%}]%%},[9,6,13,1]%%%}+%%%{%%{[%%%{-583680,[6]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[9,6,11,2]%%%}+%
%%{%%{[%%%{3328000,[7]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[9,6,9,3]%%%}+%%%{%%{[%%%{-11059200,[8]%%%},0]:[1,0,%%%{
-1,[1]%%%}]%%},[9,6,7,4]%%%}+%%%{%%{[%%%{21135360,[9]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[9,6,5,5]%%%}+%%%{%%{[%%%
{-20971520,[10]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[9,6,3,6]%%%}+%%%{%%{[%%%{7864320,[11]%%%},0]:[1,0,%%%{-1,[1]%%
%}]%%},[9,6,1,7]%%%}+%%%{%%%{3120,[4]%%%},[8,6,16,0]%%%}+%%%{%%%{-79680,[5]%%%},[8,6,14,1]%%%}+%%%{%%%{864960,
[6]%%%},[8,6,12,2]%%%}+%%%{%%%{-5168640,[7]%%%},[8,6,10,3]%%%}+%%%{%%%{18355200,[8]%%%},[8,6,8,4]%%%}+%%%{%%%{
-38830080,[9]%%%},[8,6,6,5]%%%}+%%%{%%%{45957120,[10]%%%},[8,6,4,6]%%%}+%%%{%%%{-25559040,[11]%%%},[8,6,2,7]%%
%}+%%%{%%%{3932160,[12]%%%},[8,6,0,8]%%%}+%%%{%%{[%%%{-3264,[3]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[7,6,17,0]%%%}+
%%%{%%{[%%%{86016,[4]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[7,6,15,1]%%%}+%%%{%%{[%%%{-971520,[5]%%%},0]:[1,0,%%%{-1
,[1]%%%}]%%},[7,6,13,2]%%%}+%%%{%%{[%%%{6113280,[6]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[7,6,11,3]%%%}+%%%{%%{[%%%{
-23285760,[7]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[7,6,9,4]%%%}+%%%{%%{[%%%{54460416,[8]%%%},0]:[1,0,%%%{-1,[1]%%%}
]%%},[7,6,7,5]%%%}+%%%{%%{[%%%{-75300864,[9]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[7,6,5,6]%%%}+%%%{%%{[%%%{55050240
,[10]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[7,6,3,7]%%%}+%%%{%%{[%%%{-15728640,[11]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[
7,6,1,8]%%%}+%%%{%%%{2624,[3]%%%},[6,6,18,0]%%%}+%%%{%%%{-71616,[4]%%%},[6,6,16,1]%%%}+%%%{%%%{844800,[5]%%%},
[6,6,14,2]%%%}+%%%{%%%{-5617920,[6]%%%},[6,6,12,3]%%%}+%%%{%%%{23009280,[7]%%%},[6,6,10,4]%%%}+%%%{%%%{-594370
56,[8]%%%},[6,6,8,5]%%%}+%%%{%%%{94961664,[9]%%%},[6,6,6,6]%%%}+%%%{%%%{-87490560,[10]%%%},[6,6,4,7]%%%}+%%%{%
%%{39321600,[11]%%%},[6,6,2,8]%%%}+%%%{%%%{-5242880,[12]%%%},[6,6,0,9]%%%}+%%%{%%{[%%%{-1632,[2]%%%},0]:[1,0,%
%%{-1,[1]%%%}]%%},[5,6,19,0]%%%}+%%%{%%{[%%%{46272,[3]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[5,6,17,1]%%%}+%%%{%%{[%
%%{-571776,[4]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[5,6,15,2]%%%}+%%%{%%{[%%%{4028160,[5]%%%},0]:[1,0,%%%{-1,[1]%%%
}]%%},[5,6,13,3]%%%}+%%%{%%{[%%%{-17756160,[6]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[5,6,11,4]%%%}+%%%{%%{[%%%{50515
968,[7]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[5,6,9,5]%%%}+%%%{%%{[%%%{-92110848,[8]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},
[5,6,7,6]%%%}+%%%{%%{[%%%{102825984,[9]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[5,6,5,7]%%%}+%%%{%%{[%%%{-62914560,[10
]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[5,6,3,8]%%%}+%%%{%%{[%%%{15728640,[11]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[5,6,1
,9]%%%}+%%%{%%%{780,[2]%%%},[4,6,20,0]%%%}+%%%{%%%{-23040,[3]%%%},[4,6,18,1]%%%}+%%%{%%%{299040,[4]%%%},[4,6,1
6,2]%%%}+%%%{%%%{-2236800,[5]%%%},[4,6,14,3]%%%}+%%%{%%%{10622400,[6]%%%},[4,6,12,4]%%%}+%%%{%%%{-33231360,[7]
%%%},[4,6,10,5]%%%}+%%%{%%%{68674560,[8]%%%},[4,6,8,6]%%%}+%%%{%%%{-91176960,[9]%%%},[4,6,6,7]%%%}+%%%{%%%{724
99200,[10]%%%},[4,6,4,8]%%%}+%%%{%%%{-29491200,[11]%%%},[4,6,2,9]%%%}+%%%{%%%{3932160,[12]%%%},[4,6,0,10]%%%}+
%%%{%%{[%%%{-280,[1]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3,6,21,0]%%%}+%%%{%%{[%%%{8640,[2]%%%},0]:[1,0,%%%{-1,[1]
%%%}]%%},[3,6,19,1]%%%}+%%%{%%{[%%%{-118080,[3]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3,6,17,2]%%%}+%%%{%%{[%%%{9395
20,[4]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3,6,15,3]%%%}+%%%{%%{[%%%{-4809600,[5]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[
3,6,13,4]%%%}+%%%{%%{[%%%{16512000,[6]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3,6,11,5]%%%}+%%%{%%{[%%%{-38389760,[7]
%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3,6,9,6]%%%}+%%%{%%{[%%%{59473920,[8]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3,6,7,7
]%%%}+%%%{%%{[%%%{-58490880,[9]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3,6,5,8]%%%}+%%%{%%{[%%%{32768000,[10]%%%},0]:
[1,0,%%%{-1,[1]%%%}]%%},[3,6,3,9]%%%}+%%%{%%{[%%%{-7864320,[11]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3,6,1,10]%%%}+
%%%{%%%{72,[1]%%%},[2,6,22,0]%%%}+%%%{%%%{-2328,[2]%%%},[2,6,20,1]%%%}+%%%{%%%{33600,[3]%%%},[2,6,18,2]%%%}+%%
%{%%%{-285120,[4]%%%},[2,6,16,3]%%%}+%%%{%%%{1576320,[5]%%%},[2,6,14,4]%%%}+%%%{%%%{-5941632,[6]%%%},[2,6,12,5
]%%%}+%%%{%%%{15510528,[7]%%%},[2,6,10,6]%%%}+%%%{%%%{-27863040,[8]%%%},[2,6,8,7]%%%}+%%%{%%%{33423360,[9]%%%}
,[2,6,6,8]%%%}+%%%{%%%{-25067520,[10]%%%},[2,6,4,9]%%%}+%%%{%%%{10223616,[11]%%%},[2,6,2,10]%%%}+%%%{%%%{-1572
864,[12]%%%},[2,6,0,11]%%%}+%%%{%%{[-12,0]:[1,0,%%%{-1,[1]%%%}]%%},[1,6,23,0]%%%}+%%%{%%{[%%%{408,[1]%%%},0]:[
1,0,%%%{-1,[1]%%%}]%%},[1,6,21,1]%%%}+%%%{%%{[%%%{-6240,[2]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,6,19,2]%%%}+%%%{
%%{[%%%{56640,[3]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,6,17,3]%%%}+%%%{%%{[%%%{-338880,[4]%%%},0]:[1,0,%%%{-1,[1]
%%%}]%%},[1,6,15,4]%%%}+%%%{%%{[%%%{1402752,[5]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,6,13,5]%%%}+%%%{%%{[%%%{-409
8048,[6]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,6,11,6]%%%}+%%%{%%{[%%%{8448000,[7]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},
[1,6,9,7]%%%}+%%%{%%{[%%%{-12042240,[8]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,6,7,8]%%%}+%%%{%%{[%%%{11304960,[9]%
%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,6,5,9]%%%}+%%%{%%{[%%%{-6291456,[10]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,6,3,1
0]%%%}+%%%{%%{[%%%{1572864,[11]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,6,1,11]%%%}+%%%{1,[0,6,24,0]%%%}+%%%{%%%{-36
,[1]%%%},[0,6,22,1]%%%}+%%%{%%%{588,[2]%%%},[0,6,20,2]%%%}+%%%{%%%{-5760,[3]%%%},[0,6,18,3]%%%}+%%%{%%%{37680,
[4]%%%},[0,6,16,4]%%%}+%%%{%%%{-173376,[5]%%%},[0,6,14,5]%%%}+%%%{%%%{575296,[6]%%%},[0,6,12,6]%%%}+%%%{%%%{-1
387008,[7]%%%},[0,6,10,7]%%%}+%%%{%%%{2411520,[8]%%%},[0,6,8,8]%%%}+%%%{%%%{-2949120,[9]%%%},[0,6,6,9]%%%}+%%%
{%%%{2408448,[10]%%%},[0,6,4,10]%%%}+%%%{%%%{-1179648,[11]%%%},[0,6,2,11]%%%}+%%%{%%%{262144,[12]%%%},[0,6,0,1
2]%%%} / %%%{%%%{64,[9]%%%},[12,0,0,0]%%%}+%%%{%%{poly1[%%%{-384,[8]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[11,0,1,0]
%%%}+%%%{%%%{1152,[8]%%%},[10,0,2,0]%%%}+%%%{%%%{-384,[9]%%%},[10,0,0,1]%%%}+%%%{%%{poly1[%%%{-2240,[7]%%%},0]
:[1,0,%%%{-1,[1]%%%}]%%},[9,0,3,0]%%%}+%%%{%%{[%%%{1920,[8]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[9,0,1,1]%%%}+%%%{%
%%{3120,[7]%%%},[8,0,4,0]%%%}+%%%{%%%{-4800,[8]%%%},[8,0,2,1]%%%}+%%%{%%%{960,[9]%%%},[8,0,0,2]%%%}+%%%{%%{pol
y1[%%%{-3264,[6]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[7,0,5,0]%%%}+%%%{%%{[%%%{7680,[7]%%%},0]:[1,0,%%%{-1,[1]%%%}]
%%},[7,0,3,1]%%%}+%%%{%%{poly1[%%%{-3840,[8]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[7,0,1,2]%%%}+%%%{%%%{2624,[6]%%%}
,[6,0,6,0]%%%}+%%%{%%%{-8640,[7]%%%},[6,0,4,1]%%%}+%%%{%%%{7680,[8]%%%},[6,0,2,2]%%%}+%%%{%%%{-1280,[9]%%%},[6
,0,0,3]%%%}+%%%{%%{poly1[%%%{-1632,[5]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[5,0,7,0]%%%}+%%%{%%{[%%%{7104,[6]%%%},0
]:[1,0,%%%{-1,[1]%%%}]%%},[5,0,5,1]%%%}+%%%{%%{poly1[%%%{-9600,[7]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[5,0,3,2]%%%
}+%%%{%%{[%%%{3840,[8]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[5,0,1,3]%%%}+%%%{%%%{780,[5]%%%},[4,0,8,0]%%%}+%%%{%%%{
-4320,[6]%%%},[4,0,6,1]%%%}+%%%{%%%{8160,[7]%%%},[4,0,4,2]%%%}+%%%{%%%{-5760,[8]%%%},[4,0,2,3]%%%}+%%%{%%%{960
,[9]%%%},[4,0,0,4]%%%}+%%%{%%{poly1[%%%{-280,[4]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3,0,9,0]%%%}+%%%{%%{[%%%{1920
,[5]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3,0,7,1]%%%}+%%%{%%{poly1[%%%{-4800,[6]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3
,0,5,2]%%%}+%%%{%%{[%%%{5120,[7]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3,0,3,3]%%%}+%%%{%%{poly1[%%%{-1920,[8]%%%},0
]:[1,0,%%%{-1,[1]%%%}]%%},[3,0,1,4]%%%}+%%%{%%%{72,[4]%%%},[2,0,10,0]%%%}+%%%{%%%{-600,[5]%%%},[2,0,8,1]%%%}+%
%%{%%%{1920,[6]%%%},[2,0,6,2]%%%}+%%%{%%%{-2880,[7]%%%},[2,0,4,3]%%%}+%%%{%%%{1920,[8]%%%},[2,0,2,4]%%%}+%%%{%
%%{-384,[9]%%%},[2,0,0,5]%%%}+%%%{%%{poly1[%%%{-12,[3]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,0,11,0]%%%}+%%%{%%{[%
%%{120,[4]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,0,9,1]%%%}+%%%{%%{poly1[%%%{-480,[5]%%%},0]:[1,0,%%%{-1,[1]%%%}]%
%},[1,0,7,2]%%%}+%%%{%%{[%%%{960,[6]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,0,5,3]%%%}+%%%{%%{poly1[%%%{-960,[7]%%%
},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,0,3,4]%%%}+%%%{%%{[%%%{384,[8]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,0,1,5]%%%}+%%
%{%%%{1,[3]%%%},[0,0,12,0]%%%}+%%%{%%%{-12,[4]%%%},[0,0,10,1]%%%}+%%%{%%%{60,[5]%%%},[0,0,8,2]%%%}+%%%{%%%{-16
0,[6]%%%},[0,0,6,3]%%%}+%%%{%%%{240,[7]%%%},[0,0,4,4]%%%}+%%%{%%%{-192,[8]%%%},[0,0,2,5]%%%}+%%%{%%%{64,[9]%%%
},[0,0,0,6]%%%} Error: Bad Argument Value

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maple [B]  time = 0.07, size = 1080, normalized size = 7.77 \begin {gather*} \frac {a^{3} \ln \left (\left (x +\frac {b}{2 c}\right ) \sqrt {c}+\sqrt {\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}}\right )}{\left (4 a c -b^{2}\right )^{3} \sqrt {c}\, d^{6}}-\frac {3 a^{2} b^{2} \ln \left (\left (x +\frac {b}{2 c}\right ) \sqrt {c}+\sqrt {\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}}\right )}{4 \left (4 a c -b^{2}\right )^{3} c^{\frac {3}{2}} d^{6}}+\frac {3 a \,b^{4} \ln \left (\left (x +\frac {b}{2 c}\right ) \sqrt {c}+\sqrt {\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}}\right )}{16 \left (4 a c -b^{2}\right )^{3} c^{\frac {5}{2}} d^{6}}-\frac {b^{6} \ln \left (\left (x +\frac {b}{2 c}\right ) \sqrt {c}+\sqrt {\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}}\right )}{64 \left (4 a c -b^{2}\right )^{3} c^{\frac {7}{2}} d^{6}}+\frac {\sqrt {\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}}\, a^{2} x}{\left (4 a c -b^{2}\right )^{3} d^{6}}-\frac {\sqrt {\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}}\, a \,b^{2} x}{2 \left (4 a c -b^{2}\right )^{3} c \,d^{6}}+\frac {\sqrt {\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}}\, b^{4} x}{16 \left (4 a c -b^{2}\right )^{3} c^{2} d^{6}}+\frac {\sqrt {\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}}\, a^{2} b}{2 \left (4 a c -b^{2}\right )^{3} c \,d^{6}}-\frac {\sqrt {\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}}\, a \,b^{3}}{4 \left (4 a c -b^{2}\right )^{3} c^{2} d^{6}}+\frac {2 \left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {3}{2}} a x}{3 \left (4 a c -b^{2}\right )^{3} d^{6}}+\frac {\sqrt {\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}}\, b^{5}}{32 \left (4 a c -b^{2}\right )^{3} c^{3} d^{6}}-\frac {\left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {3}{2}} b^{2} x}{6 \left (4 a c -b^{2}\right )^{3} c \,d^{6}}+\frac {\left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {3}{2}} a b}{3 \left (4 a c -b^{2}\right )^{3} c \,d^{6}}-\frac {\left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {3}{2}} b^{3}}{12 \left (4 a c -b^{2}\right )^{3} c^{2} d^{6}}+\frac {8 \left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {5}{2}} x}{15 \left (4 a c -b^{2}\right )^{3} d^{6}}+\frac {4 \left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {5}{2}} b}{15 \left (4 a c -b^{2}\right )^{3} c \,d^{6}}-\frac {8 \left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {7}{2}}}{15 \left (4 a c -b^{2}\right )^{3} \left (x +\frac {b}{2 c}\right ) c \,d^{6}}-\frac {\left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {7}{2}}}{30 \left (4 a c -b^{2}\right )^{2} \left (x +\frac {b}{2 c}\right )^{3} c^{3} d^{6}}-\frac {\left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {7}{2}}}{80 \left (4 a c -b^{2}\right ) \left (x +\frac {b}{2 c}\right )^{5} c^{5} d^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+b*x+a)^(5/2)/(2*c*d*x+b*d)^6,x)

[Out]

-1/80/d^6/c^5/(4*a*c-b^2)/(x+1/2*b/c)^5*((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(7/2)-1/30/d^6/c^3/(4*a*c-b^2)^2/(
x+1/2*b/c)^3*((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(7/2)-8/15/d^6/c/(4*a*c-b^2)^3/(x+1/2*b/c)*((x+1/2*b/c)^2*c+1
/4*(4*a*c-b^2)/c)^(7/2)+8/15/d^6/(4*a*c-b^2)^3*((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(5/2)*x+4/15/d^6/c/(4*a*c-b
^2)^3*((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(5/2)*b+2/3/d^6/(4*a*c-b^2)^3*((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(3
/2)*x*a-1/6/d^6/c/(4*a*c-b^2)^3*((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(3/2)*x*b^2+1/3/d^6/c/(4*a*c-b^2)^3*((x+1/
2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(3/2)*b*a-1/12/d^6/c^2/(4*a*c-b^2)^3*((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(3/2)*b
^3+1/d^6/(4*a*c-b^2)^3*((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(1/2)*x*a^2-1/2/d^6/c/(4*a*c-b^2)^3*((x+1/2*b/c)^2*
c+1/4*(4*a*c-b^2)/c)^(1/2)*x*a*b^2+1/16/d^6/c^2/(4*a*c-b^2)^3*((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(1/2)*x*b^4+
1/2/d^6/c/(4*a*c-b^2)^3*((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(1/2)*b*a^2-1/4/d^6/c^2/(4*a*c-b^2)^3*((x+1/2*b/c)
^2*c+1/4*(4*a*c-b^2)/c)^(1/2)*b^3*a+1/32/d^6/c^3/(4*a*c-b^2)^3*((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(1/2)*b^5+1
/d^6/c^(1/2)/(4*a*c-b^2)^3*ln((x+1/2*b/c)*c^(1/2)+((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(1/2))*a^3-3/4/d^6/c^(3/
2)/(4*a*c-b^2)^3*ln((x+1/2*b/c)*c^(1/2)+((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(1/2))*b^2*a^2+3/16/d^6/c^(5/2)/(4
*a*c-b^2)^3*ln((x+1/2*b/c)*c^(1/2)+((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(1/2))*b^4*a-1/64/d^6/c^(7/2)/(4*a*c-b^
2)^3*ln((x+1/2*b/c)*c^(1/2)+((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(1/2))*b^6

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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)^(5/2)/(2*c*d*x+b*d)^6,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 {{\left (c\,x^2+b\,x+a\right )}^{5/2}}{{\left (b\,d+2\,c\,d\,x\right )}^6} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^6,x)

[Out]

int((a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^6, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {a^{2} \sqrt {a + b x + c x^{2}}}{b^{6} + 12 b^{5} c x + 60 b^{4} c^{2} x^{2} + 160 b^{3} c^{3} x^{3} + 240 b^{2} c^{4} x^{4} + 192 b c^{5} x^{5} + 64 c^{6} x^{6}}\, dx + \int \frac {b^{2} x^{2} \sqrt {a + b x + c x^{2}}}{b^{6} + 12 b^{5} c x + 60 b^{4} c^{2} x^{2} + 160 b^{3} c^{3} x^{3} + 240 b^{2} c^{4} x^{4} + 192 b c^{5} x^{5} + 64 c^{6} x^{6}}\, dx + \int \frac {c^{2} x^{4} \sqrt {a + b x + c x^{2}}}{b^{6} + 12 b^{5} c x + 60 b^{4} c^{2} x^{2} + 160 b^{3} c^{3} x^{3} + 240 b^{2} c^{4} x^{4} + 192 b c^{5} x^{5} + 64 c^{6} x^{6}}\, dx + \int \frac {2 a b x \sqrt {a + b x + c x^{2}}}{b^{6} + 12 b^{5} c x + 60 b^{4} c^{2} x^{2} + 160 b^{3} c^{3} x^{3} + 240 b^{2} c^{4} x^{4} + 192 b c^{5} x^{5} + 64 c^{6} x^{6}}\, dx + \int \frac {2 a c x^{2} \sqrt {a + b x + c x^{2}}}{b^{6} + 12 b^{5} c x + 60 b^{4} c^{2} x^{2} + 160 b^{3} c^{3} x^{3} + 240 b^{2} c^{4} x^{4} + 192 b c^{5} x^{5} + 64 c^{6} x^{6}}\, dx + \int \frac {2 b c x^{3} \sqrt {a + b x + c x^{2}}}{b^{6} + 12 b^{5} c x + 60 b^{4} c^{2} x^{2} + 160 b^{3} c^{3} x^{3} + 240 b^{2} c^{4} x^{4} + 192 b c^{5} x^{5} + 64 c^{6} x^{6}}\, dx}{d^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+b*x+a)**(5/2)/(2*c*d*x+b*d)**6,x)

[Out]

(Integral(a**2*sqrt(a + b*x + c*x**2)/(b**6 + 12*b**5*c*x + 60*b**4*c**2*x**2 + 160*b**3*c**3*x**3 + 240*b**2*
c**4*x**4 + 192*b*c**5*x**5 + 64*c**6*x**6), x) + Integral(b**2*x**2*sqrt(a + b*x + c*x**2)/(b**6 + 12*b**5*c*
x + 60*b**4*c**2*x**2 + 160*b**3*c**3*x**3 + 240*b**2*c**4*x**4 + 192*b*c**5*x**5 + 64*c**6*x**6), x) + Integr
al(c**2*x**4*sqrt(a + b*x + c*x**2)/(b**6 + 12*b**5*c*x + 60*b**4*c**2*x**2 + 160*b**3*c**3*x**3 + 240*b**2*c*
*4*x**4 + 192*b*c**5*x**5 + 64*c**6*x**6), x) + Integral(2*a*b*x*sqrt(a + b*x + c*x**2)/(b**6 + 12*b**5*c*x +
60*b**4*c**2*x**2 + 160*b**3*c**3*x**3 + 240*b**2*c**4*x**4 + 192*b*c**5*x**5 + 64*c**6*x**6), x) + Integral(2
*a*c*x**2*sqrt(a + b*x + c*x**2)/(b**6 + 12*b**5*c*x + 60*b**4*c**2*x**2 + 160*b**3*c**3*x**3 + 240*b**2*c**4*
x**4 + 192*b*c**5*x**5 + 64*c**6*x**6), x) + Integral(2*b*c*x**3*sqrt(a + b*x + c*x**2)/(b**6 + 12*b**5*c*x +
60*b**4*c**2*x**2 + 160*b**3*c**3*x**3 + 240*b**2*c**4*x**4 + 192*b*c**5*x**5 + 64*c**6*x**6), x))/d**6

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