3.1228 \(\int \frac {(a+b x+c x^2)^{5/2}}{(b d+2 c d x)^7} \, dx\)

Optimal. Leaf size=155 \[ \frac {5 \tan ^{-1}\left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )}{1024 c^{7/2} d^7 \sqrt {b^2-4 a c}}-\frac {5 \sqrt {a+b x+c x^2}}{512 c^3 d^7 (b+2 c x)^2}-\frac {5 \left (a+b x+c x^2\right )^{3/2}}{192 c^2 d^7 (b+2 c x)^4}-\frac {\left (a+b x+c x^2\right )^{5/2}}{12 c d^7 (b+2 c x)^6} \]

[Out]

-5/192*(c*x^2+b*x+a)^(3/2)/c^2/d^7/(2*c*x+b)^4-1/12*(c*x^2+b*x+a)^(5/2)/c/d^7/(2*c*x+b)^6+5/1024*arctan(2*c^(1
/2)*(c*x^2+b*x+a)^(1/2)/(-4*a*c+b^2)^(1/2))/c^(7/2)/d^7/(-4*a*c+b^2)^(1/2)-5/512*(c*x^2+b*x+a)^(1/2)/c^3/d^7/(
2*c*x+b)^2

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Rubi [A]  time = 0.10, antiderivative size = 155, 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, 688, 205} \[ \frac {5 \tan ^{-1}\left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )}{1024 c^{7/2} d^7 \sqrt {b^2-4 a c}}-\frac {5 \left (a+b x+c x^2\right )^{3/2}}{192 c^2 d^7 (b+2 c x)^4}-\frac {5 \sqrt {a+b x+c x^2}}{512 c^3 d^7 (b+2 c x)^2}-\frac {\left (a+b x+c x^2\right )^{5/2}}{12 c d^7 (b+2 c x)^6} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^7,x]

[Out]

(-5*Sqrt[a + b*x + c*x^2])/(512*c^3*d^7*(b + 2*c*x)^2) - (5*(a + b*x + c*x^2)^(3/2))/(192*c^2*d^7*(b + 2*c*x)^
4) - (a + b*x + c*x^2)^(5/2)/(12*c*d^7*(b + 2*c*x)^6) + (5*ArcTan[(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])/Sqrt[b^2 -
4*a*c]])/(1024*c^(7/2)*Sqrt[b^2 - 4*a*c]*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]

Rubi steps

\begin {align*} \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^7} \, dx &=-\frac {\left (a+b x+c x^2\right )^{5/2}}{12 c d^7 (b+2 c x)^6}+\frac {5 \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^5} \, dx}{24 c d^2}\\ &=-\frac {5 \left (a+b x+c x^2\right )^{3/2}}{192 c^2 d^7 (b+2 c x)^4}-\frac {\left (a+b x+c x^2\right )^{5/2}}{12 c d^7 (b+2 c x)^6}+\frac {5 \int \frac {\sqrt {a+b x+c x^2}}{(b d+2 c d x)^3} \, dx}{128 c^2 d^4}\\ &=-\frac {5 \sqrt {a+b x+c x^2}}{512 c^3 d^7 (b+2 c x)^2}-\frac {5 \left (a+b x+c x^2\right )^{3/2}}{192 c^2 d^7 (b+2 c x)^4}-\frac {\left (a+b x+c x^2\right )^{5/2}}{12 c d^7 (b+2 c x)^6}+\frac {5 \int \frac {1}{(b d+2 c d x) \sqrt {a+b x+c x^2}} \, dx}{1024 c^3 d^6}\\ &=-\frac {5 \sqrt {a+b x+c x^2}}{512 c^3 d^7 (b+2 c x)^2}-\frac {5 \left (a+b x+c x^2\right )^{3/2}}{192 c^2 d^7 (b+2 c x)^4}-\frac {\left (a+b x+c x^2\right )^{5/2}}{12 c d^7 (b+2 c x)^6}+\frac {5 \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 )}{256 c^2 d^6}\\ &=-\frac {5 \sqrt {a+b x+c x^2}}{512 c^3 d^7 (b+2 c x)^2}-\frac {5 \left (a+b x+c x^2\right )^{3/2}}{192 c^2 d^7 (b+2 c x)^4}-\frac {\left (a+b x+c x^2\right )^{5/2}}{12 c d^7 (b+2 c x)^6}+\frac {5 \tan ^{-1}\left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )}{1024 c^{7/2} \sqrt {b^2-4 a c} d^7}\\ \end {align*}

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Mathematica [A]  time = 0.37, size = 233, normalized size = 1.50 \[ \frac {-2 c \left (128 a^3 c^2+8 a^2 c \left (5 b^2+68 b c x+68 c^2 x^2\right )+a \left (15 b^4+200 b^3 c x+1144 b^2 c^2 x^2+1888 b c^3 x^3+944 c^4 x^4\right )+x \left (15 b^5+175 b^4 c x+848 b^3 c^2 x^2+1744 b^2 c^3 x^3+1584 b c^4 x^4+528 c^5 x^5\right )\right )-15 (b+2 c x)^6 \sqrt {\frac {c (a+x (b+c x))}{4 a c-b^2}} \tanh ^{-1}\left (2 \sqrt {\frac {c (a+x (b+c x))}{4 a c-b^2}}\right )}{3072 c^4 d^7 (b+2 c x)^6 \sqrt {a+x (b+c x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^7,x]

[Out]

(-2*c*(128*a^3*c^2 + 8*a^2*c*(5*b^2 + 68*b*c*x + 68*c^2*x^2) + a*(15*b^4 + 200*b^3*c*x + 1144*b^2*c^2*x^2 + 18
88*b*c^3*x^3 + 944*c^4*x^4) + x*(15*b^5 + 175*b^4*c*x + 848*b^3*c^2*x^2 + 1744*b^2*c^3*x^3 + 1584*b*c^4*x^4 +
528*c^5*x^5)) - 15*(b + 2*c*x)^6*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)]*ArcTanh[2*Sqrt[(c*(a + x*(b + c*x)
))/(-b^2 + 4*a*c)]])/(3072*c^4*d^7*(b + 2*c*x)^6*Sqrt[a + x*(b + c*x)])

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fricas [B]  time = 8.26, size = 914, normalized size = 5.90 \[ \left [-\frac {15 \, {\left (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}\right )} \sqrt {-b^{2} c + 4 \, a c^{2}} \log \left (-\frac {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}}\right ) + 4 \, {\left (15 \, b^{6} c - 20 \, a b^{4} c^{2} - 32 \, a^{2} b^{2} c^{3} - 512 \, a^{3} c^{4} + 528 \, {\left (b^{2} c^{5} - 4 \, a c^{6}\right )} x^{4} + 1056 \, {\left (b^{3} c^{4} - 4 \, a b c^{5}\right )} x^{3} + 16 \, {\left (43 \, b^{4} c^{3} - 146 \, a b^{2} c^{4} - 104 \, a^{2} c^{5}\right )} x^{2} + 32 \, {\left (5 \, b^{5} c^{2} - 7 \, a b^{3} c^{3} - 52 \, a^{2} b c^{4}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{6144 \, {\left (64 \, {\left (b^{2} c^{10} - 4 \, a c^{11}\right )} d^{7} x^{6} + 192 \, {\left (b^{3} c^{9} - 4 \, a b c^{10}\right )} d^{7} x^{5} + 240 \, {\left (b^{4} c^{8} - 4 \, a b^{2} c^{9}\right )} d^{7} x^{4} + 160 \, {\left (b^{5} c^{7} - 4 \, a b^{3} c^{8}\right )} d^{7} x^{3} + 60 \, {\left (b^{6} c^{6} - 4 \, a b^{4} c^{7}\right )} d^{7} x^{2} + 12 \, {\left (b^{7} c^{5} - 4 \, a b^{5} c^{6}\right )} d^{7} x + {\left (b^{8} c^{4} - 4 \, a b^{6} c^{5}\right )} d^{7}\right )}}, -\frac {15 \, {\left (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}\right )} \sqrt {b^{2} c - 4 \, a c^{2}} \arctan \left (\frac {\sqrt {b^{2} c - 4 \, a c^{2}} \sqrt {c x^{2} + b x + a}}{2 \, {\left (c^{2} x^{2} + b c x + a c\right )}}\right ) + 2 \, {\left (15 \, b^{6} c - 20 \, a b^{4} c^{2} - 32 \, a^{2} b^{2} c^{3} - 512 \, a^{3} c^{4} + 528 \, {\left (b^{2} c^{5} - 4 \, a c^{6}\right )} x^{4} + 1056 \, {\left (b^{3} c^{4} - 4 \, a b c^{5}\right )} x^{3} + 16 \, {\left (43 \, b^{4} c^{3} - 146 \, a b^{2} c^{4} - 104 \, a^{2} c^{5}\right )} x^{2} + 32 \, {\left (5 \, b^{5} c^{2} - 7 \, a b^{3} c^{3} - 52 \, a^{2} b c^{4}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{3072 \, {\left (64 \, {\left (b^{2} c^{10} - 4 \, a c^{11}\right )} d^{7} x^{6} + 192 \, {\left (b^{3} c^{9} - 4 \, a b c^{10}\right )} d^{7} x^{5} + 240 \, {\left (b^{4} c^{8} - 4 \, a b^{2} c^{9}\right )} d^{7} x^{4} + 160 \, {\left (b^{5} c^{7} - 4 \, a b^{3} c^{8}\right )} d^{7} x^{3} + 60 \, {\left (b^{6} c^{6} - 4 \, a b^{4} c^{7}\right )} d^{7} x^{2} + 12 \, {\left (b^{7} c^{5} - 4 \, a b^{5} c^{6}\right )} d^{7} x + {\left (b^{8} c^{4} - 4 \, a b^{6} c^{5}\right )} d^{7}\right )}}\right ] \]

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)^7,x, algorithm="fricas")

[Out]

[-1/6144*(15*(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*(15*b^6*c - 20*a*b^4*c^2 - 32*a^2*b^2*c^3 - 512*a^3*c^4 + 528*(b^2*c^5
- 4*a*c^6)*x^4 + 1056*(b^3*c^4 - 4*a*b*c^5)*x^3 + 16*(43*b^4*c^3 - 146*a*b^2*c^4 - 104*a^2*c^5)*x^2 + 32*(5*b
^5*c^2 - 7*a*b^3*c^3 - 52*a^2*b*c^4)*x)*sqrt(c*x^2 + b*x + a))/(64*(b^2*c^10 - 4*a*c^11)*d^7*x^6 + 192*(b^3*c^
9 - 4*a*b*c^10)*d^7*x^5 + 240*(b^4*c^8 - 4*a*b^2*c^9)*d^7*x^4 + 160*(b^5*c^7 - 4*a*b^3*c^8)*d^7*x^3 + 60*(b^6*
c^6 - 4*a*b^4*c^7)*d^7*x^2 + 12*(b^7*c^5 - 4*a*b^5*c^6)*d^7*x + (b^8*c^4 - 4*a*b^6*c^5)*d^7), -1/3072*(15*(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*(15*b^6*c - 20*a*
b^4*c^2 - 32*a^2*b^2*c^3 - 512*a^3*c^4 + 528*(b^2*c^5 - 4*a*c^6)*x^4 + 1056*(b^3*c^4 - 4*a*b*c^5)*x^3 + 16*(43
*b^4*c^3 - 146*a*b^2*c^4 - 104*a^2*c^5)*x^2 + 32*(5*b^5*c^2 - 7*a*b^3*c^3 - 52*a^2*b*c^4)*x)*sqrt(c*x^2 + b*x
+ a))/(64*(b^2*c^10 - 4*a*c^11)*d^7*x^6 + 192*(b^3*c^9 - 4*a*b*c^10)*d^7*x^5 + 240*(b^4*c^8 - 4*a*b^2*c^9)*d^7
*x^4 + 160*(b^5*c^7 - 4*a*b^3*c^8)*d^7*x^3 + 60*(b^6*c^6 - 4*a*b^4*c^7)*d^7*x^2 + 12*(b^7*c^5 - 4*a*b^5*c^6)*d
^7*x + (b^8*c^4 - 4*a*b^6*c^5)*d^7)]

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

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)^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 = 960, normalized size = 6.19 \[ -\frac {5 a^{3} \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 {15 a^{2} 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 {15 a \,b^{4} \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 )}{256 \left (4 a c -b^{2}\right )^{3} \sqrt {\frac {4 a c -b^{2}}{c}}\, c^{3} d^{7}}+\frac {5 b^{6} \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 )}{1024 \left (4 a c -b^{2}\right )^{3} \sqrt {\frac {4 a c -b^{2}}{c}}\, c^{4} d^{7}}+\frac {5 \sqrt {4 \left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{c}}\, a^{2}}{64 \left (4 a c -b^{2}\right )^{3} c \,d^{7}}-\frac {5 \sqrt {4 \left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{c}}\, a \,b^{2}}{128 \left (4 a c -b^{2}\right )^{3} c^{2} d^{7}}+\frac {5 \sqrt {4 \left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{c}}\, b^{4}}{1024 \left (4 a c -b^{2}\right )^{3} c^{3} d^{7}}+\frac {5 \left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {3}{2}} a}{96 \left (4 a c -b^{2}\right )^{3} c \,d^{7}}-\frac {5 \left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {3}{2}} b^{2}}{384 \left (4 a c -b^{2}\right )^{3} 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 {5}{2}}}{32 \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 {7}{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 {7}{2}}}{192 \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 {7}{2}}}{192 \left (4 a c -b^{2}\right ) \left (x +\frac {b}{2 c}\right )^{6} c^{6} d^{7}} \]

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)^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)^(7/2)-1/192/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)^(7/2)-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)^(7/2)+1/32/d^7/c/(4*a*c-b^2)^3*((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(5/2)+5/96/d^7/c/(4
*a*c-b^2)^3*((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(3/2)*a-5/384/d^7/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^2+5/64/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)*a^2-5/128/d^7/c^2/(4*a*c-
b^2)^3*(4*(x+1/2*b/c)^2*c+(4*a*c-b^2)/c)^(1/2)*a*b^2+5/1024/d^7/c^3/(4*a*c-b^2)^3*(4*(x+1/2*b/c)^2*c+(4*a*c-b^
2)/c)^(1/2)*b^4-5/16/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^3+15/64/d^7/c^2/(4*a*c-b^2)^3/((4*a*c-b^2)/c)^(1/2)*l
n((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^2*b^2-1
5/256/d^7/c^3/(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*b^4+5/1024/d^7/c^4/(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^6

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

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)^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 \[ \int \frac {{\left (c\,x^2+b\,x+a\right )}^{5/2}}{{\left (b\,d+2\,c\,d\,x\right )}^7} \,d x \]

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)^7,x)

[Out]

int((a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^7, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {a^{2} \sqrt {a + b x + c x^{2}}}{b^{7} + 14 b^{6} c x + 84 b^{5} c^{2} x^{2} + 280 b^{4} c^{3} x^{3} + 560 b^{3} c^{4} x^{4} + 672 b^{2} c^{5} x^{5} + 448 b c^{6} x^{6} + 128 c^{7} x^{7}}\, dx + \int \frac {b^{2} x^{2} \sqrt {a + b x + c x^{2}}}{b^{7} + 14 b^{6} c x + 84 b^{5} c^{2} x^{2} + 280 b^{4} c^{3} x^{3} + 560 b^{3} c^{4} x^{4} + 672 b^{2} c^{5} x^{5} + 448 b c^{6} x^{6} + 128 c^{7} x^{7}}\, dx + \int \frac {c^{2} x^{4} \sqrt {a + b x + c x^{2}}}{b^{7} + 14 b^{6} c x + 84 b^{5} c^{2} x^{2} + 280 b^{4} c^{3} x^{3} + 560 b^{3} c^{4} x^{4} + 672 b^{2} c^{5} x^{5} + 448 b c^{6} x^{6} + 128 c^{7} x^{7}}\, dx + \int \frac {2 a b x \sqrt {a + b x + c x^{2}}}{b^{7} + 14 b^{6} c x + 84 b^{5} c^{2} x^{2} + 280 b^{4} c^{3} x^{3} + 560 b^{3} c^{4} x^{4} + 672 b^{2} c^{5} x^{5} + 448 b c^{6} x^{6} + 128 c^{7} x^{7}}\, dx + \int \frac {2 a c x^{2} \sqrt {a + b x + c x^{2}}}{b^{7} + 14 b^{6} c x + 84 b^{5} c^{2} x^{2} + 280 b^{4} c^{3} x^{3} + 560 b^{3} c^{4} x^{4} + 672 b^{2} c^{5} x^{5} + 448 b c^{6} x^{6} + 128 c^{7} x^{7}}\, dx + \int \frac {2 b c x^{3} \sqrt {a + b x + c x^{2}}}{b^{7} + 14 b^{6} c x + 84 b^{5} c^{2} x^{2} + 280 b^{4} c^{3} x^{3} + 560 b^{3} c^{4} x^{4} + 672 b^{2} c^{5} x^{5} + 448 b c^{6} x^{6} + 128 c^{7} x^{7}}\, dx}{d^{7}} \]

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)**7,x)

[Out]

(Integral(a**2*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) + Integral(b**2*x**2*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 + 4
48*b*c**6*x**6 + 128*c**7*x**7), x) + Integral(c**2*x**4*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
) + Integral(2*a*b*x*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) + Integral(2*a*c*x**2*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) + Integral(2*b*c*x**3*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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