3.1249 \(\int \frac {1}{(b d+2 c d x)^4 (a+b x+c x^2)^{3/2}} \, dx\)
Optimal. Leaf size=118 \[ -\frac {64 c \sqrt {a+b x+c x^2}}{3 d^4 \left (b^2-4 a c\right )^3 (b+2 c x)}-\frac {32 c \sqrt {a+b x+c x^2}}{3 d^4 \left (b^2-4 a c\right )^2 (b+2 c x)^3}-\frac {2}{d^4 \left (b^2-4 a c\right ) (b+2 c x)^3 \sqrt {a+b x+c x^2}} \]
[Out]
-2/(-4*a*c+b^2)/d^4/(2*c*x+b)^3/(c*x^2+b*x+a)^(1/2)-32/3*c*(c*x^2+b*x+a)^(1/2)/(-4*a*c+b^2)^2/d^4/(2*c*x+b)^3-
64/3*c*(c*x^2+b*x+a)^(1/2)/(-4*a*c+b^2)^3/d^4/(2*c*x+b)
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Rubi [A] time = 0.06, antiderivative size = 118, normalized size of antiderivative = 1.00,
number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used =
{687, 693, 682} \[ -\frac {64 c \sqrt {a+b x+c x^2}}{3 d^4 \left (b^2-4 a c\right )^3 (b+2 c x)}-\frac {32 c \sqrt {a+b x+c x^2}}{3 d^4 \left (b^2-4 a c\right )^2 (b+2 c x)^3}-\frac {2}{d^4 \left (b^2-4 a c\right ) (b+2 c x)^3 \sqrt {a+b x+c x^2}} \]
Antiderivative was successfully verified.
[In]
Int[1/((b*d + 2*c*d*x)^4*(a + b*x + c*x^2)^(3/2)),x]
[Out]
-2/((b^2 - 4*a*c)*d^4*(b + 2*c*x)^3*Sqrt[a + b*x + c*x^2]) - (32*c*Sqrt[a + b*x + c*x^2])/(3*(b^2 - 4*a*c)^2*d
^4*(b + 2*c*x)^3) - (64*c*Sqrt[a + b*x + c*x^2])/(3*(b^2 - 4*a*c)^3*d^4*(b + 2*c*x))
Rule 682
Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(2*c*(d + e*x)^(m +
1)*(a + b*x + c*x^2)^(p + 1))/(e*(p + 1)*(b^2 - 4*a*c)), x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*
a*c, 0] && EqQ[2*c*d - b*e, 0] && EqQ[m + 2*p + 3, 0] && NeQ[p, -1]
Rule 687
Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(2*c*(d + e*x)^(m +
1)*(a + b*x + c*x^2)^(p + 1))/(e*(p + 1)*(b^2 - 4*a*c)), x] - Dist[(2*c*e*(m + 2*p + 3))/(e*(p + 1)*(b^2 - 4*a
*c)), Int[(d + e*x)^m*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0]
&& EqQ[2*c*d - b*e, 0] && NeQ[m + 2*p + 3, 0] && LtQ[p, -1] && !GtQ[m, 1] && RationalQ[m] && IntegerQ[2*p]
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 {1}{(b d+2 c d x)^4 \left (a+b x+c x^2\right )^{3/2}} \, dx &=-\frac {2}{\left (b^2-4 a c\right ) d^4 (b+2 c x)^3 \sqrt {a+b x+c x^2}}-\frac {(16 c) \int \frac {1}{(b d+2 c d x)^4 \sqrt {a+b x+c x^2}} \, dx}{b^2-4 a c}\\ &=-\frac {2}{\left (b^2-4 a c\right ) d^4 (b+2 c x)^3 \sqrt {a+b x+c x^2}}-\frac {32 c \sqrt {a+b x+c x^2}}{3 \left (b^2-4 a c\right )^2 d^4 (b+2 c x)^3}-\frac {(32 c) \int \frac {1}{(b d+2 c d x)^2 \sqrt {a+b x+c x^2}} \, dx}{3 \left (b^2-4 a c\right )^2 d^2}\\ &=-\frac {2}{\left (b^2-4 a c\right ) d^4 (b+2 c x)^3 \sqrt {a+b x+c x^2}}-\frac {32 c \sqrt {a+b x+c x^2}}{3 \left (b^2-4 a c\right )^2 d^4 (b+2 c x)^3}-\frac {64 c \sqrt {a+b x+c x^2}}{3 \left (b^2-4 a c\right )^3 d^4 (b+2 c x)}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 108, normalized size = 0.92 \[ -\frac {2 \left (16 c^2 \left (-a^2+4 a c x^2+8 c^2 x^4\right )+8 b^2 c \left (3 a+22 c x^2\right )+64 b c^2 x \left (a+4 c x^2\right )+3 b^4+48 b^3 c x\right )}{3 d^4 \left (b^2-4 a c\right )^3 (b+2 c x)^3 \sqrt {a+x (b+c x)}} \]
Antiderivative was successfully verified.
[In]
Integrate[1/((b*d + 2*c*d*x)^4*(a + b*x + c*x^2)^(3/2)),x]
[Out]
(-2*(3*b^4 + 48*b^3*c*x + 64*b*c^2*x*(a + 4*c*x^2) + 8*b^2*c*(3*a + 22*c*x^2) + 16*c^2*(-a^2 + 4*a*c*x^2 + 8*c
^2*x^4)))/(3*(b^2 - 4*a*c)^3*d^4*(b + 2*c*x)^3*Sqrt[a + x*(b + c*x)])
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fricas [B] time = 4.82, size = 381, normalized size = 3.23 \[ -\frac {2 \, {\left (128 \, c^{4} x^{4} + 256 \, b c^{3} x^{3} + 3 \, b^{4} + 24 \, a b^{2} c - 16 \, a^{2} c^{2} + 16 \, {\left (11 \, b^{2} c^{2} + 4 \, a c^{3}\right )} x^{2} + 16 \, {\left (3 \, b^{3} c + 4 \, a b c^{2}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{3 \, {\left (8 \, {\left (b^{6} c^{4} - 12 \, a b^{4} c^{5} + 48 \, a^{2} b^{2} c^{6} - 64 \, a^{3} c^{7}\right )} d^{4} x^{5} + 20 \, {\left (b^{7} c^{3} - 12 \, a b^{5} c^{4} + 48 \, a^{2} b^{3} c^{5} - 64 \, a^{3} b c^{6}\right )} d^{4} x^{4} + 2 \, {\left (9 \, b^{8} c^{2} - 104 \, a b^{6} c^{3} + 384 \, a^{2} b^{4} c^{4} - 384 \, a^{3} b^{2} c^{5} - 256 \, a^{4} c^{6}\right )} d^{4} x^{3} + {\left (7 \, b^{9} c - 72 \, a b^{7} c^{2} + 192 \, a^{2} b^{5} c^{3} + 128 \, a^{3} b^{3} c^{4} - 768 \, a^{4} b c^{5}\right )} d^{4} x^{2} + {\left (b^{10} - 6 \, a b^{8} c - 24 \, a^{2} b^{6} c^{2} + 224 \, a^{3} b^{4} c^{3} - 384 \, a^{4} b^{2} c^{4}\right )} d^{4} x + {\left (a b^{9} - 12 \, a^{2} b^{7} c + 48 \, a^{3} b^{5} c^{2} - 64 \, a^{4} b^{3} c^{3}\right )} d^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(2*c*d*x+b*d)^4/(c*x^2+b*x+a)^(3/2),x, algorithm="fricas")
[Out]
-2/3*(128*c^4*x^4 + 256*b*c^3*x^3 + 3*b^4 + 24*a*b^2*c - 16*a^2*c^2 + 16*(11*b^2*c^2 + 4*a*c^3)*x^2 + 16*(3*b^
3*c + 4*a*b*c^2)*x)*sqrt(c*x^2 + b*x + a)/(8*(b^6*c^4 - 12*a*b^4*c^5 + 48*a^2*b^2*c^6 - 64*a^3*c^7)*d^4*x^5 +
20*(b^7*c^3 - 12*a*b^5*c^4 + 48*a^2*b^3*c^5 - 64*a^3*b*c^6)*d^4*x^4 + 2*(9*b^8*c^2 - 104*a*b^6*c^3 + 384*a^2*b
^4*c^4 - 384*a^3*b^2*c^5 - 256*a^4*c^6)*d^4*x^3 + (7*b^9*c - 72*a*b^7*c^2 + 192*a^2*b^5*c^3 + 128*a^3*b^3*c^4
- 768*a^4*b*c^5)*d^4*x^2 + (b^10 - 6*a*b^8*c - 24*a^2*b^6*c^2 + 224*a^3*b^4*c^3 - 384*a^4*b^2*c^4)*d^4*x + (a*
b^9 - 12*a^2*b^7*c + 48*a^3*b^5*c^2 - 64*a^4*b^3*c^3)*d^4)
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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(1/(2*c*d*x+b*d)^4/(c*x^2+b*x+a)^(3/2),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%%%{%%%{256,[6]%%%},[8,2,4,0]%%%}+%%%{%%%{-128,[5]%%%},[8,1,4,2]%%%
}+%%%{%%%{16,[4]%%%},[8,0,4,4]%%%}+%%%{%%{[%%%{-1024,[5]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[7,2,4,1]%%%}+%%%{%%{[
%%%{512,[4]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[7,1,4,3]%%%}+%%%{%%{[%%%{-64,[3]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[7
,0,4,5]%%%}+%%%{%%%{-1024,[6]%%%},[6,3,4,0]%%%}+%%%{%%%{2560,[5]%%%},[6,2,4,2]%%%}+%%%{%%%{-1088,[4]%%%},[6,1,
4,4]%%%}+%%%{%%%{128,[3]%%%},[6,0,4,6]%%%}+%%%{%%{[%%%{3072,[5]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[5,3,4,1]%%%}+%
%%{%%{[%%%{-4096,[4]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[5,2,4,3]%%%}+%%%{%%{[%%%{1472,[3]%%%},0]:[1,0,%%%{-1,[1]%
%%}]%%},[5,1,4,5]%%%}+%%%{%%{[%%%{-160,[2]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[5,0,4,7]%%%}+%%%{%%%{1536,[6]%%%},[
4,4,4,0]%%%}+%%%{%%%{-5376,[5]%%%},[4,3,4,2]%%%}+%%%{%%%{4576,[4]%%%},[4,2,4,4]%%%}+%%%{%%%{-1376,[3]%%%},[4,1
,4,6]%%%}+%%%{%%%{136,[2]%%%},[4,0,4,8]%%%}+%%%{%%{[%%%{-3072,[5]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3,4,4,1]%%%}
+%%%{%%{[%%%{5632,[4]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3,3,4,3]%%%}+%%%{%%{[%%%{-3520,[3]%%%},0]:[1,0,%%%{-1,[1
]%%%}]%%},[3,2,4,5]%%%}+%%%{%%{[%%%{896,[2]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3,1,4,7]%%%}+%%%{%%{[%%%{-80,[1]%%
%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3,0,4,9]%%%}+%%%{%%%{-1024,[6]%%%},[2,5,4,0]%%%}+%%%{%%%{3584,[5]%%%},[2,4,4,2]
%%%}+%%%{%%%{-3904,[4]%%%},[2,3,4,4]%%%}+%%%{%%%{1856,[3]%%%},[2,2,4,6]%%%}+%%%{%%%{-400,[2]%%%},[2,1,4,8]%%%}
+%%%{%%%{32,[1]%%%},[2,0,4,10]%%%}+%%%{%%{[%%%{1024,[5]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,5,4,1]%%%}+%%%{%%{[%
%%{-2048,[4]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,4,4,3]%%%}+%%%{%%{[%%%{1600,[3]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},
[1,3,4,5]%%%}+%%%{%%{[%%%{-608,[2]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,2,4,7]%%%}+%%%{%%{[%%%{112,[1]%%%},0]:[1,
0,%%%{-1,[1]%%%}]%%},[1,1,4,9]%%%}+%%%{%%{[-8,0]:[1,0,%%%{-1,[1]%%%}]%%},[1,0,4,11]%%%}+%%%{%%%{256,[6]%%%},[0
,6,4,0]%%%}+%%%{%%%{-640,[5]%%%},[0,5,4,2]%%%}+%%%{%%%{656,[4]%%%},[0,4,4,4]%%%}+%%%{%%%{-352,[3]%%%},[0,3,4,6
]%%%}+%%%{%%%{104,[2]%%%},[0,2,4,8]%%%}+%%%{%%%{-16,[1]%%%},[0,1,4,10]%%%}+%%%{1,[0,0,4,12]%%%} / %%%{%%%{16,[
6]%%%},[8,0,0,0]%%%}+%%%{%%{poly1[%%%{-64,[5]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[7,0,0,1]%%%}+%%%{%%%{-64,[6]%%%}
,[6,1,0,0]%%%}+%%%{%%%{128,[5]%%%},[6,0,0,2]%%%}+%%%{%%{[%%%{192,[5]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[5,1,0,1]%
%%}+%%%{%%{poly1[%%%{-160,[4]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[5,0,0,3]%%%}+%%%{%%%{96,[6]%%%},[4,2,0,0]%%%}+%%
%{%%%{-288,[5]%%%},[4,1,0,2]%%%}+%%%{%%%{136,[4]%%%},[4,0,0,4]%%%}+%%%{%%{poly1[%%%{-192,[5]%%%},0]:[1,0,%%%{-
1,[1]%%%}]%%},[3,2,0,1]%%%}+%%%{%%{[%%%{256,[4]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3,1,0,3]%%%}+%%%{%%{poly1[%%%{
-80,[3]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3,0,0,5]%%%}+%%%{%%%{-64,[6]%%%},[2,3,0,0]%%%}+%%%{%%%{192,[5]%%%},[2,
2,0,2]%%%}+%%%{%%%{-144,[4]%%%},[2,1,0,4]%%%}+%%%{%%%{32,[3]%%%},[2,0,0,6]%%%}+%%%{%%{[%%%{64,[5]%%%},0]:[1,0,
%%%{-1,[1]%%%}]%%},[1,3,0,1]%%%}+%%%{%%{poly1[%%%{-96,[4]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,2,0,3]%%%}+%%%{%%{
[%%%{48,[3]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,1,0,5]%%%}+%%%{%%{poly1[%%%{-8,[2]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%
},[1,0,0,7]%%%}+%%%{%%%{16,[6]%%%},[0,4,0,0]%%%}+%%%{%%%{-32,[5]%%%},[0,3,0,2]%%%}+%%%{%%%{24,[4]%%%},[0,2,0,4
]%%%}+%%%{%%%{-8,[3]%%%},[0,1,0,6]%%%}+%%%{%%%{1,[2]%%%},[0,0,0,8]%%%} Error: Bad Argument Value
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maple [A] time = 0.05, size = 133, normalized size = 1.13 \[ -\frac {2 \left (-128 c^{4} x^{4}-256 b \,c^{3} x^{3}-64 a \,c^{3} x^{2}-176 x^{2} b^{2} c^{2}-64 a b \,c^{2} x -48 x \,b^{3} c +16 a^{2} c^{2}-24 a \,b^{2} c -3 b^{4}\right )}{3 \left (2 c x +b \right )^{3} \left (64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right ) \sqrt {c \,x^{2}+b x +a}\, d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(2*c*d*x+b*d)^4/(c*x^2+b*x+a)^(3/2),x)
[Out]
-2/3*(-128*c^4*x^4-256*b*c^3*x^3-64*a*c^3*x^2-176*b^2*c^2*x^2-64*a*b*c^2*x-48*b^3*c*x+16*a^2*c^2-24*a*b^2*c-3*
b^4)/(2*c*x+b)^3/d^4/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)/(c*x^2+b*x+a)^(1/2)
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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(1/(2*c*d*x+b*d)^4/(c*x^2+b*x+a)^(3/2),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 positive, negative or zero?
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mupad [B] time = 2.23, size = 4588, normalized size = 38.88 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/((b*d + 2*c*d*x)^4*(a + b*x + c*x^2)^(3/2)),x)
[Out]
(8*b^6*c^2)/((a + b*x + c*x^2)^(1/2)*(4*b^11*c^2*d^4 - 80*a*b^9*c^3*d^4 - 4096*a^5*b*c^7*d^4 - 8192*a^5*c^8*d^
4*x + 8*b^10*c^3*d^4*x + 640*a^2*b^7*c^4*d^4 - 2560*a^3*b^5*c^5*d^4 + 5120*a^4*b^3*c^6*d^4 - 160*a*b^8*c^4*d^4
*x + 1280*a^2*b^6*c^5*d^4*x - 5120*a^3*b^4*c^6*d^4*x + 10240*a^4*b^2*c^7*d^4*x)) - (128*b^8*c^5)/((a + b*x + c
*x^2)^(1/2)*(32*b^13*c^5*d^4 - 768*a*b^11*c^6*d^4 + 131072*a^6*b*c^11*d^4 + 262144*a^6*c^12*d^4*x + 64*b^12*c^
6*d^4*x + 7680*a^2*b^9*c^7*d^4 - 40960*a^3*b^7*c^8*d^4 + 122880*a^4*b^5*c^9*d^4 - 196608*a^5*b^3*c^10*d^4 - 15
36*a*b^10*c^7*d^4*x + 15360*a^2*b^8*c^8*d^4*x - 81920*a^3*b^6*c^9*d^4*x + 245760*a^4*b^4*c^10*d^4*x - 393216*a
^5*b^2*c^11*d^4*x)) - (2560*a^3*c^5)/(3*(a + b*x + c*x^2)^(1/2)*(4*b^11*c^2*d^4 - 80*a*b^9*c^3*d^4 - 4096*a^5*
b*c^7*d^4 - 8192*a^5*c^8*d^4*x + 8*b^10*c^3*d^4*x + 640*a^2*b^7*c^4*d^4 - 2560*a^3*b^5*c^5*d^4 + 5120*a^4*b^3*
c^6*d^4 - 160*a*b^8*c^4*d^4*x + 1280*a^2*b^6*c^5*d^4*x - 5120*a^3*b^4*c^6*d^4*x + 10240*a^4*b^2*c^7*d^4*x)) -
(32*c^4*(a + b*x + c*x^2)^(1/2))/(12*b^7*c^3*d^4 - 96*a*b^5*c^4*d^4 + 72*b^6*c^4*d^4*x + 192*a^2*b^3*c^5*d^4 +
1536*a^2*c^8*d^4*x^3 + 144*b^5*c^5*d^4*x^2 + 96*b^4*c^6*d^4*x^3 - 576*a*b^4*c^5*d^4*x + 1152*a^2*b^2*c^6*d^4*
x - 1152*a*b^3*c^6*d^4*x^2 + 2304*a^2*b*c^7*d^4*x^2 - 768*a*b^2*c^7*d^4*x^3) - (352*a*b^4*c^3)/(3*(a + b*x + c
*x^2)^(1/2)*(4*b^11*c^2*d^4 - 80*a*b^9*c^3*d^4 - 4096*a^5*b*c^7*d^4 - 8192*a^5*c^8*d^4*x + 8*b^10*c^3*d^4*x +
640*a^2*b^7*c^4*d^4 - 2560*a^3*b^5*c^5*d^4 + 5120*a^4*b^3*c^6*d^4 - 160*a*b^8*c^4*d^4*x + 1280*a^2*b^6*c^5*d^4
*x - 5120*a^3*b^4*c^6*d^4*x + 10240*a^4*b^2*c^7*d^4*x)) - (64*b^5*c^3*x)/(3*(a + b*x + c*x^2)^(1/2)*(4*b^11*c^
2*d^4 - 80*a*b^9*c^3*d^4 - 4096*a^5*b*c^7*d^4 - 8192*a^5*c^8*d^4*x + 8*b^10*c^3*d^4*x + 640*a^2*b^7*c^4*d^4 -
2560*a^3*b^5*c^5*d^4 + 5120*a^4*b^3*c^6*d^4 - 160*a*b^8*c^4*d^4*x + 1280*a^2*b^6*c^5*d^4*x - 5120*a^3*b^4*c^6*
d^4*x + 10240*a^4*b^2*c^7*d^4*x)) + (1536*a*b^6*c^6)/((a + b*x + c*x^2)^(1/2)*(32*b^13*c^5*d^4 - 768*a*b^11*c^
6*d^4 + 131072*a^6*b*c^11*d^4 + 262144*a^6*c^12*d^4*x + 64*b^12*c^6*d^4*x + 7680*a^2*b^9*c^7*d^4 - 40960*a^3*b
^7*c^8*d^4 + 122880*a^4*b^5*c^9*d^4 - 196608*a^5*b^3*c^10*d^4 - 1536*a*b^10*c^7*d^4*x + 15360*a^2*b^8*c^8*d^4*
x - 81920*a^3*b^6*c^9*d^4*x + 245760*a^4*b^4*c^10*d^4*x - 393216*a^5*b^2*c^11*d^4*x)) - (512*b^7*c^6*x)/((a +
b*x + c*x^2)^(1/2)*(32*b^13*c^5*d^4 - 768*a*b^11*c^6*d^4 + 131072*a^6*b*c^11*d^4 + 262144*a^6*c^12*d^4*x + 64*
b^12*c^6*d^4*x + 7680*a^2*b^9*c^7*d^4 - 40960*a^3*b^7*c^8*d^4 + 122880*a^4*b^5*c^9*d^4 - 196608*a^5*b^3*c^10*d
^4 - 1536*a*b^10*c^7*d^4*x + 15360*a^2*b^8*c^8*d^4*x - 81920*a^3*b^6*c^9*d^4*x + 245760*a^4*b^4*c^10*d^4*x - 3
93216*a^5*b^2*c^11*d^4*x)) + (1664*a^2*b^2*c^4)/(3*(a + b*x + c*x^2)^(1/2)*(4*b^11*c^2*d^4 - 80*a*b^9*c^3*d^4
- 4096*a^5*b*c^7*d^4 - 8192*a^5*c^8*d^4*x + 8*b^10*c^3*d^4*x + 640*a^2*b^7*c^4*d^4 - 2560*a^3*b^5*c^5*d^4 + 51
20*a^4*b^3*c^6*d^4 - 160*a*b^8*c^4*d^4*x + 1280*a^2*b^6*c^5*d^4*x - 5120*a^3*b^4*c^6*d^4*x + 10240*a^4*b^2*c^7
*d^4*x)) - (1024*a^2*c^6*x^2)/(3*(a + b*x + c*x^2)^(1/2)*(4*b^11*c^2*d^4 - 80*a*b^9*c^3*d^4 - 4096*a^5*b*c^7*d
^4 - 8192*a^5*c^8*d^4*x + 8*b^10*c^3*d^4*x + 640*a^2*b^7*c^4*d^4 - 2560*a^3*b^5*c^5*d^4 + 5120*a^4*b^3*c^6*d^4
- 160*a*b^8*c^4*d^4*x + 1280*a^2*b^6*c^5*d^4*x - 5120*a^3*b^4*c^6*d^4*x + 10240*a^4*b^2*c^7*d^4*x)) - (64*b^4
*c^4*x^2)/(3*(a + b*x + c*x^2)^(1/2)*(4*b^11*c^2*d^4 - 80*a*b^9*c^3*d^4 - 4096*a^5*b*c^7*d^4 - 8192*a^5*c^8*d^
4*x + 8*b^10*c^3*d^4*x + 640*a^2*b^7*c^4*d^4 - 2560*a^3*b^5*c^5*d^4 + 5120*a^4*b^3*c^6*d^4 - 160*a*b^8*c^4*d^4
*x + 1280*a^2*b^6*c^5*d^4*x - 5120*a^3*b^4*c^6*d^4*x + 10240*a^4*b^2*c^7*d^4*x)) - (6144*a^2*b^4*c^7)/((a + b*
x + c*x^2)^(1/2)*(32*b^13*c^5*d^4 - 768*a*b^11*c^6*d^4 + 131072*a^6*b*c^11*d^4 + 262144*a^6*c^12*d^4*x + 64*b^
12*c^6*d^4*x + 7680*a^2*b^9*c^7*d^4 - 40960*a^3*b^7*c^8*d^4 + 122880*a^4*b^5*c^9*d^4 - 196608*a^5*b^3*c^10*d^4
- 1536*a*b^10*c^7*d^4*x + 15360*a^2*b^8*c^8*d^4*x - 81920*a^3*b^6*c^9*d^4*x + 245760*a^4*b^4*c^10*d^4*x - 393
216*a^5*b^2*c^11*d^4*x)) + (8192*a^3*b^2*c^8)/((a + b*x + c*x^2)^(1/2)*(32*b^13*c^5*d^4 - 768*a*b^11*c^6*d^4 +
131072*a^6*b*c^11*d^4 + 262144*a^6*c^12*d^4*x + 64*b^12*c^6*d^4*x + 7680*a^2*b^9*c^7*d^4 - 40960*a^3*b^7*c^8*
d^4 + 122880*a^4*b^5*c^9*d^4 - 196608*a^5*b^3*c^10*d^4 - 1536*a*b^10*c^7*d^4*x + 15360*a^2*b^8*c^8*d^4*x - 819
20*a^3*b^6*c^9*d^4*x + 245760*a^4*b^4*c^10*d^4*x - 393216*a^5*b^2*c^11*d^4*x)) + (32768*a^3*c^10*x^2)/((a + b*
x + c*x^2)^(1/2)*(32*b^13*c^5*d^4 - 768*a*b^11*c^6*d^4 + 131072*a^6*b*c^11*d^4 + 262144*a^6*c^12*d^4*x + 64*b^
12*c^6*d^4*x + 7680*a^2*b^9*c^7*d^4 - 40960*a^3*b^7*c^8*d^4 + 122880*a^4*b^5*c^9*d^4 - 196608*a^5*b^3*c^10*d^4
- 1536*a*b^10*c^7*d^4*x + 15360*a^2*b^8*c^8*d^4*x - 81920*a^3*b^6*c^9*d^4*x + 245760*a^4*b^4*c^10*d^4*x - 393
216*a^5*b^2*c^11*d^4*x)) - (512*b^6*c^7*x^2)/((a + b*x + c*x^2)^(1/2)*(32*b^13*c^5*d^4 - 768*a*b^11*c^6*d^4 +
131072*a^6*b*c^11*d^4 + 262144*a^6*c^12*d^4*x + 64*b^12*c^6*d^4*x + 7680*a^2*b^9*c^7*d^4 - 40960*a^3*b^7*c^8*d
^4 + 122880*a^4*b^5*c^9*d^4 - 196608*a^5*b^3*c^10*d^4 - 1536*a*b^10*c^7*d^4*x + 15360*a^2*b^8*c^8*d^4*x - 8192
0*a^3*b^6*c^9*d^4*x + 245760*a^4*b^4*c^10*d^4*x - 393216*a^5*b^2*c^11*d^4*x)) + (512*a*b^2*c^5*x^2)/(3*(a + b*
x + c*x^2)^(1/2)*(4*b^11*c^2*d^4 - 80*a*b^9*c^3*d^4 - 4096*a^5*b*c^7*d^4 - 8192*a^5*c^8*d^4*x + 8*b^10*c^3*d^4
*x + 640*a^2*b^7*c^4*d^4 - 2560*a^3*b^5*c^5*d^4 + 5120*a^4*b^3*c^6*d^4 - 160*a*b^8*c^4*d^4*x + 1280*a^2*b^6*c^
5*d^4*x - 5120*a^3*b^4*c^6*d^4*x + 10240*a^4*b^2*c^7*d^4*x)) - (24576*a^2*b^3*c^8*x)/((a + b*x + c*x^2)^(1/2)*
(32*b^13*c^5*d^4 - 768*a*b^11*c^6*d^4 + 131072*a^6*b*c^11*d^4 + 262144*a^6*c^12*d^4*x + 64*b^12*c^6*d^4*x + 76
80*a^2*b^9*c^7*d^4 - 40960*a^3*b^7*c^8*d^4 + 122880*a^4*b^5*c^9*d^4 - 196608*a^5*b^3*c^10*d^4 - 1536*a*b^10*c^
7*d^4*x + 15360*a^2*b^8*c^8*d^4*x - 81920*a^3*b^6*c^9*d^4*x + 245760*a^4*b^4*c^10*d^4*x - 393216*a^5*b^2*c^11*
d^4*x)) + (6144*a*b^4*c^8*x^2)/((a + b*x + c*x^2)^(1/2)*(32*b^13*c^5*d^4 - 768*a*b^11*c^6*d^4 + 131072*a^6*b*c
^11*d^4 + 262144*a^6*c^12*d^4*x + 64*b^12*c^6*d^4*x + 7680*a^2*b^9*c^7*d^4 - 40960*a^3*b^7*c^8*d^4 + 122880*a^
4*b^5*c^9*d^4 - 196608*a^5*b^3*c^10*d^4 - 1536*a*b^10*c^7*d^4*x + 15360*a^2*b^8*c^8*d^4*x - 81920*a^3*b^6*c^9*
d^4*x + 245760*a^4*b^4*c^10*d^4*x - 393216*a^5*b^2*c^11*d^4*x)) + (512*a*b^3*c^4*x)/(3*(a + b*x + c*x^2)^(1/2)
*(4*b^11*c^2*d^4 - 80*a*b^9*c^3*d^4 - 4096*a^5*b*c^7*d^4 - 8192*a^5*c^8*d^4*x + 8*b^10*c^3*d^4*x + 640*a^2*b^7
*c^4*d^4 - 2560*a^3*b^5*c^5*d^4 + 5120*a^4*b^3*c^6*d^4 - 160*a*b^8*c^4*d^4*x + 1280*a^2*b^6*c^5*d^4*x - 5120*a
^3*b^4*c^6*d^4*x + 10240*a^4*b^2*c^7*d^4*x)) - (1024*a^2*b*c^5*x)/(3*(a + b*x + c*x^2)^(1/2)*(4*b^11*c^2*d^4 -
80*a*b^9*c^3*d^4 - 4096*a^5*b*c^7*d^4 - 8192*a^5*c^8*d^4*x + 8*b^10*c^3*d^4*x + 640*a^2*b^7*c^4*d^4 - 2560*a^
3*b^5*c^5*d^4 + 5120*a^4*b^3*c^6*d^4 - 160*a*b^8*c^4*d^4*x + 1280*a^2*b^6*c^5*d^4*x - 5120*a^3*b^4*c^6*d^4*x +
10240*a^4*b^2*c^7*d^4*x)) - (24576*a^2*b^2*c^9*x^2)/((a + b*x + c*x^2)^(1/2)*(32*b^13*c^5*d^4 - 768*a*b^11*c^
6*d^4 + 131072*a^6*b*c^11*d^4 + 262144*a^6*c^12*d^4*x + 64*b^12*c^6*d^4*x + 7680*a^2*b^9*c^7*d^4 - 40960*a^3*b
^7*c^8*d^4 + 122880*a^4*b^5*c^9*d^4 - 196608*a^5*b^3*c^10*d^4 - 1536*a*b^10*c^7*d^4*x + 15360*a^2*b^8*c^8*d^4*
x - 81920*a^3*b^6*c^9*d^4*x + 245760*a^4*b^4*c^10*d^4*x - 393216*a^5*b^2*c^11*d^4*x)) + (6144*a*b^5*c^7*x)/((a
+ b*x + c*x^2)^(1/2)*(32*b^13*c^5*d^4 - 768*a*b^11*c^6*d^4 + 131072*a^6*b*c^11*d^4 + 262144*a^6*c^12*d^4*x +
64*b^12*c^6*d^4*x + 7680*a^2*b^9*c^7*d^4 - 40960*a^3*b^7*c^8*d^4 + 122880*a^4*b^5*c^9*d^4 - 196608*a^5*b^3*c^1
0*d^4 - 1536*a*b^10*c^7*d^4*x + 15360*a^2*b^8*c^8*d^4*x - 81920*a^3*b^6*c^9*d^4*x + 245760*a^4*b^4*c^10*d^4*x
- 393216*a^5*b^2*c^11*d^4*x)) + (32768*a^3*b*c^9*x)/((a + b*x + c*x^2)^(1/2)*(32*b^13*c^5*d^4 - 768*a*b^11*c^6
*d^4 + 131072*a^6*b*c^11*d^4 + 262144*a^6*c^12*d^4*x + 64*b^12*c^6*d^4*x + 7680*a^2*b^9*c^7*d^4 - 40960*a^3*b^
7*c^8*d^4 + 122880*a^4*b^5*c^9*d^4 - 196608*a^5*b^3*c^10*d^4 - 1536*a*b^10*c^7*d^4*x + 15360*a^2*b^8*c^8*d^4*x
- 81920*a^3*b^6*c^9*d^4*x + 245760*a^4*b^4*c^10*d^4*x - 393216*a^5*b^2*c^11*d^4*x))
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {1}{a b^{4} \sqrt {a + b x + c x^{2}} + 8 a b^{3} c x \sqrt {a + b x + c x^{2}} + 24 a b^{2} c^{2} x^{2} \sqrt {a + b x + c x^{2}} + 32 a b c^{3} x^{3} \sqrt {a + b x + c x^{2}} + 16 a c^{4} x^{4} \sqrt {a + b x + c x^{2}} + b^{5} x \sqrt {a + b x + c x^{2}} + 9 b^{4} c x^{2} \sqrt {a + b x + c x^{2}} + 32 b^{3} c^{2} x^{3} \sqrt {a + b x + c x^{2}} + 56 b^{2} c^{3} x^{4} \sqrt {a + b x + c x^{2}} + 48 b c^{4} x^{5} \sqrt {a + b x + c x^{2}} + 16 c^{5} x^{6} \sqrt {a + b x + c x^{2}}}\, dx}{d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(2*c*d*x+b*d)**4/(c*x**2+b*x+a)**(3/2),x)
[Out]
Integral(1/(a*b**4*sqrt(a + b*x + c*x**2) + 8*a*b**3*c*x*sqrt(a + b*x + c*x**2) + 24*a*b**2*c**2*x**2*sqrt(a +
b*x + c*x**2) + 32*a*b*c**3*x**3*sqrt(a + b*x + c*x**2) + 16*a*c**4*x**4*sqrt(a + b*x + c*x**2) + b**5*x*sqrt
(a + b*x + c*x**2) + 9*b**4*c*x**2*sqrt(a + b*x + c*x**2) + 32*b**3*c**2*x**3*sqrt(a + b*x + c*x**2) + 56*b**2
*c**3*x**4*sqrt(a + b*x + c*x**2) + 48*b*c**4*x**5*sqrt(a + b*x + c*x**2) + 16*c**5*x**6*sqrt(a + b*x + c*x**2
)), x)/d**4
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