3.11.43 \(\int \frac {(a+b x+c x^2)^{5/2}}{(b d+2 c d x)^{10}} \, dx\)
Optimal. Leaf size=79 \[ \frac {4 \left (a+b x+c x^2\right )^{7/2}}{63 d^{10} \left (b^2-4 a c\right )^2 (b+2 c x)^7}+\frac {2 \left (a+b x+c x^2\right )^{7/2}}{9 d^{10} \left (b^2-4 a c\right ) (b+2 c x)^9} \]
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Rubi [A] time = 0.03, antiderivative size = 79, normalized size of antiderivative = 1.00,
number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used =
{693, 682} \begin {gather*} \frac {4 \left (a+b x+c x^2\right )^{7/2}}{63 d^{10} \left (b^2-4 a c\right )^2 (b+2 c x)^7}+\frac {2 \left (a+b x+c x^2\right )^{7/2}}{9 d^{10} \left (b^2-4 a c\right ) (b+2 c x)^9} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^10,x]
[Out]
(2*(a + b*x + c*x^2)^(7/2))/(9*(b^2 - 4*a*c)*d^10*(b + 2*c*x)^9) + (4*(a + b*x + c*x^2)^(7/2))/(63*(b^2 - 4*a*
c)^2*d^10*(b + 2*c*x)^7)
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 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 {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^{10}} \, dx &=\frac {2 \left (a+b x+c x^2\right )^{7/2}}{9 \left (b^2-4 a c\right ) d^{10} (b+2 c x)^9}+\frac {2 \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^8} \, dx}{9 \left (b^2-4 a c\right ) d^2}\\ &=\frac {2 \left (a+b x+c x^2\right )^{7/2}}{9 \left (b^2-4 a c\right ) d^{10} (b+2 c x)^9}+\frac {4 \left (a+b x+c x^2\right )^{7/2}}{63 \left (b^2-4 a c\right )^2 d^{10} (b+2 c x)^7}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 62, normalized size = 0.78 \begin {gather*} \frac {2 (a+x (b+c x))^{7/2} \left (4 c \left (2 c x^2-7 a\right )+9 b^2+8 b c x\right )}{63 d^{10} \left (b^2-4 a c\right )^2 (b+2 c x)^9} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^10,x]
[Out]
(2*(a + x*(b + c*x))^(7/2)*(9*b^2 + 8*b*c*x + 4*c*(-7*a + 2*c*x^2)))/(63*(b^2 - 4*a*c)^2*d^10*(b + 2*c*x)^9)
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IntegrateAlgebraic [F] time = 180.00, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}
Verification is not applicable to the result.
[In]
IntegrateAlgebraic[(a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^10,x]
[Out]
$Aborted
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fricas [B] time = 28.17, size = 530, normalized size = 6.71 \begin {gather*} \frac {2 \, {\left (8 \, c^{5} x^{8} + 32 \, b c^{4} x^{7} + {\left (57 \, b^{2} c^{3} - 4 \, a c^{4}\right )} x^{6} + {\left (59 \, b^{3} c^{2} - 12 \, a b c^{3}\right )} x^{5} + 9 \, a^{3} b^{2} - 28 \, a^{4} c + 5 \, {\left (7 \, b^{4} c + 3 \, a b^{2} c^{2} - 12 \, a^{2} c^{3}\right )} x^{4} + {\left (9 \, b^{5} + 50 \, a b^{3} c - 120 \, a^{2} b c^{2}\right )} x^{3} + {\left (27 \, a b^{4} - 33 \, a^{2} b^{2} c - 76 \, a^{3} c^{2}\right )} x^{2} + {\left (27 \, a^{2} b^{3} - 76 \, a^{3} b c\right )} x\right )} \sqrt {c x^{2} + b x + a}}{63 \, {\left (512 \, {\left (b^{4} c^{9} - 8 \, a b^{2} c^{10} + 16 \, a^{2} c^{11}\right )} d^{10} x^{9} + 2304 \, {\left (b^{5} c^{8} - 8 \, a b^{3} c^{9} + 16 \, a^{2} b c^{10}\right )} d^{10} x^{8} + 4608 \, {\left (b^{6} c^{7} - 8 \, a b^{4} c^{8} + 16 \, a^{2} b^{2} c^{9}\right )} d^{10} x^{7} + 5376 \, {\left (b^{7} c^{6} - 8 \, a b^{5} c^{7} + 16 \, a^{2} b^{3} c^{8}\right )} d^{10} x^{6} + 4032 \, {\left (b^{8} c^{5} - 8 \, a b^{6} c^{6} + 16 \, a^{2} b^{4} c^{7}\right )} d^{10} x^{5} + 2016 \, {\left (b^{9} c^{4} - 8 \, a b^{7} c^{5} + 16 \, a^{2} b^{5} c^{6}\right )} d^{10} x^{4} + 672 \, {\left (b^{10} c^{3} - 8 \, a b^{8} c^{4} + 16 \, a^{2} b^{6} c^{5}\right )} d^{10} x^{3} + 144 \, {\left (b^{11} c^{2} - 8 \, a b^{9} c^{3} + 16 \, a^{2} b^{7} c^{4}\right )} d^{10} x^{2} + 18 \, {\left (b^{12} c - 8 \, a b^{10} c^{2} + 16 \, a^{2} b^{8} c^{3}\right )} d^{10} x + {\left (b^{13} - 8 \, a b^{11} c + 16 \, a^{2} b^{9} c^{2}\right )} d^{10}\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)^10,x, algorithm="fricas")
[Out]
2/63*(8*c^5*x^8 + 32*b*c^4*x^7 + (57*b^2*c^3 - 4*a*c^4)*x^6 + (59*b^3*c^2 - 12*a*b*c^3)*x^5 + 9*a^3*b^2 - 28*a
^4*c + 5*(7*b^4*c + 3*a*b^2*c^2 - 12*a^2*c^3)*x^4 + (9*b^5 + 50*a*b^3*c - 120*a^2*b*c^2)*x^3 + (27*a*b^4 - 33*
a^2*b^2*c - 76*a^3*c^2)*x^2 + (27*a^2*b^3 - 76*a^3*b*c)*x)*sqrt(c*x^2 + b*x + a)/(512*(b^4*c^9 - 8*a*b^2*c^10
+ 16*a^2*c^11)*d^10*x^9 + 2304*(b^5*c^8 - 8*a*b^3*c^9 + 16*a^2*b*c^10)*d^10*x^8 + 4608*(b^6*c^7 - 8*a*b^4*c^8
+ 16*a^2*b^2*c^9)*d^10*x^7 + 5376*(b^7*c^6 - 8*a*b^5*c^7 + 16*a^2*b^3*c^8)*d^10*x^6 + 4032*(b^8*c^5 - 8*a*b^6*
c^6 + 16*a^2*b^4*c^7)*d^10*x^5 + 2016*(b^9*c^4 - 8*a*b^7*c^5 + 16*a^2*b^5*c^6)*d^10*x^4 + 672*(b^10*c^3 - 8*a*
b^8*c^4 + 16*a^2*b^6*c^5)*d^10*x^3 + 144*(b^11*c^2 - 8*a*b^9*c^3 + 16*a^2*b^7*c^4)*d^10*x^2 + 18*(b^12*c - 8*a
*b^10*c^2 + 16*a^2*b^8*c^3)*d^10*x + (b^13 - 8*a*b^11*c + 16*a^2*b^9*c^2)*d^10)
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giac [B] time = 2.79, size = 1849, normalized size = 23.41
result too large to display
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)^10,x, algorithm="giac")
[Out]
1/2016*(4032*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^14*c^(15/2) + 28224*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^13*b*
c^7 + 90048*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^12*b^2*c^(13/2) + 6720*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^12*
a*c^(15/2) + 173376*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^11*b^3*c^6 + 40320*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))
^11*a*b*c^7 + 225792*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^10*b^4*c^(11/2) + 100800*(sqrt(c)*x - sqrt(c*x^2 + b*
x + a))^10*a*b^2*c^(13/2) + 20160*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^10*a^2*c^(15/2) + 212352*(sqrt(c)*x - sq
rt(c*x^2 + b*x + a))^9*b^5*c^5 + 134400*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*a*b^3*c^6 + 100800*(sqrt(c)*x -
sqrt(c*x^2 + b*x + a))^9*a^2*b*c^7 + 151200*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*b^6*c^(9/2) + 96768*(sqrt(c)
*x - sqrt(c*x^2 + b*x + a))^8*a*b^4*c^(11/2) + 217728*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*a^2*b^2*c^(13/2) +
12096*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*a^3*c^(15/2) + 84672*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*b^7*c^
4 + 24192*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*a*b^5*c^5 + 266112*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*a^2*b
^3*c^6 + 48384*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*a^3*b*c^7 + 38304*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*b
^8*c^(7/2) - 20160*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*a*b^6*c^(9/2) + 205632*(sqrt(c)*x - sqrt(c*x^2 + b*x
+ a))^6*a^2*b^4*c^(11/2) + 72576*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*a^3*b^2*c^(13/2) + 12096*(sqrt(c)*x - s
qrt(c*x^2 + b*x + a))^6*a^4*c^(15/2) + 14112*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*b^9*c^3 - 24192*(sqrt(c)*x
- sqrt(c*x^2 + b*x + a))^5*a*b^7*c^4 + 108864*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a^2*b^5*c^5 + 48384*(sqrt(
c)*x - sqrt(c*x^2 + b*x + a))^5*a^3*b^3*c^6 + 36288*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a^4*b*c^7 + 4176*(sq
rt(c)*x - sqrt(c*x^2 + b*x + a))^4*b^10*c^(5/2) - 12960*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a*b^8*c^(7/2) +
43200*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a^2*b^6*c^(9/2) + 8640*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a^3*b
^4*c^(11/2) + 43200*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a^4*b^2*c^(13/2) + 1728*(sqrt(c)*x - sqrt(c*x^2 + b*
x + a))^4*a^5*c^(15/2) + 960*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*b^11*c^2 - 4416*(sqrt(c)*x - sqrt(c*x^2 + b
*x + a))^3*a*b^9*c^3 + 13824*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^2*b^7*c^4 - 6912*(sqrt(c)*x - sqrt(c*x^2
+ b*x + a))^3*a^3*b^5*c^5 + 25920*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^4*b^3*c^6 + 3456*(sqrt(c)*x - sqrt(c
*x^2 + b*x + a))^3*a^5*b*c^7 + 162*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b^12*c^(3/2) - 1008*(sqrt(c)*x - sqrt
(c*x^2 + b*x + a))^2*a*b^10*c^(5/2) + 3456*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^2*b^8*c^(7/2) - 4608*(sqrt(
c)*x - sqrt(c*x^2 + b*x + a))^2*a^3*b^6*c^(9/2) + 8640*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^4*b^4*c^(11/2)
+ 1728*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^5*b^2*c^(13/2) + 576*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^6*
c^(15/2) + 18*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b^13*c - 144*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a*b^11*c^2
+ 576*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^2*b^9*c^3 - 1152*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^3*b^7*c^4 +
1728*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^4*b^5*c^5 + 576*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^6*b*c^7 + b^
14*sqrt(c) - 10*a*b^12*c^(3/2) + 48*a^2*b^10*c^(5/2) - 128*a^3*b^8*c^(7/2) + 224*a^4*b^6*c^(9/2) - 192*a^5*b^4
*c^(11/2) + 256*a^6*b^2*c^(13/2) - 64*a^7*c^(15/2))/((2*(sqrt(c)*x - sqrt(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)^9*c^4*d^10)
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maple [A] time = 0.05, size = 70, normalized size = 0.89 \begin {gather*} -\frac {2 \left (-8 c^{2} x^{2}-8 b c x +28 a c -9 b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {7}{2}}}{63 \left (2 c x +b \right )^{9} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) d^{10}} \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)^10,x)
[Out]
-2/63*(-8*c^2*x^2-8*b*c*x+28*a*c-9*b^2)*(c*x^2+b*x+a)^(7/2)/(2*c*x+b)^9/d^10/(16*a^2*c^2-8*a*b^2*c+b^4)
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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)^10,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 [B] time = 7.03, size = 5511, normalized size = 69.76
result too large to display
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)^10,x)
[Out]
(((b*((b*((b*((b*((b*((4*c^4*(16*a*c + 3*b^2))/(3*d^10*(4*a*c - b^2)*(128*a*c^3 - 32*b^2*c^2)) - (8*b^2*c^4)/(
3*d^10*(4*a*c - b^2)*(128*a*c^3 - 32*b^2*c^2))))/(2*c) - (10*b*c^3*(48*a*c - 5*b^2))/(9*d^10*(4*a*c - b^2)*(12
8*a*c^3 - 32*b^2*c^2))))/(2*c) + (960*a^2*c^4 - 110*b^4*c^2 + 480*a*b^2*c^3)/(18*d^10*(4*a*c - b^2)*(128*a*c^3
- 32*b^2*c^2))))/(2*c) + (b*c*(3*b^4 - 480*a^2*c^2 + 80*a*b^2*c))/(6*d^10*(4*a*c - b^2)*(128*a*c^3 - 32*b^2*c
^2))))/(2*c) + (9*b^6 + 608*a^3*c^3 + 264*a^2*b^2*c^2 - 126*a*b^4*c)/(18*d^10*(4*a*c - b^2)*(128*a*c^3 - 32*b^
2*c^2))))/(2*c) - (9*a*b^5 - 108*a^2*b^3*c + 304*a^3*b*c^2)/(18*d^10*(4*a*c - b^2)*(128*a*c^3 - 32*b^2*c^2)))*
(a + b*x + c*x^2)^(1/2))/(b + 2*c*x)^8 + (((b*((b*((b*((b*((b*((8*c^4*(18*a*c - b^2))/(315*d^10*(4*a*c - b^2)^
4*(32*a*c^3 - 8*b^2*c^2)) - (8*b^2*c^4)/(315*d^10*(4*a*c - b^2)^4*(32*a*c^3 - 8*b^2*c^2))))/(2*c) - (8*b*c^3*(
27*a*c - 5*b^2))/(189*d^10*(4*a*c - b^2)^4*(32*a*c^3 - 8*b^2*c^2))))/(2*c) + (4496*a^2*c^4 + 46*b^4*c^2 - 1168
*a*b^2*c^3)/(945*d^10*(4*a*c - b^2)^4*(32*a*c^3 - 8*b^2*c^2))))/(2*c) - (b*c*(99*b^4 + 2248*a^2*c^2 - 944*a*b^
2*c))/(315*d^10*(4*a*c - b^2)^4*(32*a*c^3 - 8*b^2*c^2))))/(2*c) + (115*b^6 + 4080*a^3*c^3 + 312*a^2*b^2*c^2 -
786*a*b^4*c)/(945*d^10*(4*a*c - b^2)^4*(32*a*c^3 - 8*b^2*c^2))))/(2*c) - (115*a*b^5 - 968*a^2*b^3*c + 2040*a^3
*b*c^2)/(945*d^10*(4*a*c - b^2)^4*(32*a*c^3 - 8*b^2*c^2)))*(a + b*x + c*x^2)^(1/2))/(b + 2*c*x)^2 + (((b*((b*(
(b*((b*((b*((8*c^4*(38*a*c + b^2))/(63*d^10*(4*a*c - b^2)^2*(96*a*c^3 - 24*b^2*c^2)) - (8*b^2*c^4)/(21*d^10*(4
*a*c - b^2)^2*(96*a*c^3 - 24*b^2*c^2))))/(2*c) - (40*b*c^3*(19*a*c - 3*b^2))/(63*d^10*(4*a*c - b^2)^2*(96*a*c^
3 - 24*b^2*c^2))))/(2*c) - (82*b^4*c^2 - 3648*a^2*c^4 + 304*a*b^2*c^3)/(126*d^10*(4*a*c - b^2)^2*(96*a*c^3 - 2
4*b^2*c^2))))/(2*c) - (173*b^5*c - 1976*a*b^3*c^2 + 5472*a^2*b*c^3)/(126*d^10*(4*a*c - b^2)^2*(96*a*c^3 - 24*b
^2*c^2))))/(2*c) + (79*b^6 + 3072*a^3*c^3 + 432*a^2*b^2*c^2 - 602*a*b^4*c)/(126*d^10*(4*a*c - b^2)^2*(96*a*c^3
- 24*b^2*c^2))))/(2*c) - (79*a*b^5 - 696*a^2*b^3*c + 1536*a^3*b*c^2)/(126*d^10*(4*a*c - b^2)^2*(96*a*c^3 - 24
*b^2*c^2)))*(a + b*x + c*x^2)^(1/2))/(b + 2*c*x)^6 + (((b*((b*((8*a*c - b^2)/(240*c*d^10*(4*a*c - b^2)^4) - b^
2/(480*c*d^10*(4*a*c - b^2)^4)))/(2*c) - (b*(24*a*c - 5*b^2))/(720*c^2*d^10*(4*a*c - b^2)^4)))/(2*c) + (2*b^4
+ 54*a^2*c^2 - 21*a*b^2*c)/(720*c^3*d^10*(4*a*c - b^2)^4))*(a + b*x + c*x^2)^(1/2))/(b + 2*c*x) - (((b*((b*((1
6*a*c - b^2)/(720*c*d^10*(4*a*c - b^2)^4) - b^2/(480*c*d^10*(4*a*c - b^2)^4)))/(2*c) - (b*(16*a*c - 3*b^2))/(7
20*c^2*d^10*(4*a*c - b^2)^4)))/(2*c) + (12*a^2*c^2 - b^4 + 2*a*b^2*c)/(1440*c^3*d^10*(4*a*c - b^2)^4))*(a + b*
x + c*x^2)^(1/2))/(b + 2*c*x) - (((b*((b*((b*((b*((2*(96*a*c^4 - 9*b^2*c^3))/(315*d^10*(4*a*c - b^2)^3*(48*a*c
^3 - 12*b^2*c^2)) - (2*b^2*c^3)/(63*d^10*(4*a*c - b^2)^3*(48*a*c^3 - 12*b^2*c^2))))/(2*c) - (4*b*c^2*(96*a*c -
19*b^2))/(315*d^10*(4*a*c - b^2)^3*(48*a*c^3 - 12*b^2*c^2))))/(2*c) + (2*(20*b^4*c + 836*a^2*c^3 - 274*a*b^2*
c^2))/(315*d^10*(4*a*c - b^2)^3*(48*a*c^3 - 12*b^2*c^2))))/(2*c) - (2*(41*b^5 + 836*a^2*b*c^2 - 370*a*b^3*c))/
(315*d^10*(4*a*c - b^2)^3*(48*a*c^3 - 12*b^2*c^2))))/(2*c) + (2*(41*a*b^4 + 744*a^3*c^2 - 349*a^2*b^2*c))/(315
*d^10*(4*a*c - b^2)^3*(48*a*c^3 - 12*b^2*c^2)))*(a + b*x + c*x^2)^(1/2))/(b + 2*c*x)^3 + (((b*((b*((b*((b*((c^
3*(16*a*c + b^2))/(30*d^10*(4*a*c - b^2)^3*(48*a*c^3 - 12*b^2*c^2)) - (b^2*c^3)/(18*d^10*(4*a*c - b^2)^3*(48*a
*c^3 - 12*b^2*c^2))))/(2*c) - (b*c^2*(48*a*c - 7*b^2))/(45*d^10*(4*a*c - b^2)^3*(48*a*c^3 - 12*b^2*c^2))))/(2*
c) + (560*a^2*c^4 - 22*b^4*c^2 + 8*a*b^2*c^3)/(360*c*d^10*(4*a*c - b^2)^3*(48*a*c^3 - 12*b^2*c^2))))/(2*c) - (
b*(7*b^4 + 280*a^2*c^2 - 92*a*b^2*c))/(180*d^10*(4*a*c - b^2)^3*(48*a*c^3 - 12*b^2*c^2))))/(2*c) + (3*b^6 + 19
2*a^3*c^3 - 4*a^2*b^2*c^2 - 22*a*b^4*c)/(360*c*d^10*(4*a*c - b^2)^3*(48*a*c^3 - 12*b^2*c^2)))*(a + b*x + c*x^2
)^(1/2))/(b + 2*c*x)^3 + (((b*((b*((c*(8*a*c - b^2))/(21*d^10*(4*a*c - b^2)^2*(48*a*c^3 - 12*b^2*c^2)) - (b^2*
c)/(42*d^10*(4*a*c - b^2)^2*(48*a*c^3 - 12*b^2*c^2))))/(2*c) - (b*(24*a*c - 5*b^2))/(63*d^10*(4*a*c - b^2)^2*(
48*a*c^3 - 12*b^2*c^2))))/(2*c) + (19*b^4 + 480*a^2*c^2 - 192*a*b^2*c)/(504*c*d^10*(4*a*c - b^2)^2*(48*a*c^3 -
12*b^2*c^2)))*(a + b*x + c*x^2)^(1/2))/(b + 2*c*x)^3 - (((6*a*c - b^2)/(720*c^3*d^10*(4*a*c - b^2)^3) - b^2/(
1440*c^3*d^10*(4*a*c - b^2)^3))*(a + b*x + c*x^2)^(1/2))/(b + 2*c*x) - (((32*a*c - 9*b^2)/(2016*c^3*d^10*(4*a*
c - b^2)^3) + b^2/(2016*c^3*d^10*(4*a*c - b^2)^3))*(a + b*x + c*x^2)^(1/2))/(b + 2*c*x) + (((b*((b*((b*((b*((b
*((4*c^4*(32*a*c - b^2))/(105*d^10*(4*a*c - b^2)^3*(64*a*c^3 - 16*b^2*c^2)) - (8*b^2*c^4)/(105*d^10*(4*a*c - b
^2)^3*(64*a*c^3 - 16*b^2*c^2))))/(2*c) - (2*b*c^3*(96*a*c - 17*b^2))/(63*d^10*(4*a*c - b^2)^3*(64*a*c^3 - 16*b
^2*c^2))))/(2*c) + (4*c^2*(b^4 + 836*a^2*c^2 - 178*a*b^2*c))/(315*d^10*(4*a*c - b^2)^3*(64*a*c^3 - 16*b^2*c^2)
)))/(2*c) - (4*b*c*(17*b^4 + 418*a^2*c^2 - 169*a*b^2*c))/(105*d^10*(4*a*c - b^2)^3*(64*a*c^3 - 16*b^2*c^2))))/
(2*c) + (2*(41*b^6 + 1488*a^3*c^3 + 138*a^2*b^2*c^2 - 288*a*b^4*c))/(315*d^10*(4*a*c - b^2)^3*(64*a*c^3 - 16*b
^2*c^2))))/(2*c) - (2*(41*a*b^5 - 349*a^2*b^3*c + 744*a^3*b*c^2))/(315*d^10*(4*a*c - b^2)^3*(64*a*c^3 - 16*b^2
*c^2)))*(a + b*x + c*x^2)^(1/2))/(b + 2*c*x)^4 - (((b*((b*(b^2/(840*c*d^10*(4*a*c - b^2)^4) - (76*a*c^3 - 13*b
^2*c^2)/(2520*c^3*d^10*(4*a*c - b^2)^4)))/(2*c) + (b*(76*a*c - 17*b^2))/(2520*c^2*d^10*(4*a*c - b^2)^4)))/(2*c
) - (24*b^4 + 456*a^2*c^2 - 209*a*b^2*c)/(2520*c^3*d^10*(4*a*c - b^2)^4))*(a + b*x + c*x^2)^(1/2))/(b + 2*c*x)
- (((b*((b*((b*((2*c^2*(16*a*c + b^2))/(45*d^10*(4*a*c - b^2)^2*(64*a*c^3 - 16*b^2*c^2)) - (4*b^2*c^2)/(45*d^
10*(4*a*c - b^2)^2*(64*a*c^3 - 16*b^2*c^2))))/(2*c) - (b*c*(48*a*c - 7*b^2))/(45*d^10*(4*a*c - b^2)^2*(64*a*c^
3 - 16*b^2*c^2))))/(2*c) + (24*a^2*c^3 - 8*b^4*c + 36*a*b^2*c^2)/(90*c*d^10*(4*a*c - b^2)^2*(64*a*c^3 - 16*b^2
*c^2))))/(2*c) - (12*a^2*b*c^2 - b^5 + 2*a*b^3*c)/(90*c*d^10*(4*a*c - b^2)^2*(64*a*c^3 - 16*b^2*c^2)))*(a + b*
x + c*x^2)^(1/2))/(b + 2*c*x)^4 - (((b*((b*((b*((b*((2*c^3*(16*a*c + b^2))/(3*d^10*(4*a*c - b^2)*(112*a*c^3 -
28*b^2*c^2)) - (10*b^2*c^3)/(9*d^10*(4*a*c - b^2)*(112*a*c^3 - 28*b^2*c^2))))/(2*c) - (4*b*c^2*(48*a*c - 7*b^2
))/(9*d^10*(4*a*c - b^2)*(112*a*c^3 - 28*b^2*c^2))))/(2*c) + (480*a^2*c^3 - 27*b^4*c + 48*a*b^2*c^2)/(18*d^10*
(4*a*c - b^2)*(112*a*c^3 - 28*b^2*c^2))))/(2*c) - (9*b^5 + 480*a^2*b*c^2 - 144*a*b^3*c)/(18*d^10*(4*a*c - b^2)
*(112*a*c^3 - 28*b^2*c^2))))/(2*c) + (9*a*b^4 + 304*a^3*c^2 - 108*a^2*b^2*c)/(18*d^10*(4*a*c - b^2)*(112*a*c^3
- 28*b^2*c^2)))*(a + b*x + c*x^2)^(1/2))/(b + 2*c*x)^7 - (((b*((b*((b*((b*((24*c^3*(a*c + b^2))/(d^10*(144*a*
c^3 - 36*b^2*c^2)) - (10*b^2*c^3)/(d^10*(144*a*c^3 - 36*b^2*c^2))))/(2*c) - (4*c^2*(2*b^3 + 12*a*b*c))/(d^10*(
144*a*c^3 - 36*b^2*c^2))))/(2*c) + (24*a*c^2*(a*c + b^2))/(d^10*(144*a*c^3 - 36*b^2*c^2))))/(2*c) - (24*a^2*b*
c^2)/(d^10*(144*a*c^3 - 36*b^2*c^2))))/(2*c) + (8*a^3*c^2)/(d^10*(144*a*c^3 - 36*b^2*c^2)))*(a + b*x + c*x^2)^
(1/2))/(b + 2*c*x)^9 - (((b*((b*((b^2*c)/(6*d^10*(4*a*c - b^2)*(80*a*c^3 - 20*b^2*c^2)) - (c*(4*a*c + b^2))/(6
*d^10*(4*a*c - b^2)*(80*a*c^3 - 20*b^2*c^2))))/(2*c) + (b*(12*a*c - b^2))/(18*d^10*(4*a*c - b^2)*(80*a*c^3 - 2
0*b^2*c^2))))/(2*c) - (b^4 + 48*a^2*c^2 - 12*a*b^2*c)/(72*c*d^10*(4*a*c - b^2)*(80*a*c^3 - 20*b^2*c^2)))*(a +
b*x + c*x^2)^(1/2))/(b + 2*c*x)^5 + (((b*(b^2/(126*d^10*(4*a*c - b^2)^2*(32*a*c^3 - 8*b^2*c^2)) + (64*a*c^2 -
22*b^2*c)/(504*c*d^10*(4*a*c - b^2)^2*(32*a*c^3 - 8*b^2*c^2))))/(2*c) + (9*b^3 - 32*a*b*c)/(504*c*d^10*(4*a*c
- b^2)^2*(32*a*c^3 - 8*b^2*c^2)))*(a + b*x + c*x^2)^(1/2))/(b + 2*c*x)^2 - (((b*((b*((b*((b*((2*c*(9*a*c - b^2
))/(315*d^10*(4*a*c - b^2)^5) - (b^2*c)/(378*d^10*(4*a*c - b^2)^5)))/(2*c) - (2*b*(54*a*c - 11*b^2))/(945*d^10
*(4*a*c - b^2)^5)))/(2*c) + (67*b^4*c + 2248*a^2*c^3 - 800*a*b^2*c^2)/(3780*c^2*d^10*(4*a*c - b^2)^5)))/(2*c)
- (115*b^5 + 2248*a^2*b*c^2 - 1016*a*b^3*c)/(3780*c^2*d^10*(4*a*c - b^2)^5)))/(2*c) + (115*a*b^4 + 2040*a^3*c^
2 - 968*a^2*b^2*c)/(3780*c^2*d^10*(4*a*c - b^2)^5))*(a + b*x + c*x^2)^(1/2))/(b + 2*c*x) - (((b*((b*((b*((c^2*
(24*a*c - b^2))/(90*d^10*(4*a*c - b^2)^3*(32*a*c^3 - 8*b^2*c^2)) - (b^2*c^2)/(45*d^10*(4*a*c - b^2)^3*(32*a*c^
3 - 8*b^2*c^2))))/(2*c) - (b*c*(72*a*c - 13*b^2))/(180*d^10*(4*a*c - b^2)^3*(32*a*c^3 - 8*b^2*c^2))))/(2*c) -
(b^4*c - 108*a^2*c^3 + 18*a*b^2*c^2)/(180*c*d^10*(4*a*c - b^2)^3*(32*a*c^3 - 8*b^2*c^2))))/(2*c) - (2*b^5 + 54
*a^2*b*c^2 - 21*a*b^3*c)/(180*c*d^10*(4*a*c - b^2)^3*(32*a*c^3 - 8*b^2*c^2)))*(a + b*x + c*x^2)^(1/2))/(b + 2*
c*x)^2 - (((b*((b*((b*((b*((8*c^3*(19*a*c - b^2))/(63*d^10*(4*a*c - b^2)^2*(80*a*c^3 - 20*b^2*c^2)) - (10*b^2*
c^3)/(63*d^10*(4*a*c - b^2)^2*(80*a*c^3 - 20*b^2*c^2))))/(2*c) - (8*b*c^2*(38*a*c - 7*b^2))/(63*d^10*(4*a*c -
b^2)^2*(80*a*c^3 - 20*b^2*c^2))))/(2*c) + (15*b^4*c + 1824*a^2*c^3 - 456*a*b^2*c^2)/(126*d^10*(4*a*c - b^2)^2*
(80*a*c^3 - 20*b^2*c^2))))/(2*c) - (79*b^5 + 1824*a^2*b*c^2 - 760*a*b^3*c)/(126*d^10*(4*a*c - b^2)^2*(80*a*c^3
- 20*b^2*c^2))))/(2*c) + (79*a*b^4 + 1536*a^3*c^2 - 696*a^2*b^2*c)/(126*d^10*(4*a*c - b^2)^2*(80*a*c^3 - 20*b
^2*c^2)))*(a + b*x + c*x^2)^(1/2))/(b + 2*c*x)^5
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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^{10} + 20 b^{9} c x + 180 b^{8} c^{2} x^{2} + 960 b^{7} c^{3} x^{3} + 3360 b^{6} c^{4} x^{4} + 8064 b^{5} c^{5} x^{5} + 13440 b^{4} c^{6} x^{6} + 15360 b^{3} c^{7} x^{7} + 11520 b^{2} c^{8} x^{8} + 5120 b c^{9} x^{9} + 1024 c^{10} x^{10}}\, dx + \int \frac {b^{2} x^{2} \sqrt {a + b x + c x^{2}}}{b^{10} + 20 b^{9} c x + 180 b^{8} c^{2} x^{2} + 960 b^{7} c^{3} x^{3} + 3360 b^{6} c^{4} x^{4} + 8064 b^{5} c^{5} x^{5} + 13440 b^{4} c^{6} x^{6} + 15360 b^{3} c^{7} x^{7} + 11520 b^{2} c^{8} x^{8} + 5120 b c^{9} x^{9} + 1024 c^{10} x^{10}}\, dx + \int \frac {c^{2} x^{4} \sqrt {a + b x + c x^{2}}}{b^{10} + 20 b^{9} c x + 180 b^{8} c^{2} x^{2} + 960 b^{7} c^{3} x^{3} + 3360 b^{6} c^{4} x^{4} + 8064 b^{5} c^{5} x^{5} + 13440 b^{4} c^{6} x^{6} + 15360 b^{3} c^{7} x^{7} + 11520 b^{2} c^{8} x^{8} + 5120 b c^{9} x^{9} + 1024 c^{10} x^{10}}\, dx + \int \frac {2 a b x \sqrt {a + b x + c x^{2}}}{b^{10} + 20 b^{9} c x + 180 b^{8} c^{2} x^{2} + 960 b^{7} c^{3} x^{3} + 3360 b^{6} c^{4} x^{4} + 8064 b^{5} c^{5} x^{5} + 13440 b^{4} c^{6} x^{6} + 15360 b^{3} c^{7} x^{7} + 11520 b^{2} c^{8} x^{8} + 5120 b c^{9} x^{9} + 1024 c^{10} x^{10}}\, dx + \int \frac {2 a c x^{2} \sqrt {a + b x + c x^{2}}}{b^{10} + 20 b^{9} c x + 180 b^{8} c^{2} x^{2} + 960 b^{7} c^{3} x^{3} + 3360 b^{6} c^{4} x^{4} + 8064 b^{5} c^{5} x^{5} + 13440 b^{4} c^{6} x^{6} + 15360 b^{3} c^{7} x^{7} + 11520 b^{2} c^{8} x^{8} + 5120 b c^{9} x^{9} + 1024 c^{10} x^{10}}\, dx + \int \frac {2 b c x^{3} \sqrt {a + b x + c x^{2}}}{b^{10} + 20 b^{9} c x + 180 b^{8} c^{2} x^{2} + 960 b^{7} c^{3} x^{3} + 3360 b^{6} c^{4} x^{4} + 8064 b^{5} c^{5} x^{5} + 13440 b^{4} c^{6} x^{6} + 15360 b^{3} c^{7} x^{7} + 11520 b^{2} c^{8} x^{8} + 5120 b c^{9} x^{9} + 1024 c^{10} x^{10}}\, dx}{d^{10}} \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)**10,x)
[Out]
(Integral(a**2*sqrt(a + b*x + c*x**2)/(b**10 + 20*b**9*c*x + 180*b**8*c**2*x**2 + 960*b**7*c**3*x**3 + 3360*b*
*6*c**4*x**4 + 8064*b**5*c**5*x**5 + 13440*b**4*c**6*x**6 + 15360*b**3*c**7*x**7 + 11520*b**2*c**8*x**8 + 5120
*b*c**9*x**9 + 1024*c**10*x**10), x) + Integral(b**2*x**2*sqrt(a + b*x + c*x**2)/(b**10 + 20*b**9*c*x + 180*b*
*8*c**2*x**2 + 960*b**7*c**3*x**3 + 3360*b**6*c**4*x**4 + 8064*b**5*c**5*x**5 + 13440*b**4*c**6*x**6 + 15360*b
**3*c**7*x**7 + 11520*b**2*c**8*x**8 + 5120*b*c**9*x**9 + 1024*c**10*x**10), x) + Integral(c**2*x**4*sqrt(a +
b*x + c*x**2)/(b**10 + 20*b**9*c*x + 180*b**8*c**2*x**2 + 960*b**7*c**3*x**3 + 3360*b**6*c**4*x**4 + 8064*b**5
*c**5*x**5 + 13440*b**4*c**6*x**6 + 15360*b**3*c**7*x**7 + 11520*b**2*c**8*x**8 + 5120*b*c**9*x**9 + 1024*c**1
0*x**10), x) + Integral(2*a*b*x*sqrt(a + b*x + c*x**2)/(b**10 + 20*b**9*c*x + 180*b**8*c**2*x**2 + 960*b**7*c*
*3*x**3 + 3360*b**6*c**4*x**4 + 8064*b**5*c**5*x**5 + 13440*b**4*c**6*x**6 + 15360*b**3*c**7*x**7 + 11520*b**2
*c**8*x**8 + 5120*b*c**9*x**9 + 1024*c**10*x**10), x) + Integral(2*a*c*x**2*sqrt(a + b*x + c*x**2)/(b**10 + 20
*b**9*c*x + 180*b**8*c**2*x**2 + 960*b**7*c**3*x**3 + 3360*b**6*c**4*x**4 + 8064*b**5*c**5*x**5 + 13440*b**4*c
**6*x**6 + 15360*b**3*c**7*x**7 + 11520*b**2*c**8*x**8 + 5120*b*c**9*x**9 + 1024*c**10*x**10), x) + Integral(2
*b*c*x**3*sqrt(a + b*x + c*x**2)/(b**10 + 20*b**9*c*x + 180*b**8*c**2*x**2 + 960*b**7*c**3*x**3 + 3360*b**6*c*
*4*x**4 + 8064*b**5*c**5*x**5 + 13440*b**4*c**6*x**6 + 15360*b**3*c**7*x**7 + 11520*b**2*c**8*x**8 + 5120*b*c*
*9*x**9 + 1024*c**10*x**10), x))/d**10
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