3.1220 \(\int \frac{\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^{10}} \, dx\)

Optimal. Leaf size=79 \[ \frac{4 \left (a+b x+c x^2\right )^{7/2}}{63 d^{10} \left (b^2-4 a c\right )^2 (b+2 c x)^7}+\frac{2 \left (a+b x+c x^2\right )^{7/2}}{9 d^{10} \left (b^2-4 a c\right ) (b+2 c x)^9} \]

[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)

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Rubi [A]  time = 0.111718, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{4 \left (a+b x+c x^2\right )^{7/2}}{63 d^{10} \left (b^2-4 a c\right )^2 (b+2 c x)^7}+\frac{2 \left (a+b x+c x^2\right )^{7/2}}{9 d^{10} \left (b^2-4 a c\right ) (b+2 c x)^9} \]

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)

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Rubi in Sympy [A]  time = 27.2821, size = 75, normalized size = 0.95 \[ \frac{4 \left (a + b x + c x^{2}\right )^{\frac{7}{2}}}{63 d^{10} \left (b + 2 c x\right )^{7} \left (- 4 a c + b^{2}\right )^{2}} + \frac{2 \left (a + b x + c x^{2}\right )^{\frac{7}{2}}}{9 d^{10} \left (b + 2 c x\right )^{9} \left (- 4 a c + b^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+b*x+a)**(5/2)/(2*c*d*x+b*d)**10,x)

[Out]

4*(a + b*x + c*x**2)**(7/2)/(63*d**10*(b + 2*c*x)**7*(-4*a*c + b**2)**2) + 2*(a
+ b*x + c*x**2)**(7/2)/(9*d**10*(b + 2*c*x)**9*(-4*a*c + b**2))

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Mathematica [A]  time = 0.325424, size = 62, normalized size = 0.78 \[ \frac{2 (a+x (b+c x))^{7/2} \left (4 c \left (2 c x^2-7 a\right )+9 b^2+8 b c x\right )}{63 d^{10} \left (b^2-4 a c\right )^2 (b+2 c x)^9} \]

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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Maple [A]  time = 0.01, size = 70, normalized size = 0.9 \[ -{\frac{-16\,{c}^{2}{x}^{2}-16\,bxc+56\,ac-18\,{b}^{2}}{63\, \left ( 2\,cx+b \right ) ^{9}{d}^{10} \left ( 16\,{a}^{2}{c}^{2}-8\,ac{b}^{2}+{b}^{4} \right ) } \left ( c{x}^{2}+bx+a \right ) ^{{\frac{7}{2}}}} \]

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]  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)^(5/2)/(2*c*d*x + b*d)^10,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 7.69072, size = 716, normalized size = 9.06 \[ \frac{2 \,{\left (8 \, c^{5} x^{8} + 32 \, b c^{4} x^{7} +{\left (57 \, b^{2} c^{3} - 4 \, a c^{4}\right )} x^{6} +{\left (59 \, b^{3} c^{2} - 12 \, a b c^{3}\right )} x^{5} + 9 \, a^{3} b^{2} - 28 \, a^{4} c + 5 \,{\left (7 \, b^{4} c + 3 \, a b^{2} c^{2} - 12 \, a^{2} c^{3}\right )} x^{4} +{\left (9 \, b^{5} + 50 \, a b^{3} c - 120 \, a^{2} b c^{2}\right )} x^{3} +{\left (27 \, a b^{4} - 33 \, a^{2} b^{2} c - 76 \, a^{3} c^{2}\right )} x^{2} +{\left (27 \, a^{2} b^{3} - 76 \, a^{3} b c\right )} x\right )} \sqrt{c x^{2} + b x + a}}{63 \,{\left (512 \,{\left (b^{4} c^{9} - 8 \, a b^{2} c^{10} + 16 \, a^{2} c^{11}\right )} d^{10} x^{9} + 2304 \,{\left (b^{5} c^{8} - 8 \, a b^{3} c^{9} + 16 \, a^{2} b c^{10}\right )} d^{10} x^{8} + 4608 \,{\left (b^{6} c^{7} - 8 \, a b^{4} c^{8} + 16 \, a^{2} b^{2} c^{9}\right )} d^{10} x^{7} + 5376 \,{\left (b^{7} c^{6} - 8 \, a b^{5} c^{7} + 16 \, a^{2} b^{3} c^{8}\right )} d^{10} x^{6} + 4032 \,{\left (b^{8} c^{5} - 8 \, a b^{6} c^{6} + 16 \, a^{2} b^{4} c^{7}\right )} d^{10} x^{5} + 2016 \,{\left (b^{9} c^{4} - 8 \, a b^{7} c^{5} + 16 \, a^{2} b^{5} c^{6}\right )} d^{10} x^{4} + 672 \,{\left (b^{10} c^{3} - 8 \, a b^{8} c^{4} + 16 \, a^{2} b^{6} c^{5}\right )} d^{10} x^{3} + 144 \,{\left (b^{11} c^{2} - 8 \, a b^{9} c^{3} + 16 \, a^{2} b^{7} c^{4}\right )} d^{10} x^{2} + 18 \,{\left (b^{12} c - 8 \, a b^{10} c^{2} + 16 \, a^{2} b^{8} c^{3}\right )} d^{10} x +{\left (b^{13} - 8 \, a b^{11} c + 16 \, a^{2} b^{9} c^{2}\right )} d^{10}\right )}} \]

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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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)**(5/2)/(2*c*d*x+b*d)**10,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 1.42125, size = 4, normalized size = 0.05 \[ \mathit{sage}_{0} x \]

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]

sage0*x