3.13.16 \(\int \frac {(a+b x+c x^2)^{3/2}}{(b d+2 c d x)^{10}} \, dx\) [1216]

Optimal. Leaf size=118 \[ \frac {2 \left (a+b x+c x^2\right )^{5/2}}{9 \left (b^2-4 a c\right ) d^{10} (b+2 c x)^9}+\frac {8 \left (a+b x+c x^2\right )^{5/2}}{63 \left (b^2-4 a c\right )^2 d^{10} (b+2 c x)^7}+\frac {16 \left (a+b x+c x^2\right )^{5/2}}{315 \left (b^2-4 a c\right )^3 d^{10} (b+2 c x)^5} \]

[Out]

2/9*(c*x^2+b*x+a)^(5/2)/(-4*a*c+b^2)/d^10/(2*c*x+b)^9+8/63*(c*x^2+b*x+a)^(5/2)/(-4*a*c+b^2)^2/d^10/(2*c*x+b)^7
+16/315*(c*x^2+b*x+a)^(5/2)/(-4*a*c+b^2)^3/d^10/(2*c*x+b)^5

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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 )^{5/2}}{315 d^{10} \left (b^2-4 a c\right )^3 (b+2 c x)^5}+\frac {8 \left (a+b x+c x^2\right )^{5/2}}{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 )^{5/2}}{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)^(3/2)/(b*d + 2*c*d*x)^10,x]

[Out]

(2*(a + b*x + c*x^2)^(5/2))/(9*(b^2 - 4*a*c)*d^10*(b + 2*c*x)^9) + (8*(a + b*x + c*x^2)^(5/2))/(63*(b^2 - 4*a*
c)^2*d^10*(b + 2*c*x)^7) + (16*(a + b*x + c*x^2)^(5/2))/(315*(b^2 - 4*a*c)^3*d^10*(b + 2*c*x)^5)

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 )^{3/2}}{(b d+2 c d x)^{10}} \, dx &=\frac {2 \left (a+b x+c x^2\right )^{5/2}}{9 \left (b^2-4 a c\right ) d^{10} (b+2 c x)^9}+\frac {4 \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(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 )^{5/2}}{9 \left (b^2-4 a c\right ) d^{10} (b+2 c x)^9}+\frac {8 \left (a+b x+c x^2\right )^{5/2}}{63 \left (b^2-4 a c\right )^2 d^{10} (b+2 c x)^7}+\frac {8 \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^6} \, dx}{63 \left (b^2-4 a c\right )^2 d^4}\\ &=\frac {2 \left (a+b x+c x^2\right )^{5/2}}{9 \left (b^2-4 a c\right ) d^{10} (b+2 c x)^9}+\frac {8 \left (a+b x+c x^2\right )^{5/2}}{63 \left (b^2-4 a c\right )^2 d^{10} (b+2 c x)^7}+\frac {16 \left (a+b x+c x^2\right )^{5/2}}{315 \left (b^2-4 a c\right )^3 d^{10} (b+2 c x)^5}\\ \end {align*}

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Mathematica [A]
time = 10.06, size = 110, normalized size = 0.93 \begin {gather*} \frac {2 (a+x (b+c x))^{5/2} \left (63 b^4+144 b^3 c x+64 b c^2 x \left (-5 a+4 c x^2\right )+8 b^2 c \left (-45 a+34 c x^2\right )+16 c^2 \left (35 a^2-20 a c x^2+8 c^2 x^4\right )\right )}{315 \left (b^2-4 a c\right )^3 d^{10} (b+2 c x)^9} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x + c*x^2)^(3/2)/(b*d + 2*c*d*x)^10,x]

[Out]

(2*(a + x*(b + c*x))^(5/2)*(63*b^4 + 144*b^3*c*x + 64*b*c^2*x*(-5*a + 4*c*x^2) + 8*b^2*c*(-45*a + 34*c*x^2) +
16*c^2*(35*a^2 - 20*a*c*x^2 + 8*c^2*x^4)))/(315*(b^2 - 4*a*c)^3*d^10*(b + 2*c*x)^9)

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Maple [A]
time = 0.69, size = 195, normalized size = 1.65

method	result	size
gosper	\(-\frac {2 \left (128 c^{4} x^{4}+256 b \,c^{3} x^{3}-320 x^{2} c^{3} a +272 b^{2} c^{2} x^{2}-320 x a b \,c^{2}+144 b^{3} c x +560 a^{2} c^{2}-360 a c \,b^{2}+63 b^{4}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}{315 \left (2 c x +b \right )^{9} d^{10} \left (64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right )}\)	\(133\)
default	\(\frac {-\frac {4 c \left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {5}{2}}}{9 \left (4 a c -b^{2}\right ) \left (x +\frac {b}{2 c}\right )^{9}}-\frac {16 c^{2} \left (-\frac {4 c \left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {5}{2}}}{7 \left (4 a c -b^{2}\right ) \left (x +\frac {b}{2 c}\right )^{7}}+\frac {32 c^{3} \left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {5}{2}}}{35 \left (4 a c -b^{2}\right )^{2} \left (x +\frac {b}{2 c}\right )^{5}}\right )}{9 \left (4 a c -b^{2}\right )}}{1024 d^{10} c^{10}}\)	\(195\)
trager	\(-\frac {2 \left (128 c^{6} x^{8}+512 b \,c^{5} x^{7}-64 c^{5} a \,x^{6}+912 b^{2} c^{4} x^{6}-192 a b \,c^{4} x^{5}+944 b^{3} c^{3} x^{5}+48 a^{2} c^{4} x^{4}-264 a \,b^{2} c^{3} x^{4}+623 b^{4} c^{2} x^{4}+96 a^{2} b \,c^{3} x^{3}-208 a \,b^{3} c^{2} x^{3}+270 b^{5} c \,x^{3}+800 a^{3} c^{3} x^{2}-528 a^{2} b^{2} c^{2} x^{2}+54 a \,b^{4} c \,x^{2}+63 b^{6} x^{2}+800 a^{3} b \,c^{2} x -576 a^{2} b^{3} c x +126 a \,b^{5} x +560 a^{4} c^{2}-360 a^{3} b^{2} c +63 a^{2} b^{4}\right ) \sqrt {c \,x^{2}+b x +a}}{315 d^{10} \left (64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right ) \left (2 c x +b \right )^{9}}\)	\(281\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+b*x+a)^(3/2)/(2*c*d*x+b*d)^10,x,method=_RETURNVERBOSE)

[Out]

1/1024/d^10/c^10*(-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)^(5/2)-16/9*c^2/(4*a*c-b
^2)*(-4/7/(4*a*c-b^2)*c/(x+1/2*b/c)^7*((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(5/2)+32/35*c^3/(4*a*c-b^2)^2/(x+1/2
*b/c)^5*((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(5/2)))

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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)^(3/2)/(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 deta

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 676 vs. \(2 (106) = 212\).
time = 26.42, size = 676, normalized size = 5.73 \begin {gather*} \frac {2 \, {\left (128 \, c^{6} x^{8} + 512 \, b c^{5} x^{7} + 16 \, {\left (57 \, b^{2} c^{4} - 4 \, a c^{5}\right )} x^{6} + 63 \, a^{2} b^{4} - 360 \, a^{3} b^{2} c + 560 \, a^{4} c^{2} + 16 \, {\left (59 \, b^{3} c^{3} - 12 \, a b c^{4}\right )} x^{5} + {\left (623 \, b^{4} c^{2} - 264 \, a b^{2} c^{3} + 48 \, a^{2} c^{4}\right )} x^{4} + 2 \, {\left (135 \, b^{5} c - 104 \, a b^{3} c^{2} + 48 \, a^{2} b c^{3}\right )} x^{3} + {\left (63 \, b^{6} + 54 \, a b^{4} c - 528 \, a^{2} b^{2} c^{2} + 800 \, a^{3} c^{3}\right )} x^{2} + 2 \, {\left (63 \, a b^{5} - 288 \, a^{2} b^{3} c + 400 \, a^{3} b c^{2}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{315 \, {\left (512 \, {\left (b^{6} c^{9} - 12 \, a b^{4} c^{10} + 48 \, a^{2} b^{2} c^{11} - 64 \, a^{3} c^{12}\right )} d^{10} x^{9} + 2304 \, {\left (b^{7} c^{8} - 12 \, a b^{5} c^{9} + 48 \, a^{2} b^{3} c^{10} - 64 \, a^{3} b c^{11}\right )} d^{10} x^{8} + 4608 \, {\left (b^{8} c^{7} - 12 \, a b^{6} c^{8} + 48 \, a^{2} b^{4} c^{9} - 64 \, a^{3} b^{2} c^{10}\right )} d^{10} x^{7} + 5376 \, {\left (b^{9} c^{6} - 12 \, a b^{7} c^{7} + 48 \, a^{2} b^{5} c^{8} - 64 \, a^{3} b^{3} c^{9}\right )} d^{10} x^{6} + 4032 \, {\left (b^{10} c^{5} - 12 \, a b^{8} c^{6} + 48 \, a^{2} b^{6} c^{7} - 64 \, a^{3} b^{4} c^{8}\right )} d^{10} x^{5} + 2016 \, {\left (b^{11} c^{4} - 12 \, a b^{9} c^{5} + 48 \, a^{2} b^{7} c^{6} - 64 \, a^{3} b^{5} c^{7}\right )} d^{10} x^{4} + 672 \, {\left (b^{12} c^{3} - 12 \, a b^{10} c^{4} + 48 \, a^{2} b^{8} c^{5} - 64 \, a^{3} b^{6} c^{6}\right )} d^{10} x^{3} + 144 \, {\left (b^{13} c^{2} - 12 \, a b^{11} c^{3} + 48 \, a^{2} b^{9} c^{4} - 64 \, a^{3} b^{7} c^{5}\right )} d^{10} x^{2} + 18 \, {\left (b^{14} c - 12 \, a b^{12} c^{2} + 48 \, a^{2} b^{10} c^{3} - 64 \, a^{3} b^{8} c^{4}\right )} d^{10} x + {\left (b^{15} - 12 \, a b^{13} c + 48 \, a^{2} b^{11} c^{2} - 64 \, a^{3} b^{9} c^{3}\right )} d^{10}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^(3/2)/(2*c*d*x+b*d)^10,x, algorithm="fricas")

[Out]

2/315*(128*c^6*x^8 + 512*b*c^5*x^7 + 16*(57*b^2*c^4 - 4*a*c^5)*x^6 + 63*a^2*b^4 - 360*a^3*b^2*c + 560*a^4*c^2
+ 16*(59*b^3*c^3 - 12*a*b*c^4)*x^5 + (623*b^4*c^2 - 264*a*b^2*c^3 + 48*a^2*c^4)*x^4 + 2*(135*b^5*c - 104*a*b^3
*c^2 + 48*a^2*b*c^3)*x^3 + (63*b^6 + 54*a*b^4*c - 528*a^2*b^2*c^2 + 800*a^3*c^3)*x^2 + 2*(63*a*b^5 - 288*a^2*b
^3*c + 400*a^3*b*c^2)*x)*sqrt(c*x^2 + b*x + a)/(512*(b^6*c^9 - 12*a*b^4*c^10 + 48*a^2*b^2*c^11 - 64*a^3*c^12)*
d^10*x^9 + 2304*(b^7*c^8 - 12*a*b^5*c^9 + 48*a^2*b^3*c^10 - 64*a^3*b*c^11)*d^10*x^8 + 4608*(b^8*c^7 - 12*a*b^6
*c^8 + 48*a^2*b^4*c^9 - 64*a^3*b^2*c^10)*d^10*x^7 + 5376*(b^9*c^6 - 12*a*b^7*c^7 + 48*a^2*b^5*c^8 - 64*a^3*b^3
*c^9)*d^10*x^6 + 4032*(b^10*c^5 - 12*a*b^8*c^6 + 48*a^2*b^6*c^7 - 64*a^3*b^4*c^8)*d^10*x^5 + 2016*(b^11*c^4 -
12*a*b^9*c^5 + 48*a^2*b^7*c^6 - 64*a^3*b^5*c^7)*d^10*x^4 + 672*(b^12*c^3 - 12*a*b^10*c^4 + 48*a^2*b^8*c^5 - 64
*a^3*b^6*c^6)*d^10*x^3 + 144*(b^13*c^2 - 12*a*b^11*c^3 + 48*a^2*b^9*c^4 - 64*a^3*b^7*c^5)*d^10*x^2 + 18*(b^14*
c - 12*a*b^12*c^2 + 48*a^2*b^10*c^3 - 64*a^3*b^8*c^4)*d^10*x + (b^15 - 12*a*b^13*c + 48*a^2*b^11*c^2 - 64*a^3*
b^9*c^3)*d^10)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {a \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 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 {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}{d^{10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+b*x+a)**(3/2)/(2*c*d*x+b*d)**10,x)

[Out]

(Integral(a*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*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(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)
)/d**10

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1393 vs. \(2 (106) = 212\).
time = 3.52, size = 1393, normalized size = 11.81 \begin {gather*} \frac {3360 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{12} c^{\frac {13}{2}} + 20160 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{11} b c^{6} + 54180 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{10} b^{2} c^{\frac {11}{2}} + 5040 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{10} a c^{\frac {13}{2}} + 86100 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{9} b^{3} c^{5} + 25200 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{9} a b c^{6} + 90216 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{8} b^{4} c^{\frac {9}{2}} + 53172 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{8} a b^{2} c^{\frac {11}{2}} + 7056 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{8} a^{2} c^{\frac {13}{2}} + 66024 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{7} b^{5} c^{4} + 61488 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{7} a b^{3} c^{5} + 28224 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{7} a^{2} b c^{6} + 35028 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{6} b^{6} c^{\frac {7}{2}} + 41832 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{6} a b^{4} c^{\frac {9}{2}} + 47880 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{6} a^{2} b^{2} c^{\frac {11}{2}} + 2016 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{6} a^{3} c^{\frac {13}{2}} + 13860 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{5} b^{7} c^{3} + 16128 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{5} a b^{5} c^{4} + 44856 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{5} a^{2} b^{3} c^{5} + 6048 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{5} a^{3} b c^{6} + 4176 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{4} b^{8} c^{\frac {5}{2}} + 2484 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{4} a b^{6} c^{\frac {7}{2}} + 25416 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{4} a^{2} b^{4} c^{\frac {9}{2}} + 6984 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{4} a^{3} b^{2} c^{\frac {11}{2}} + 576 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{4} a^{4} c^{\frac {13}{2}} + 960 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} b^{9} c^{2} - 576 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} a b^{7} c^{3} + 9000 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} a^{2} b^{5} c^{4} + 3888 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} a^{3} b^{3} c^{5} + 1152 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} a^{4} b c^{6} + 162 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} b^{10} c^{\frac {3}{2}} - 360 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} a b^{8} c^{\frac {5}{2}} + 2016 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} a^{2} b^{6} c^{\frac {7}{2}} + 936 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} a^{3} b^{4} c^{\frac {9}{2}} + 1044 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} a^{4} b^{2} c^{\frac {11}{2}} - 144 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} a^{5} c^{\frac {13}{2}} + 18 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} b^{11} c - 72 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} a b^{9} c^{2} + 288 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} a^{2} b^{7} c^{3} + 468 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} a^{4} b^{3} c^{5} - 144 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} a^{5} b c^{6} + b^{12} \sqrt {c} - 6 \, a b^{10} c^{\frac {3}{2}} + 24 \, a^{2} b^{8} c^{\frac {5}{2}} - 32 \, a^{3} b^{6} c^{\frac {7}{2}} + 96 \, a^{4} b^{4} c^{\frac {9}{2}} - 60 \, a^{5} b^{2} c^{\frac {11}{2}} + 16 \, a^{6} c^{\frac {13}{2}}}{630 \, {\left (2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} c + 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} b \sqrt {c} + b^{2} - 2 \, a c\right )}^{9} c^{3} d^{10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^(3/2)/(2*c*d*x+b*d)^10,x, algorithm="giac")

[Out]

1/630*(3360*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^12*c^(13/2) + 20160*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^11*b*c
^6 + 54180*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^10*b^2*c^(11/2) + 5040*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^10*a
*c^(13/2) + 86100*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*b^3*c^5 + 25200*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*
a*b*c^6 + 90216*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*b^4*c^(9/2) + 53172*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^
8*a*b^2*c^(11/2) + 7056*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*a^2*c^(13/2) + 66024*(sqrt(c)*x - sqrt(c*x^2 + b
*x + a))^7*b^5*c^4 + 61488*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*a*b^3*c^5 + 28224*(sqrt(c)*x - sqrt(c*x^2 + b
*x + a))^7*a^2*b*c^6 + 35028*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*b^6*c^(7/2) + 41832*(sqrt(c)*x - sqrt(c*x^2
+ b*x + a))^6*a*b^4*c^(9/2) + 47880*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*a^2*b^2*c^(11/2) + 2016*(sqrt(c)*x
- sqrt(c*x^2 + b*x + a))^6*a^3*c^(13/2) + 13860*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*b^7*c^3 + 16128*(sqrt(c)
*x - sqrt(c*x^2 + b*x + a))^5*a*b^5*c^4 + 44856*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a^2*b^3*c^5 + 6048*(sqrt
(c)*x - sqrt(c*x^2 + b*x + a))^5*a^3*b*c^6 + 4176*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*b^8*c^(5/2) + 2484*(sq
rt(c)*x - sqrt(c*x^2 + b*x + a))^4*a*b^6*c^(7/2) + 25416*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a^2*b^4*c^(9/2)
+ 6984*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a^3*b^2*c^(11/2) + 576*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a^4
*c^(13/2) + 960*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*b^9*c^2 - 576*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*b^
7*c^3 + 9000*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^2*b^5*c^4 + 3888*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^
3*b^3*c^5 + 1152*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^4*b*c^6 + 162*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b
^10*c^(3/2) - 360*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a*b^8*c^(5/2) + 2016*(sqrt(c)*x - sqrt(c*x^2 + b*x + a
))^2*a^2*b^6*c^(7/2) + 936*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^3*b^4*c^(9/2) + 1044*(sqrt(c)*x - sqrt(c*x^
2 + b*x + a))^2*a^4*b^2*c^(11/2) - 144*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^5*c^(13/2) + 18*(sqrt(c)*x - sq
rt(c*x^2 + b*x + a))*b^11*c - 72*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a*b^9*c^2 + 288*(sqrt(c)*x - sqrt(c*x^2 +
b*x + a))*a^2*b^7*c^3 + 468*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^4*b^3*c^5 - 144*(sqrt(c)*x - sqrt(c*x^2 + b
*x + a))*a^5*b*c^6 + b^12*sqrt(c) - 6*a*b^10*c^(3/2) + 24*a^2*b^8*c^(5/2) - 32*a^3*b^6*c^(7/2) + 96*a^4*b^4*c^
(9/2) - 60*a^5*b^2*c^(11/2) + 16*a^6*c^(13/2))/((2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*c + 2*(sqrt(c)*x - sq
rt(c*x^2 + b*x + a))*b*sqrt(c) + b^2 - 2*a*c)^9*c^3*d^10)

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Mupad [B]
time = 4.26, 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)^(3/2)/(b*d + 2*c*d*x)^10,x)

[Out]

(((b*((b*((b*((4*c^3*(44*a*c - b^2))/(9*d^10*(4*a*c - b^2)*(128*a*c^3 - 32*b^2*c^2)) - (16*b^2*c^3)/(9*d^10*(4
*a*c - b^2)*(128*a*c^3 - 32*b^2*c^2))))/(2*c) + (2*c*(23*b^3*c - 132*a*b*c^2))/(9*d^10*(4*a*c - b^2)*(128*a*c^
3 - 32*b^2*c^2))))/(2*c) + (2*c*(80*a^2*c^2 - 9*b^4 + 26*a*b^2*c))/(9*d^10*(4*a*c - b^2)*(128*a*c^3 - 32*b^2*c
^2))))/(2*c) + (2*c*(9*a*b^3 - 40*a^2*b*c))/(9*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^2/(18*d^10*(4*a*c - b^2)*(80*a*c^3 - 20*b^2*c^2)) - (8*a*c - b^2)/(18*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^2/(126*d^10*(4*a*c - b^2)^2*(48*
a*c^3 - 12*b^2*c^2)) - (22*a*c - 5*b^2)/(63*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 - (((b*((c*(6*a*c - b^2))/(30*d^10*(4*a*c - b^2)^3*(32*a*c^3 - 8*b^2*c^2)) - (b^2*c)/(90*d
^10*(4*a*c - b^2)^3*(32*a*c^3 - 8*b^2*c^2))))/(2*c) + (2*b^3 - 9*a*b*c)/(90*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*((4*c*(104*a*c^3 - 16*b^2*c^2))/(945*d^10*(4
*a*c - b^2)^4*(32*a*c^3 - 8*b^2*c^2)) - (16*b^2*c^3)/(945*d^10*(4*a*c - b^2)^4*(32*a*c^3 - 8*b^2*c^2))))/(2*c)
- (8*b*c^2*(78*a*c - 17*b^2))/(945*d^10*(4*a*c - b^2)^4*(32*a*c^3 - 8*b^2*c^2))))/(2*c) + (4*c*(100*a^2*c^2 -
12*b^4 + 28*a*b^2*c))/(945*d^10*(4*a*c - b^2)^4*(32*a*c^3 - 8*b^2*c^2))))/(2*c) - (8*a*b*c*(25*a*c - 6*b^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*((8*c^2*(
6*a*c - b^2))/(21*d^10*(4*a*c - b^2)^2*(80*a*c^3 - 20*b^2*c^2)) - (2*b^2*c^2)/(21*d^10*(4*a*c - b^2)^2*(80*a*c
^3 - 20*b^2*c^2))))/(2*c) + (4*c*(8*b^3 - 36*a*b*c))/(63*d^10*(4*a*c - b^2)^2*(80*a*c^3 - 20*b^2*c^2))))/(2*c)
- (4*c*(8*a*b^2 - 34*a^2*c))/(63*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 + (((3*a*c - b^2)/(360*c^2*d^10*(4*a*c - b^2)^4) + b^2/(1440*c^2*d^10*(4*a*c - b^2)^4))*(a + b*x + c
*x^2)^(1/2))/(b + 2*c*x) + (((9*a*c - 2*b^2)/(360*c^2*d^10*(4*a*c - b^2)^4) - b^2/(1440*c^2*d^10*(4*a*c - b^2)
^4))*(a + b*x + c*x^2)^(1/2))/(b + 2*c*x) + (((72*a*c - 17*b^2)/(2520*c^2*d^10*(4*a*c - b^2)^4) - b^2/(2520*c^
2*d^10*(4*a*c - b^2)^4))*(a + b*x + c*x^2)^(1/2))/(b + 2*c*x) + (((b*((b*(b^2/(630*d^10*(4*a*c - b^2)^5) - (52
*a*c^2 - 10*b^2*c)/(945*c*d^10*(4*a*c - b^2)^5)))/(2*c) - (12*b^3 - 52*a*b*c)/(945*c*d^10*(4*a*c - b^2)^5)))/(
2*c) + (12*a*b^2 - 50*a^2*c)/(945*c*d^10*(4*a*c - b^2)^5))*(a + b*x + c*x^2)^(1/2))/(b + 2*c*x) + (((b*((b*((b
*((32*c^3*(9*a*c - b^2))/(63*d^10*(4*a*c - b^2)^2*(96*a*c^3 - 24*b^2*c^2)) - (16*b^2*c^3)/(63*d^10*(4*a*c - b^
2)^2*(96*a*c^3 - 24*b^2*c^2))))/(2*c) - (8*b*c^2*(54*a*c - 11*b^2))/(63*d^10*(4*a*c - b^2)^2*(96*a*c^3 - 24*b^
2*c^2))))/(2*c) + (4*c*(68*a^2*c^2 - 8*b^4 + 20*a*b^2*c))/(63*d^10*(4*a*c - b^2)^2*(96*a*c^3 - 24*b^2*c^2))))/
(2*c) - (8*a*b*c*(17*a*c - 4*b^2))/(63*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*((2*c*(44*a*c^2 - 5*b^2*c))/(9*d^10*(4*a*c - b^2)*(112*a*c^3 - 28*b^2*c^2)) - (2*b^2
*c^2)/(3*d^10*(4*a*c - b^2)*(112*a*c^3 - 28*b^2*c^2))))/(2*c) + (2*c*(9*b^3 - 44*a*b*c))/(9*d^10*(4*a*c - b^2)
*(112*a*c^3 - 28*b^2*c^2))))/(2*c) - (2*c*(9*a*b^2 - 40*a^2*c))/(9*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*((2*c*(92*a*c^2 - 17*b^2*c))/(315*d^10*(4*a*c - b^2)^3*(4
8*a*c^3 - 12*b^2*c^2)) - (2*b^2*c^2)/(105*d^10*(4*a*c - b^2)^3*(48*a*c^3 - 12*b^2*c^2))))/(2*c) + (2*c*(21*b^3
- 92*a*b*c))/(315*d^10*(4*a*c - b^2)^3*(48*a*c^3 - 12*b^2*c^2))))/(2*c) - (2*c*(21*a*b^2 - 88*a^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*((4*a*c^3 + 5*b
^2*c^2)/(90*d^10*(4*a*c - b^2)^3*(48*a*c^3 - 12*b^2*c^2)) - (b^2*c^2)/(30*d^10*(4*a*c - b^2)^3*(48*a*c^3 - 12*
b^2*c^2))))/(2*c) - (b*c*(4*a*c + b^2))/(90*d^10*(4*a*c - b^2)^3*(48*a*c^3 - 12*b^2*c^2))))/(2*c) - (3*b^4 + 4
8*a^2*c^2 - 25*a*b^2*c)/(90*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*((2*c*(2*a*c - b^2))/(15*d^10*(4*a*c - b^2)^2*(64*a*c^3 - 16*b^2*c^2)) + (2*b^2*c)/(45*d^10*(4*a*c
- b^2)^2*(64*a*c^3 - 16*b^2*c^2))))/(2*c) + (2*(b^3 - 3*a*b*c))/(45*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*((2*c*(184*a*c^3 - 26*b^2*c^2))/(315*d^10*(4*a*c -
b^2)^3*(64*a*c^3 - 16*b^2*c^2)) - (16*b^2*c^3)/(315*d^10*(4*a*c - b^2)^3*(64*a*c^3 - 16*b^2*c^2))))/(2*c) + (2
*c*(59*b^3*c - 276*a*b*c^2))/(315*d^10*(4*a*c - b^2)^3*(64*a*c^3 - 16*b^2*c^2))))/(2*c) + (2*c*(176*a^2*c^2 -
21*b^4 + 50*a*b^2*c))/(315*d^10*(4*a*c - b^2)^3*(64*a*c^3 - 16*b^2*c^2))))/(2*c) + (2*c*(21*a*b^3 - 88*a^2*b*c
))/(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*((4*c
^2*(4*a*c + 2*b^2))/(d^10*(144*a*c^3 - 36*b^2*c^2)) - (6*b^2*c^2)/(d^10*(144*a*c^3 - 36*b^2*c^2))))/(2*c) - (1
6*a*b*c^2)/(d^10*(144*a*c^3 - 36*b^2*c^2))))/(2...

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