3.13.33 \(\int \frac {(a+b x+c x^2)^{5/2}}{(b d+2 c d x)^{12}} \, dx\) [1233]
Optimal. Leaf size=118 \[ \frac {2 \left (a+b x+c x^2\right )^{7/2}}{11 \left (b^2-4 a c\right ) d^{12} (b+2 c x)^{11}}+\frac {8 \left (a+b x+c x^2\right )^{7/2}}{99 \left (b^2-4 a c\right )^2 d^{12} (b+2 c x)^9}+\frac {16 \left (a+b x+c x^2\right )^{7/2}}{693 \left (b^2-4 a c\right )^3 d^{12} (b+2 c x)^7} \]
[Out]
2/11*(c*x^2+b*x+a)^(7/2)/(-4*a*c+b^2)/d^12/(2*c*x+b)^11+8/99*(c*x^2+b*x+a)^(7/2)/(-4*a*c+b^2)^2/d^12/(2*c*x+b)
^9+16/693*(c*x^2+b*x+a)^(7/2)/(-4*a*c+b^2)^3/d^12/(2*c*x+b)^7
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Rubi [A]
time = 0.04, antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {707, 696}
\begin {gather*} \frac {16 \left (a+b x+c x^2\right )^{7/2}}{693 d^{12} \left (b^2-4 a c\right )^3 (b+2 c x)^7}+\frac {8 \left (a+b x+c x^2\right )^{7/2}}{99 d^{12} \left (b^2-4 a c\right )^2 (b+2 c x)^9}+\frac {2 \left (a+b x+c x^2\right )^{7/2}}{11 d^{12} \left (b^2-4 a c\right ) (b+2 c x)^{11}} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^12,x]
[Out]
(2*(a + b*x + c*x^2)^(7/2))/(11*(b^2 - 4*a*c)*d^12*(b + 2*c*x)^11) + (8*(a + b*x + c*x^2)^(7/2))/(99*(b^2 - 4*
a*c)^2*d^12*(b + 2*c*x)^9) + (16*(a + b*x + c*x^2)^(7/2))/(693*(b^2 - 4*a*c)^3*d^12*(b + 2*c*x)^7)
Rule 696
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 707
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)^{12}} \, dx &=\frac {2 \left (a+b x+c x^2\right )^{7/2}}{11 \left (b^2-4 a c\right ) d^{12} (b+2 c x)^{11}}+\frac {4 \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^{10}} \, dx}{11 \left (b^2-4 a c\right ) d^2}\\ &=\frac {2 \left (a+b x+c x^2\right )^{7/2}}{11 \left (b^2-4 a c\right ) d^{12} (b+2 c x)^{11}}+\frac {8 \left (a+b x+c x^2\right )^{7/2}}{99 \left (b^2-4 a c\right )^2 d^{12} (b+2 c x)^9}+\frac {8 \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^8} \, dx}{99 \left (b^2-4 a c\right )^2 d^4}\\ &=\frac {2 \left (a+b x+c x^2\right )^{7/2}}{11 \left (b^2-4 a c\right ) d^{12} (b+2 c x)^{11}}+\frac {8 \left (a+b x+c x^2\right )^{7/2}}{99 \left (b^2-4 a c\right )^2 d^{12} (b+2 c x)^9}+\frac {16 \left (a+b x+c x^2\right )^{7/2}}{693 \left (b^2-4 a c\right )^3 d^{12} (b+2 c x)^7}\\ \end {align*}
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Mathematica [A]
time = 10.07, size = 110, normalized size = 0.93 \begin {gather*} \frac {2 (a+x (b+c x))^{7/2} \left (99 b^4+176 b^3 c x+64 b c^2 x \left (-7 a+4 c x^2\right )+8 b^2 c \left (-77 a+38 c x^2\right )+16 c^2 \left (63 a^2-28 a c x^2+8 c^2 x^4\right )\right )}{693 \left (b^2-4 a c\right )^3 d^{12} (b+2 c x)^{11}} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^12,x]
[Out]
(2*(a + x*(b + c*x))^(7/2)*(99*b^4 + 176*b^3*c*x + 64*b*c^2*x*(-7*a + 4*c*x^2) + 8*b^2*c*(-77*a + 38*c*x^2) +
16*c^2*(63*a^2 - 28*a*c*x^2 + 8*c^2*x^4)))/(693*(b^2 - 4*a*c)^3*d^12*(b + 2*c*x)^11)
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Maple [A]
time = 0.73, size = 195, normalized size = 1.65
| | |
| method |
result |
size |
| | |
| gosper |
\(-\frac {2 \left (128 c^{4} x^{4}+256 b \,c^{3} x^{3}-448 x^{2} c^{3} a +304 b^{2} c^{2} x^{2}-448 x a b \,c^{2}+176 b^{3} c x +1008 a^{2} c^{2}-616 a c \,b^{2}+99 b^{4}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {7}{2}}}{693 \left (2 c x +b \right )^{11} d^{12} \left (64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right )}\) |
\(133\) |
| default |
\(\frac {-\frac {4 c \left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {7}{2}}}{11 \left (4 a c -b^{2}\right ) \left (x +\frac {b}{2 c}\right )^{11}}-\frac {16 c^{2} \left (-\frac {4 c \left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {7}{2}}}{9 \left (4 a c -b^{2}\right ) \left (x +\frac {b}{2 c}\right )^{9}}+\frac {32 c^{3} \left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {7}{2}}}{63 \left (4 a c -b^{2}\right )^{2} \left (x +\frac {b}{2 c}\right )^{7}}\right )}{11 \left (4 a c -b^{2}\right )}}{4096 d^{12} c^{12}}\) |
\(195\) |
| trager |
\(-\frac {2 \left (128 c^{7} x^{10}+640 b \,c^{6} x^{9}-64 c^{6} a \,x^{8}+1456 b^{2} c^{5} x^{8}-256 a b \,c^{5} x^{7}+1984 b^{3} c^{4} x^{7}+48 a^{2} c^{5} x^{6}-472 a \,b^{2} c^{4} x^{6}+1795 b^{4} c^{3} x^{6}+144 a^{2} b \,c^{4} x^{5}-520 a \,b^{3} c^{3} x^{5}+1129 b^{5} c^{2} x^{5}+1808 a^{3} c^{4} x^{4}-1176 a^{2} b^{2} c^{3} x^{4}-31 a \,b^{4} c^{2} x^{4}+473 b^{6} c \,x^{4}+3616 x^{3} a^{3} b \,c^{3}-2592 a^{2} b^{3} c^{2} x^{3}+506 a \,b^{5} c \,x^{3}+99 b^{7} x^{3}+2576 a^{4} c^{3} x^{2}+136 a^{3} b^{2} c^{2} x^{2}-1023 a^{2} b^{4} c \,x^{2}+297 a \,b^{6} x^{2}+2576 a^{4} b \,c^{2} x -1672 c \,a^{3} b^{3} x +297 a^{2} b^{5} x +1008 a^{5} c^{2}-616 a^{4} b^{2} c +99 a^{3} b^{4}\right ) \sqrt {c \,x^{2}+b x +a}}{693 d^{12} \left (64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right ) \left (2 c x +b \right )^{11}}\) |
\(378\) |
| | |
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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)^12,x,method=_RETURNVERBOSE)
[Out]
1/4096/d^12/c^12*(-4/11/(4*a*c-b^2)*c/(x+1/2*b/c)^11*((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(7/2)-16/11*c^2/(4*a*
c-b^2)*(-4/9/(4*a*c-b^2)*c/(x+1/2*b/c)^9*((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(7/2)+32/63*c^3/(4*a*c-b^2)^2/(x+
1/2*b/c)^7*((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(7/2)))
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((c*x^2+b*x+a)^(5/2)/(2*c*d*x+b*d)^12,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 deta
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 851 vs.
\(2 (106) = 212\).
time = 108.65, size = 851, normalized size = 7.21 \begin {gather*} \frac {2 \, {\left (128 \, c^{7} x^{10} + 640 \, b c^{6} x^{9} + 16 \, {\left (91 \, b^{2} c^{5} - 4 \, a c^{6}\right )} x^{8} + 64 \, {\left (31 \, b^{3} c^{4} - 4 \, a b c^{5}\right )} x^{7} + 99 \, a^{3} b^{4} - 616 \, a^{4} b^{2} c + 1008 \, a^{5} c^{2} + {\left (1795 \, b^{4} c^{3} - 472 \, a b^{2} c^{4} + 48 \, a^{2} c^{5}\right )} x^{6} + {\left (1129 \, b^{5} c^{2} - 520 \, a b^{3} c^{3} + 144 \, a^{2} b c^{4}\right )} x^{5} + {\left (473 \, b^{6} c - 31 \, a b^{4} c^{2} - 1176 \, a^{2} b^{2} c^{3} + 1808 \, a^{3} c^{4}\right )} x^{4} + {\left (99 \, b^{7} + 506 \, a b^{5} c - 2592 \, a^{2} b^{3} c^{2} + 3616 \, a^{3} b c^{3}\right )} x^{3} + {\left (297 \, a b^{6} - 1023 \, a^{2} b^{4} c + 136 \, a^{3} b^{2} c^{2} + 2576 \, a^{4} c^{3}\right )} x^{2} + {\left (297 \, a^{2} b^{5} - 1672 \, a^{3} b^{3} c + 2576 \, a^{4} b c^{2}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{693 \, {\left (2048 \, {\left (b^{6} c^{11} - 12 \, a b^{4} c^{12} + 48 \, a^{2} b^{2} c^{13} - 64 \, a^{3} c^{14}\right )} d^{12} x^{11} + 11264 \, {\left (b^{7} c^{10} - 12 \, a b^{5} c^{11} + 48 \, a^{2} b^{3} c^{12} - 64 \, a^{3} b c^{13}\right )} d^{12} x^{10} + 28160 \, {\left (b^{8} c^{9} - 12 \, a b^{6} c^{10} + 48 \, a^{2} b^{4} c^{11} - 64 \, a^{3} b^{2} c^{12}\right )} d^{12} x^{9} + 42240 \, {\left (b^{9} c^{8} - 12 \, a b^{7} c^{9} + 48 \, a^{2} b^{5} c^{10} - 64 \, a^{3} b^{3} c^{11}\right )} d^{12} x^{8} + 42240 \, {\left (b^{10} c^{7} - 12 \, a b^{8} c^{8} + 48 \, a^{2} b^{6} c^{9} - 64 \, a^{3} b^{4} c^{10}\right )} d^{12} x^{7} + 29568 \, {\left (b^{11} c^{6} - 12 \, a b^{9} c^{7} + 48 \, a^{2} b^{7} c^{8} - 64 \, a^{3} b^{5} c^{9}\right )} d^{12} x^{6} + 14784 \, {\left (b^{12} c^{5} - 12 \, a b^{10} c^{6} + 48 \, a^{2} b^{8} c^{7} - 64 \, a^{3} b^{6} c^{8}\right )} d^{12} x^{5} + 5280 \, {\left (b^{13} c^{4} - 12 \, a b^{11} c^{5} + 48 \, a^{2} b^{9} c^{6} - 64 \, a^{3} b^{7} c^{7}\right )} d^{12} x^{4} + 1320 \, {\left (b^{14} c^{3} - 12 \, a b^{12} c^{4} + 48 \, a^{2} b^{10} c^{5} - 64 \, a^{3} b^{8} c^{6}\right )} d^{12} x^{3} + 220 \, {\left (b^{15} c^{2} - 12 \, a b^{13} c^{3} + 48 \, a^{2} b^{11} c^{4} - 64 \, a^{3} b^{9} c^{5}\right )} d^{12} x^{2} + 22 \, {\left (b^{16} c - 12 \, a b^{14} c^{2} + 48 \, a^{2} b^{12} c^{3} - 64 \, a^{3} b^{10} c^{4}\right )} d^{12} x + {\left (b^{17} - 12 \, a b^{15} c + 48 \, a^{2} b^{13} c^{2} - 64 \, a^{3} b^{11} c^{3}\right )} d^{12}\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)^12,x, algorithm="fricas")
[Out]
2/693*(128*c^7*x^10 + 640*b*c^6*x^9 + 16*(91*b^2*c^5 - 4*a*c^6)*x^8 + 64*(31*b^3*c^4 - 4*a*b*c^5)*x^7 + 99*a^3
*b^4 - 616*a^4*b^2*c + 1008*a^5*c^2 + (1795*b^4*c^3 - 472*a*b^2*c^4 + 48*a^2*c^5)*x^6 + (1129*b^5*c^2 - 520*a*
b^3*c^3 + 144*a^2*b*c^4)*x^5 + (473*b^6*c - 31*a*b^4*c^2 - 1176*a^2*b^2*c^3 + 1808*a^3*c^4)*x^4 + (99*b^7 + 50
6*a*b^5*c - 2592*a^2*b^3*c^2 + 3616*a^3*b*c^3)*x^3 + (297*a*b^6 - 1023*a^2*b^4*c + 136*a^3*b^2*c^2 + 2576*a^4*
c^3)*x^2 + (297*a^2*b^5 - 1672*a^3*b^3*c + 2576*a^4*b*c^2)*x)*sqrt(c*x^2 + b*x + a)/(2048*(b^6*c^11 - 12*a*b^4
*c^12 + 48*a^2*b^2*c^13 - 64*a^3*c^14)*d^12*x^11 + 11264*(b^7*c^10 - 12*a*b^5*c^11 + 48*a^2*b^3*c^12 - 64*a^3*
b*c^13)*d^12*x^10 + 28160*(b^8*c^9 - 12*a*b^6*c^10 + 48*a^2*b^4*c^11 - 64*a^3*b^2*c^12)*d^12*x^9 + 42240*(b^9*
c^8 - 12*a*b^7*c^9 + 48*a^2*b^5*c^10 - 64*a^3*b^3*c^11)*d^12*x^8 + 42240*(b^10*c^7 - 12*a*b^8*c^8 + 48*a^2*b^6
*c^9 - 64*a^3*b^4*c^10)*d^12*x^7 + 29568*(b^11*c^6 - 12*a*b^9*c^7 + 48*a^2*b^7*c^8 - 64*a^3*b^5*c^9)*d^12*x^6
+ 14784*(b^12*c^5 - 12*a*b^10*c^6 + 48*a^2*b^8*c^7 - 64*a^3*b^6*c^8)*d^12*x^5 + 5280*(b^13*c^4 - 12*a*b^11*c^5
+ 48*a^2*b^9*c^6 - 64*a^3*b^7*c^7)*d^12*x^4 + 1320*(b^14*c^3 - 12*a*b^12*c^4 + 48*a^2*b^10*c^5 - 64*a^3*b^8*c
^6)*d^12*x^3 + 220*(b^15*c^2 - 12*a*b^13*c^3 + 48*a^2*b^11*c^4 - 64*a^3*b^9*c^5)*d^12*x^2 + 22*(b^16*c - 12*a*
b^14*c^2 + 48*a^2*b^12*c^3 - 64*a^3*b^10*c^4)*d^12*x + (b^17 - 12*a*b^15*c + 48*a^2*b^13*c^2 - 64*a^3*b^11*c^3
)*d^12)
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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^{12} + 24 b^{11} c x + 264 b^{10} c^{2} x^{2} + 1760 b^{9} c^{3} x^{3} + 7920 b^{8} c^{4} x^{4} + 25344 b^{7} c^{5} x^{5} + 59136 b^{6} c^{6} x^{6} + 101376 b^{5} c^{7} x^{7} + 126720 b^{4} c^{8} x^{8} + 112640 b^{3} c^{9} x^{9} + 67584 b^{2} c^{10} x^{10} + 24576 b c^{11} x^{11} + 4096 c^{12} x^{12}}\, dx + \int \frac {b^{2} x^{2} \sqrt {a + b x + c x^{2}}}{b^{12} + 24 b^{11} c x + 264 b^{10} c^{2} x^{2} + 1760 b^{9} c^{3} x^{3} + 7920 b^{8} c^{4} x^{4} + 25344 b^{7} c^{5} x^{5} + 59136 b^{6} c^{6} x^{6} + 101376 b^{5} c^{7} x^{7} + 126720 b^{4} c^{8} x^{8} + 112640 b^{3} c^{9} x^{9} + 67584 b^{2} c^{10} x^{10} + 24576 b c^{11} x^{11} + 4096 c^{12} x^{12}}\, dx + \int \frac {c^{2} x^{4} \sqrt {a + b x + c x^{2}}}{b^{12} + 24 b^{11} c x + 264 b^{10} c^{2} x^{2} + 1760 b^{9} c^{3} x^{3} + 7920 b^{8} c^{4} x^{4} + 25344 b^{7} c^{5} x^{5} + 59136 b^{6} c^{6} x^{6} + 101376 b^{5} c^{7} x^{7} + 126720 b^{4} c^{8} x^{8} + 112640 b^{3} c^{9} x^{9} + 67584 b^{2} c^{10} x^{10} + 24576 b c^{11} x^{11} + 4096 c^{12} x^{12}}\, dx + \int \frac {2 a b x \sqrt {a + b x + c x^{2}}}{b^{12} + 24 b^{11} c x + 264 b^{10} c^{2} x^{2} + 1760 b^{9} c^{3} x^{3} + 7920 b^{8} c^{4} x^{4} + 25344 b^{7} c^{5} x^{5} + 59136 b^{6} c^{6} x^{6} + 101376 b^{5} c^{7} x^{7} + 126720 b^{4} c^{8} x^{8} + 112640 b^{3} c^{9} x^{9} + 67584 b^{2} c^{10} x^{10} + 24576 b c^{11} x^{11} + 4096 c^{12} x^{12}}\, dx + \int \frac {2 a c x^{2} \sqrt {a + b x + c x^{2}}}{b^{12} + 24 b^{11} c x + 264 b^{10} c^{2} x^{2} + 1760 b^{9} c^{3} x^{3} + 7920 b^{8} c^{4} x^{4} + 25344 b^{7} c^{5} x^{5} + 59136 b^{6} c^{6} x^{6} + 101376 b^{5} c^{7} x^{7} + 126720 b^{4} c^{8} x^{8} + 112640 b^{3} c^{9} x^{9} + 67584 b^{2} c^{10} x^{10} + 24576 b c^{11} x^{11} + 4096 c^{12} x^{12}}\, dx + \int \frac {2 b c x^{3} \sqrt {a + b x + c x^{2}}}{b^{12} + 24 b^{11} c x + 264 b^{10} c^{2} x^{2} + 1760 b^{9} c^{3} x^{3} + 7920 b^{8} c^{4} x^{4} + 25344 b^{7} c^{5} x^{5} + 59136 b^{6} c^{6} x^{6} + 101376 b^{5} c^{7} x^{7} + 126720 b^{4} c^{8} x^{8} + 112640 b^{3} c^{9} x^{9} + 67584 b^{2} c^{10} x^{10} + 24576 b c^{11} x^{11} + 4096 c^{12} x^{12}}\, dx}{d^{12}} \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)**12,x)
[Out]
(Integral(a**2*sqrt(a + b*x + c*x**2)/(b**12 + 24*b**11*c*x + 264*b**10*c**2*x**2 + 1760*b**9*c**3*x**3 + 7920
*b**8*c**4*x**4 + 25344*b**7*c**5*x**5 + 59136*b**6*c**6*x**6 + 101376*b**5*c**7*x**7 + 126720*b**4*c**8*x**8
+ 112640*b**3*c**9*x**9 + 67584*b**2*c**10*x**10 + 24576*b*c**11*x**11 + 4096*c**12*x**12), x) + Integral(b**2
*x**2*sqrt(a + b*x + c*x**2)/(b**12 + 24*b**11*c*x + 264*b**10*c**2*x**2 + 1760*b**9*c**3*x**3 + 7920*b**8*c**
4*x**4 + 25344*b**7*c**5*x**5 + 59136*b**6*c**6*x**6 + 101376*b**5*c**7*x**7 + 126720*b**4*c**8*x**8 + 112640*
b**3*c**9*x**9 + 67584*b**2*c**10*x**10 + 24576*b*c**11*x**11 + 4096*c**12*x**12), x) + Integral(c**2*x**4*sqr
t(a + b*x + c*x**2)/(b**12 + 24*b**11*c*x + 264*b**10*c**2*x**2 + 1760*b**9*c**3*x**3 + 7920*b**8*c**4*x**4 +
25344*b**7*c**5*x**5 + 59136*b**6*c**6*x**6 + 101376*b**5*c**7*x**7 + 126720*b**4*c**8*x**8 + 112640*b**3*c**9
*x**9 + 67584*b**2*c**10*x**10 + 24576*b*c**11*x**11 + 4096*c**12*x**12), x) + Integral(2*a*b*x*sqrt(a + b*x +
c*x**2)/(b**12 + 24*b**11*c*x + 264*b**10*c**2*x**2 + 1760*b**9*c**3*x**3 + 7920*b**8*c**4*x**4 + 25344*b**7*
c**5*x**5 + 59136*b**6*c**6*x**6 + 101376*b**5*c**7*x**7 + 126720*b**4*c**8*x**8 + 112640*b**3*c**9*x**9 + 675
84*b**2*c**10*x**10 + 24576*b*c**11*x**11 + 4096*c**12*x**12), x) + Integral(2*a*c*x**2*sqrt(a + b*x + c*x**2)
/(b**12 + 24*b**11*c*x + 264*b**10*c**2*x**2 + 1760*b**9*c**3*x**3 + 7920*b**8*c**4*x**4 + 25344*b**7*c**5*x**
5 + 59136*b**6*c**6*x**6 + 101376*b**5*c**7*x**7 + 126720*b**4*c**8*x**8 + 112640*b**3*c**9*x**9 + 67584*b**2*
c**10*x**10 + 24576*b*c**11*x**11 + 4096*c**12*x**12), x) + Integral(2*b*c*x**3*sqrt(a + b*x + c*x**2)/(b**12
+ 24*b**11*c*x + 264*b**10*c**2*x**2 + 1760*b**9*c**3*x**3 + 7920*b**8*c**4*x**4 + 25344*b**7*c**5*x**5 + 5913
6*b**6*c**6*x**6 + 101376*b**5*c**7*x**7 + 126720*b**4*c**8*x**8 + 112640*b**3*c**9*x**9 + 67584*b**2*c**10*x*
*10 + 24576*b*c**11*x**11 + 4096*c**12*x**12), x))/d**12
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 2371 vs.
\(2 (106) = 212\).
time = 6.54, size = 2371, normalized size = 20.09 \begin {gather*} \text {Too large to display} \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)^12,x, algorithm="giac")
[Out]
1/5544*(29568*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^16*c^(17/2) + 236544*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^15*
b*c^8 + 868560*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^14*b^2*c^(15/2) + 73920*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))
^14*a*c^(17/2) + 1940400*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^13*b^3*c^7 + 517440*(sqrt(c)*x - sqrt(c*x^2 + b*x
+ a))^13*a*b*c^8 + 2953104*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^12*b^4*c^(13/2) + 1600368*(sqrt(c)*x - sqrt(c*
x^2 + b*x + a))^12*a*b^2*c^(15/2) + 162624*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^12*a^2*c^(17/2) + 3256176*(sqrt
(c)*x - sqrt(c*x^2 + b*x + a))^11*b^5*c^6 + 2875488*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^11*a*b^3*c^7 + 975744*
(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^11*a^2*b*c^8 + 2709168*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^10*b^6*c^(11/2)
+ 3307920*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^10*a*b^4*c^(13/2) + 2583504*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))
^10*a^2*b^2*c^(15/2) + 133056*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^10*a^3*c^(17/2) + 1755600*(sqrt(c)*x - sqrt(
c*x^2 + b*x + a))^9*b^7*c^5 + 2513280*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*a*b^5*c^6 + 3973200*(sqrt(c)*x - s
qrt(c*x^2 + b*x + a))^9*a^2*b^3*c^7 + 665280*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*a^3*b*c^8 + 910800*(sqrt(c)
*x - sqrt(c*x^2 + b*x + a))^8*b^8*c^(9/2) + 1227600*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*a*b^6*c^(11/2) + 394
4160*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*a^2*b^4*c^(13/2) + 1401840*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*a^
3*b^2*c^(15/2) + 95040*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*a^4*c^(17/2) + 387024*(sqrt(c)*x - sqrt(c*x^2 + b
*x + a))^7*b^9*c^4 + 319968*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*a*b^7*c^5 + 2670624*(sqrt(c)*x - sqrt(c*x^2
+ b*x + a))^7*a^2*b^5*c^6 + 1615680*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*a^3*b^3*c^7 + 380160*(sqrt(c)*x - sq
rt(c*x^2 + b*x + a))^7*a^4*b*c^8 + 136488*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*b^10*c^(7/2) - 20592*(sqrt(c)*
x - sqrt(c*x^2 + b*x + a))^6*a*b^8*c^(9/2) + 1284624*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*a^2*b^6*c^(11/2) +
1092960*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*a^3*b^4*c^(13/2) + 641520*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*
a^4*b^2*c^(15/2) + 19008*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*a^5*c^(17/2) + 39864*(sqrt(c)*x - sqrt(c*x^2 +
b*x + a))^5*b^11*c^3 - 58080*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a*b^9*c^4 + 460944*(sqrt(c)*x - sqrt(c*x^2
+ b*x + a))^5*a^2*b^7*c^5 + 418176*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a^3*b^5*c^6 + 594000*(sqrt(c)*x - sqr
t(c*x^2 + b*x + a))^5*a^4*b^3*c^7 + 57024*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a^5*b*c^8 + 9460*(sqrt(c)*x -
sqrt(c*x^2 + b*x + a))^4*b^12*c^(5/2) - 27720*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a*b^10*c^(7/2) + 132000*(s
qrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a^2*b^8*c^(9/2) + 64240*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a^3*b^6*c^(1
1/2) + 330000*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a^4*b^4*c^(13/2) + 66000*(sqrt(c)*x - sqrt(c*x^2 + b*x + a
))^4*a^5*b^2*c^(15/2) + 3520*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a^6*c^(17/2) + 1760*(sqrt(c)*x - sqrt(c*x^2
+ b*x + a))^3*b^13*c^2 - 7920*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*b^11*c^3 + 31680*(sqrt(c)*x - sqrt(c*x^
2 + b*x + a))^3*a^2*b^9*c^4 - 14080*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^3*b^7*c^5 + 113520*(sqrt(c)*x - sq
rt(c*x^2 + b*x + a))^3*a^4*b^5*c^6 + 36960*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^5*b^3*c^7 + 7040*(sqrt(c)*x
- sqrt(c*x^2 + b*x + a))^3*a^6*b*c^8 + 242*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b^14*c^(3/2) - 1496*(sqrt(c)
*x - sqrt(c*x^2 + b*x + a))^2*a*b^12*c^(5/2) + 6072*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^2*b^10*c^(7/2) - 8
800*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^3*b^8*c^(9/2) + 24640*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^4*b^
6*c^(11/2) + 8976*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^5*b^4*c^(13/2) + 6512*(sqrt(c)*x - sqrt(c*x^2 + b*x
+ a))^2*a^6*b^2*c^(15/2) - 704*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^7*c^(17/2) + 22*(sqrt(c)*x - sqrt(c*x^2
+ b*x + a))*b^15*c - 176*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a*b^13*c^2 + 792*(sqrt(c)*x - sqrt(c*x^2 + b*x +
a))*a^2*b^11*c^3 - 1760*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^3*b^9*c^4 + 3520*(sqrt(c)*x - sqrt(c*x^2 + b*x
+ a))*a^4*b^7*c^5 + 2992*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^6*b^3*c^7 - 704*(sqrt(c)*x - sqrt(c*x^2 + b*x +
a))*a^7*b*c^8 + b^16*sqrt(c) - 10*a*b^14*c^(3/2) + 52*a^2*b^12*c^(5/2) - 152*a^3*b^10*c^(7/2) + 320*a^4*b^8*c
^(9/2) - 320*a^5*b^6*c^(11/2) + 640*a^6*b^4*c^(13/2) - 304*a^7*b^2*c^(15/2) + 64*a^8*c^(17/2))/((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)^11*c^4*d^12)
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Mupad [B]
time = 15.67, size = 2500, normalized size = 21.19 \begin {gather*} \text {Too large to display} \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)^12,x)
[Out]
(((b*((b*((b*((b*((b*((4*c^4*(40*a*c - 3*b^2))/(231*d^12*(4*a*c - b^2)^3*(96*a*c^3 - 24*b^2*c^2)) - (8*b^2*c^4
)/(231*d^12*(4*a*c - b^2)^3*(96*a*c^3 - 24*b^2*c^2))))/(2*c) - (10*b*c^3*(120*a*c - 23*b^2))/(693*d^12*(4*a*c
- b^2)^3*(96*a*c^3 - 24*b^2*c^2))))/(2*c) + (5168*a^2*c^4 + 58*b^4*c^2 - 1384*a*b^2*c^3)/(693*d^12*(4*a*c - b^
2)^3*(96*a*c^3 - 24*b^2*c^2))))/(2*c) - (b*c*(115*b^4 + 2584*a^2*c^2 - 1092*a*b^2*c))/(231*d^12*(4*a*c - b^2)^
3*(96*a*c^3 - 24*b^2*c^2))))/(2*c) + (133*b^6 + 4704*a^3*c^3 + 348*a^2*b^2*c^2 - 906*a*b^4*c)/(693*d^12*(4*a*c
- b^2)^3*(96*a*c^3 - 24*b^2*c^2))))/(2*c) - (133*a*b^5 - 1118*a^2*b^3*c + 2352*a^3*b*c^2)/(693*d^12*(4*a*c -
b^2)^3*(96*a*c^3 - 24*b^2*c^2)))*(a + b*x + c*x^2)^(1/2))/(b + 2*c*x)^6 + (((b*((b*(b^2/(6160*c*d^12*(4*a*c -
b^2)^5) - (32*a*c^3 - 5*b^2*c^2)/(9240*c^3*d^12*(4*a*c - b^2)^5)))/(2*c) - (7*b^3*c - 32*a*b*c^2)/(9240*c^3*d^
12*(4*a*c - b^2)^5)))/(2*c) - (6*b^4 + 126*a^2*c^2 - 55*a*b^2*c)/(9240*c^3*d^12*(4*a*c - b^2)^5))*(a + b*x + c
*x^2)^(1/2))/(b + 2*c*x) + (((b*((b*(b^2/(5280*c*d^12*(4*a*c - b^2)^5) - (76*a*c^3 - 13*b^2*c^2)/(15840*c^3*d^
12*(4*a*c - b^2)^5)))/(2*c) - (17*b^3*c - 76*a*b*c^2)/(15840*c^3*d^12*(4*a*c - b^2)^5)))/(2*c) + (23*b^4 + 296
*a^2*c^2 - 167*a*b^2*c)/(15840*c^3*d^12*(4*a*c - b^2)^5))*(a + b*x + c*x^2)^(1/2))/(b + 2*c*x) + (((b*((b*((5*
a*c - b^2)/(1155*c*d^12*(4*a*c - b^2)^5) - b^2/(9240*c*d^12*(4*a*c - b^2)^5)))/(2*c) - (b*(30*a*c - 7*b^2))/(6
930*c^2*d^12*(4*a*c - b^2)^5)))/(2*c) + (147*b^4 + 2584*a^2*c^2 - 1232*a*b^2*c)/(55440*c^3*d^12*(4*a*c - b^2)^
5))*(a + b*x + c*x^2)^(1/2))/(b + 2*c*x) - (((b*((b*((b*((b*((368*a*c^4 - 32*b^2*c^3)/(198*d^12*(4*a*c - b^2)^
2*(112*a*c^3 - 28*b^2*c^2)) - (10*b^2*c^3)/(99*d^12*(4*a*c - b^2)^2*(112*a*c^3 - 28*b^2*c^2))))/(2*c) - (8*b*c
^2*(46*a*c - 9*b^2))/(99*d^12*(4*a*c - b^2)^2*(112*a*c^3 - 28*b^2*c^2))))/(2*c) + (39*b^4*c + 2592*a^2*c^3 - 7
44*a*b^2*c^2)/(198*d^12*(4*a*c - b^2)^2*(112*a*c^3 - 28*b^2*c^2))))/(2*c) - (119*b^5 + 2592*a^2*b*c^2 - 1112*a
*b^3*c)/(198*d^12*(4*a*c - b^2)^2*(112*a*c^3 - 28*b^2*c^2))))/(2*c) + (119*a*b^4 + 2240*a^3*c^2 - 1032*a^2*b^2
*c)/(198*d^12*(4*a*c - b^2)^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*((b^2*c)/(4158*d^12*(4*a*c - b^2)^6) - (608*a*c^4 - 92*b^2*c^3)/(83160*c^2*d^12*(4*a*c - b^2)^6)))/(2*c)
+ (b*(152*a*c - 33*b^2))/(10395*d^12*(4*a*c - b^2)^6)))/(2*c) - (365*b^4*c + 9248*a^2*c^3 - 3712*a*b^2*c^2)/(8
3160*c^2*d^12*(4*a*c - b^2)^6)))/(2*c) + (505*b^5 + 9248*a^2*b*c^2 - 4320*a*b^3*c)/(83160*c^2*d^12*(4*a*c - b^
2)^6)))/(2*c) - (505*a*b^4 + 8656*a^3*c^2 - 4180*a^2*b^2*c)/(83160*c^2*d^12*(4*a*c - b^2)^6))*(a + b*x + c*x^2
)^(1/2))/(b + 2*c*x) + (((b*((b*((4*c*(7*a*c - b^2))/(99*d^12*(4*a*c - b^2)^2*(80*a*c^3 - 20*b^2*c^2)) - (b^2*
c)/(66*d^12*(4*a*c - b^2)^2*(80*a*c^3 - 20*b^2*c^2))))/(2*c) - (2*b*(14*a*c - 3*b^2))/(99*d^12*(4*a*c - b^2)^2
*(80*a*c^3 - 20*b^2*c^2))))/(2*c) + (23*b^4 + 576*a^2*c^2 - 232*a*b^2*c)/(792*c*d^12*(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 - (((b*((b*((c*(10*a*c - b^2))/(231*d^12*(4*a*c - b^2)
^3*(48*a*c^3 - 12*b^2*c^2)) - (b^2*c)/(308*d^12*(4*a*c - b^2)^3*(48*a*c^3 - 12*b^2*c^2))))/(2*c) - (2*b*(5*a*c
- b^2))/(231*d^12*(4*a*c - b^2)^3*(48*a*c^3 - 12*b^2*c^2))))/(2*c) + (24*a^2*c^2 - 3*b^4 + 8*a*b^2*c)/(1848*c
*d^12*(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 - (((32*a*c - 7*b^2)/(1
4784*c^3*d^12*(4*a*c - b^2)^4) - b^2/(14784*c^3*d^12*(4*a*c - b^2)^4))*(a + b*x + c*x^2)^(1/2))/(b + 2*c*x) -
(((56*a*c - 13*b^2)/(15840*c^3*d^12*(4*a*c - b^2)^4) - b^2/(15840*c^3*d^12*(4*a*c - b^2)^4))*(a + b*x + c*x^2)
^(1/2))/(b + 2*c*x) - (((170*a*c - 43*b^2)/(11088*c^3*d^12*(4*a*c - b^2)^4) + b^2/(22176*c^3*d^12*(4*a*c - b^2
)^4))*(a + b*x + c*x^2)^(1/2))/(b + 2*c*x) - (((b*((b*(b^2/(5280*c*d^12*(4*a*c - b^2)^5) - (276*a*c^3 - 51*b^2
*c^2)/(47520*c^3*d^12*(4*a*c - b^2)^5)))/(2*c) + (b*(92*a*c - 21*b^2))/(15840*c^2*d^12*(4*a*c - b^2)^5)))/(2*c
) - (133*b^4 + 2392*a^2*c^2 - 1127*a*b^2*c)/(47520*c^3*d^12*(4*a*c - b^2)^5))*(a + b*x + c*x^2)^(1/2))/(b + 2*
c*x) + (((b*((b*((b*((b*((b*((8*c^4*(46*a*c - b^2))/(99*d^12*(4*a*c - b^2)^2*(128*a*c^3 - 32*b^2*c^2)) - (8*b^
2*c^4)/(33*d^12*(4*a*c - b^2)^2*(128*a*c^3 - 32*b^2*c^2))))/(2*c) - (40*b*c^3*(23*a*c - 4*b^2))/(99*d^12*(4*a*
c - b^2)^2*(128*a*c^3 - 32*b^2*c^2))))/(2*c) - (66*b^4*c^2 - 5184*a^2*c^4 + 752*a*b^2*c^3)/(198*d^12*(4*a*c -
b^2)^2*(128*a*c^3 - 32*b^2*c^2))))/(2*c) - (277*b^5*c - 2968*a*b^3*c^2 + 7776*a^2*b*c^3)/(198*d^12*(4*a*c - b^
2)^2*(128*a*c^3 - 32*b^2*c^2))))/(2*c) + (119*b^6 + 4480*a^3*c^3 + 528*a^2*b^2*c^2 - 874*a*b^4*c)/(198*d^12*(4
*a*c - b^2)^2*(128*a*c^3 - 32*b^2*c^2))))/(2*c) - (119*a*b^5 - 1032*a^2*b^3*c + 2240*a^3*b*c^2)/(198*d^12*(4*a
*c - b^2)^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*(
10*a*c - b^2))/(495*d^12*(4*a*c - b^2)^4*(64*a*c^3 - 16*b^2*c^2)) - (8*b^2*c^4)/(1155*d^12*(4*a*c - b^2)^4*(64
*a*c^3 - 16*b^2*c^2))))/(2*c) - (8*b*c^3*(5*a*c...
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