3.1906 \(\int \frac{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^6} \, dx\)

Optimal. Leaf size=231 \[ \frac{32 c^3 d^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{315 (d+e x)^3 \left (c d^2-a e^2\right )^4}+\frac{16 c^2 d^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{105 (d+e x)^4 \left (c d^2-a e^2\right )^3}+\frac{4 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{21 (d+e x)^5 \left (c d^2-a e^2\right )^2}+\frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{9 (d+e x)^6 \left (c d^2-a e^2\right )} \]

[Out]

(2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(9*(c*d^2 - a*e^2)*(d + e*x)^6
) + (4*c*d*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(21*(c*d^2 - a*e^2)^2*
(d + e*x)^5) + (16*c^2*d^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(105*(
c*d^2 - a*e^2)^3*(d + e*x)^4) + (32*c^3*d^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x
^2)^(3/2))/(315*(c*d^2 - a*e^2)^4*(d + e*x)^3)

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Rubi [A]  time = 0.394276, antiderivative size = 231, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.054 \[ \frac{32 c^3 d^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{315 (d+e x)^3 \left (c d^2-a e^2\right )^4}+\frac{16 c^2 d^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{105 (d+e x)^4 \left (c d^2-a e^2\right )^3}+\frac{4 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{21 (d+e x)^5 \left (c d^2-a e^2\right )^2}+\frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{9 (d+e x)^6 \left (c d^2-a e^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(d + e*x)^6,x]

[Out]

(2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(9*(c*d^2 - a*e^2)*(d + e*x)^6
) + (4*c*d*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(21*(c*d^2 - a*e^2)^2*
(d + e*x)^5) + (16*c^2*d^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(105*(
c*d^2 - a*e^2)^3*(d + e*x)^4) + (32*c^3*d^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x
^2)^(3/2))/(315*(c*d^2 - a*e^2)^4*(d + e*x)^3)

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Rubi in Sympy [A]  time = 79.7739, size = 218, normalized size = 0.94 \[ \frac{32 c^{3} d^{3} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}}{315 \left (d + e x\right )^{3} \left (a e^{2} - c d^{2}\right )^{4}} - \frac{16 c^{2} d^{2} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}}{105 \left (d + e x\right )^{4} \left (a e^{2} - c d^{2}\right )^{3}} + \frac{4 c d \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}}{21 \left (d + e x\right )^{5} \left (a e^{2} - c d^{2}\right )^{2}} - \frac{2 \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}}{9 \left (d + e x\right )^{6} \left (a e^{2} - c d^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/2)/(e*x+d)**6,x)

[Out]

32*c**3*d**3*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))**(3/2)/(315*(d + e*x)**3
*(a*e**2 - c*d**2)**4) - 16*c**2*d**2*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))
**(3/2)/(105*(d + e*x)**4*(a*e**2 - c*d**2)**3) + 4*c*d*(a*d*e + c*d*e*x**2 + x*
(a*e**2 + c*d**2))**(3/2)/(21*(d + e*x)**5*(a*e**2 - c*d**2)**2) - 2*(a*d*e + c*
d*e*x**2 + x*(a*e**2 + c*d**2))**(3/2)/(9*(d + e*x)**6*(a*e**2 - c*d**2))

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Mathematica [A]  time = 0.365493, size = 144, normalized size = 0.62 \[ \frac{2 \sqrt{(d+e x) (a e+c d x)} \left (\frac{16 c^4 d^4 (d+e x)^4}{\left (c d^2-a e^2\right )^4}+\frac{8 c^3 d^3 (d+e x)^3}{\left (c d^2-a e^2\right )^3}+\frac{6 c^2 d^2 (d+e x)^2}{\left (c d^2-a e^2\right )^2}+\frac{5 c d (d+e x)}{c d^2-a e^2}-35\right )}{315 e (d+e x)^5} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(d + e*x)^6,x]

[Out]

(2*Sqrt[(a*e + c*d*x)*(d + e*x)]*(-35 + (5*c*d*(d + e*x))/(c*d^2 - a*e^2) + (6*c
^2*d^2*(d + e*x)^2)/(c*d^2 - a*e^2)^2 + (8*c^3*d^3*(d + e*x)^3)/(c*d^2 - a*e^2)^
3 + (16*c^4*d^4*(d + e*x)^4)/(c*d^2 - a*e^2)^4))/(315*e*(d + e*x)^5)

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Maple [A]  time = 0.013, size = 217, normalized size = 0.9 \[ -{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( -16\,{x}^{3}{c}^{3}{d}^{3}{e}^{3}+24\,{x}^{2}a{c}^{2}{d}^{2}{e}^{4}-72\,{x}^{2}{c}^{3}{d}^{4}{e}^{2}-30\,x{a}^{2}cd{e}^{5}+108\,xa{c}^{2}{d}^{3}{e}^{3}-126\,{c}^{3}{d}^{5}ex+35\,{a}^{3}{e}^{6}-135\,{a}^{2}c{d}^{2}{e}^{4}+189\,{c}^{2}{d}^{4}a{e}^{2}-105\,{c}^{3}{d}^{6} \right ) }{315\, \left ( ex+d \right ) ^{5} \left ({a}^{4}{e}^{8}-4\,{a}^{3}c{d}^{2}{e}^{6}+6\,{a}^{2}{c}^{2}{d}^{4}{e}^{4}-4\,a{c}^{3}{d}^{6}{e}^{2}+{c}^{4}{d}^{8} \right ) }\sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+aed}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(e*x+d)^6,x)

[Out]

-2/315*(c*d*x+a*e)*(-16*c^3*d^3*e^3*x^3+24*a*c^2*d^2*e^4*x^2-72*c^3*d^4*e^2*x^2-
30*a^2*c*d*e^5*x+108*a*c^2*d^3*e^3*x-126*c^3*d^5*e*x+35*a^3*e^6-135*a^2*c*d^2*e^
4+189*a*c^2*d^4*e^2-105*c^3*d^6)*(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(1/2)/(e*x+d)
^5/(a^4*e^8-4*a^3*c*d^2*e^6+6*a^2*c^2*d^4*e^4-4*a*c^3*d^6*e^2+c^4*d^8)

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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(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)/(e*x + d)^6,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 4.39507, size = 772, normalized size = 3.34 \[ \frac{2 \,{\left (16 \, c^{4} d^{4} e^{3} x^{4} + 105 \, a c^{3} d^{6} e - 189 \, a^{2} c^{2} d^{4} e^{3} + 135 \, a^{3} c d^{2} e^{5} - 35 \, a^{4} e^{7} + 8 \,{\left (9 \, c^{4} d^{5} e^{2} - a c^{3} d^{3} e^{4}\right )} x^{3} + 6 \,{\left (21 \, c^{4} d^{6} e - 6 \, a c^{3} d^{4} e^{3} + a^{2} c^{2} d^{2} e^{5}\right )} x^{2} +{\left (105 \, c^{4} d^{7} - 63 \, a c^{3} d^{5} e^{2} + 27 \, a^{2} c^{2} d^{3} e^{4} - 5 \, a^{3} c d e^{6}\right )} x\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}}{315 \,{\left (c^{4} d^{13} - 4 \, a c^{3} d^{11} e^{2} + 6 \, a^{2} c^{2} d^{9} e^{4} - 4 \, a^{3} c d^{7} e^{6} + a^{4} d^{5} e^{8} +{\left (c^{4} d^{8} e^{5} - 4 \, a c^{3} d^{6} e^{7} + 6 \, a^{2} c^{2} d^{4} e^{9} - 4 \, a^{3} c d^{2} e^{11} + a^{4} e^{13}\right )} x^{5} + 5 \,{\left (c^{4} d^{9} e^{4} - 4 \, a c^{3} d^{7} e^{6} + 6 \, a^{2} c^{2} d^{5} e^{8} - 4 \, a^{3} c d^{3} e^{10} + a^{4} d e^{12}\right )} x^{4} + 10 \,{\left (c^{4} d^{10} e^{3} - 4 \, a c^{3} d^{8} e^{5} + 6 \, a^{2} c^{2} d^{6} e^{7} - 4 \, a^{3} c d^{4} e^{9} + a^{4} d^{2} e^{11}\right )} x^{3} + 10 \,{\left (c^{4} d^{11} e^{2} - 4 \, a c^{3} d^{9} e^{4} + 6 \, a^{2} c^{2} d^{7} e^{6} - 4 \, a^{3} c d^{5} e^{8} + a^{4} d^{3} e^{10}\right )} x^{2} + 5 \,{\left (c^{4} d^{12} e - 4 \, a c^{3} d^{10} e^{3} + 6 \, a^{2} c^{2} d^{8} e^{5} - 4 \, a^{3} c d^{6} e^{7} + a^{4} d^{4} e^{9}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)/(e*x + d)^6,x, algorithm="fricas")

[Out]

2/315*(16*c^4*d^4*e^3*x^4 + 105*a*c^3*d^6*e - 189*a^2*c^2*d^4*e^3 + 135*a^3*c*d^
2*e^5 - 35*a^4*e^7 + 8*(9*c^4*d^5*e^2 - a*c^3*d^3*e^4)*x^3 + 6*(21*c^4*d^6*e - 6
*a*c^3*d^4*e^3 + a^2*c^2*d^2*e^5)*x^2 + (105*c^4*d^7 - 63*a*c^3*d^5*e^2 + 27*a^2
*c^2*d^3*e^4 - 5*a^3*c*d*e^6)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)/(c^
4*d^13 - 4*a*c^3*d^11*e^2 + 6*a^2*c^2*d^9*e^4 - 4*a^3*c*d^7*e^6 + a^4*d^5*e^8 +
(c^4*d^8*e^5 - 4*a*c^3*d^6*e^7 + 6*a^2*c^2*d^4*e^9 - 4*a^3*c*d^2*e^11 + a^4*e^13
)*x^5 + 5*(c^4*d^9*e^4 - 4*a*c^3*d^7*e^6 + 6*a^2*c^2*d^5*e^8 - 4*a^3*c*d^3*e^10
+ a^4*d*e^12)*x^4 + 10*(c^4*d^10*e^3 - 4*a*c^3*d^8*e^5 + 6*a^2*c^2*d^6*e^7 - 4*a
^3*c*d^4*e^9 + a^4*d^2*e^11)*x^3 + 10*(c^4*d^11*e^2 - 4*a*c^3*d^9*e^4 + 6*a^2*c^
2*d^7*e^6 - 4*a^3*c*d^5*e^8 + a^4*d^3*e^10)*x^2 + 5*(c^4*d^12*e - 4*a*c^3*d^10*e
^3 + 6*a^2*c^2*d^8*e^5 - 4*a^3*c*d^6*e^7 + a^4*d^4*e^9)*x)

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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((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/2)/(e*x+d)**6,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 10.0651, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)/(e*x + d)^6,x, algorithm="giac")

[Out]

Done