3.1205 \(\int \frac{\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^{10}} \, dx\)

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

[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)

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Rubi [A]  time = 0.174811, antiderivative size = 118, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{16 \left (a+b x+c x^2\right )^{5/2}}{315 d^{10} \left (b^2-4 a c\right )^3 (b+2 c x)^5}+\frac{8 \left (a+b x+c x^2\right )^{5/2}}{63 d^{10} \left (b^2-4 a c\right )^2 (b+2 c x)^7}+\frac{2 \left (a+b x+c x^2\right )^{5/2}}{9 d^{10} \left (b^2-4 a c\right ) (b+2 c x)^9} \]

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)

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Rubi in Sympy [A]  time = 43.7846, size = 114, normalized size = 0.97 \[ \frac{16 \left (a + b x + c x^{2}\right )^{\frac{5}{2}}}{315 d^{10} \left (b + 2 c x\right )^{5} \left (- 4 a c + b^{2}\right )^{3}} + \frac{8 \left (a + b x + c x^{2}\right )^{\frac{5}{2}}}{63 d^{10} \left (b + 2 c x\right )^{7} \left (- 4 a c + b^{2}\right )^{2}} + \frac{2 \left (a + b x + c x^{2}\right )^{\frac{5}{2}}}{9 d^{10} \left (b + 2 c x\right )^{9} \left (- 4 a c + b^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.268224, size = 110, normalized size = 0.93 \[ \frac{2 (a+x (b+c x))^{5/2} \left (16 c^2 \left (35 a^2-20 a c x^2+8 c^2 x^4\right )+8 b^2 c \left (34 c x^2-45 a\right )+64 b c^2 x \left (4 c x^2-5 a\right )+63 b^4+144 b^3 c x\right )}{315 d^{10} \left (b^2-4 a c\right )^3 (b+2 c x)^9} \]

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 = 0.013, size = 133, normalized size = 1.1 \[ -{\frac{256\,{c}^{4}{x}^{4}+512\,b{c}^{3}{x}^{3}-640\,a{c}^{3}{x}^{2}+544\,{b}^{2}{c}^{2}{x}^{2}-640\,ab{c}^{2}x+288\,{b}^{3}cx+1120\,{a}^{2}{c}^{2}-720\,ac{b}^{2}+126\,{b}^{4}}{315\, \left ( 2\,cx+b \right ) ^{9}{d}^{10} \left ( 64\,{a}^{3}{c}^{3}-48\,{a}^{2}{b}^{2}{c}^{2}+12\,a{b}^{4}c-{b}^{6} \right ) } \left ( c{x}^{2}+bx+a \right ) ^{{\frac{5}{2}}}} \]

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)

[Out]

-2/315*(128*c^4*x^4+256*b*c^3*x^3-320*a*c^3*x^2+272*b^2*c^2*x^2-320*a*b*c^2*x+14
4*b^3*c*x+560*a^2*c^2-360*a*b^2*c+63*b^4)*(c*x^2+b*x+a)^(5/2)/(2*c*x+b)^9/d^10/(
64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

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

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Fricas [A]  time = 7.50432, size = 913, normalized size = 7.74 \[ \frac{2 \,{\left (128 \, c^{6} x^{8} + 512 \, b c^{5} x^{7} + 16 \,{\left (57 \, b^{2} c^{4} - 4 \, a c^{5}\right )} x^{6} + 63 \, a^{2} b^{4} - 360 \, a^{3} b^{2} c + 560 \, a^{4} c^{2} + 16 \,{\left (59 \, b^{3} c^{3} - 12 \, a b c^{4}\right )} x^{5} +{\left (623 \, b^{4} c^{2} - 264 \, a b^{2} c^{3} + 48 \, a^{2} c^{4}\right )} x^{4} + 2 \,{\left (135 \, b^{5} c - 104 \, a b^{3} c^{2} + 48 \, a^{2} b c^{3}\right )} x^{3} +{\left (63 \, b^{6} + 54 \, a b^{4} c - 528 \, a^{2} b^{2} c^{2} + 800 \, a^{3} c^{3}\right )} x^{2} + 2 \,{\left (63 \, a b^{5} - 288 \, a^{2} b^{3} c + 400 \, a^{3} b c^{2}\right )} x\right )} \sqrt{c x^{2} + b x + a}}{315 \,{\left (512 \,{\left (b^{6} c^{9} - 12 \, a b^{4} c^{10} + 48 \, a^{2} b^{2} c^{11} - 64 \, a^{3} c^{12}\right )} d^{10} x^{9} + 2304 \,{\left (b^{7} c^{8} - 12 \, a b^{5} c^{9} + 48 \, a^{2} b^{3} c^{10} - 64 \, a^{3} b c^{11}\right )} d^{10} x^{8} + 4608 \,{\left (b^{8} c^{7} - 12 \, a b^{6} c^{8} + 48 \, a^{2} b^{4} c^{9} - 64 \, a^{3} b^{2} c^{10}\right )} d^{10} x^{7} + 5376 \,{\left (b^{9} c^{6} - 12 \, a b^{7} c^{7} + 48 \, a^{2} b^{5} c^{8} - 64 \, a^{3} b^{3} c^{9}\right )} d^{10} x^{6} + 4032 \,{\left (b^{10} c^{5} - 12 \, a b^{8} c^{6} + 48 \, a^{2} b^{6} c^{7} - 64 \, a^{3} b^{4} c^{8}\right )} d^{10} x^{5} + 2016 \,{\left (b^{11} c^{4} - 12 \, a b^{9} c^{5} + 48 \, a^{2} b^{7} c^{6} - 64 \, a^{3} b^{5} c^{7}\right )} d^{10} x^{4} + 672 \,{\left (b^{12} c^{3} - 12 \, a b^{10} c^{4} + 48 \, a^{2} b^{8} c^{5} - 64 \, a^{3} b^{6} c^{6}\right )} d^{10} x^{3} + 144 \,{\left (b^{13} c^{2} - 12 \, a b^{11} c^{3} + 48 \, a^{2} b^{9} c^{4} - 64 \, a^{3} b^{7} c^{5}\right )} d^{10} x^{2} + 18 \,{\left (b^{14} c - 12 \, a b^{12} c^{2} + 48 \, a^{2} b^{10} c^{3} - 64 \, a^{3} b^{8} c^{4}\right )} d^{10} x +{\left (b^{15} - 12 \, a b^{13} c + 48 \, a^{2} b^{11} c^{2} - 64 \, a^{3} b^{9} c^{3}\right )} d^{10}\right )}} \]

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(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

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]

Timed out

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GIAC/XCAS [A]  time = 1.02451, size = 4, normalized size = 0.03 \[ \mathit{sage}_{0} x \]

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]

sage0*x