3.2046 \(\int \frac{(d+e x)^{7/2}}{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx\)

Optimal. Leaf size=233 \[ \frac{32 \left (c d^2-a e^2\right )^3 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{35 c^4 d^4 \sqrt{d+e x}}+\frac{16 \sqrt{d+e x} \left (c d^2-a e^2\right )^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{35 c^3 d^3}+\frac{12 (d+e x)^{3/2} \left (c d^2-a e^2\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{35 c^2 d^2}+\frac{2 (d+e x)^{5/2} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{7 c d} \]

[Out]

(32*(c*d^2 - a*e^2)^3*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(35*c^4*d^4*S
qrt[d + e*x]) + (16*(c*d^2 - a*e^2)^2*Sqrt[d + e*x]*Sqrt[a*d*e + (c*d^2 + a*e^2)
*x + c*d*e*x^2])/(35*c^3*d^3) + (12*(c*d^2 - a*e^2)*(d + e*x)^(3/2)*Sqrt[a*d*e +
(c*d^2 + a*e^2)*x + c*d*e*x^2])/(35*c^2*d^2) + (2*(d + e*x)^(5/2)*Sqrt[a*d*e +
(c*d^2 + a*e^2)*x + c*d*e*x^2])/(7*c*d)

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Rubi [A]  time = 0.511376, antiderivative size = 233, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.051 \[ \frac{32 \left (c d^2-a e^2\right )^3 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{35 c^4 d^4 \sqrt{d+e x}}+\frac{16 \sqrt{d+e x} \left (c d^2-a e^2\right )^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{35 c^3 d^3}+\frac{12 (d+e x)^{3/2} \left (c d^2-a e^2\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{35 c^2 d^2}+\frac{2 (d+e x)^{5/2} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{7 c d} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^(7/2)/Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2],x]

[Out]

(32*(c*d^2 - a*e^2)^3*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(35*c^4*d^4*S
qrt[d + e*x]) + (16*(c*d^2 - a*e^2)^2*Sqrt[d + e*x]*Sqrt[a*d*e + (c*d^2 + a*e^2)
*x + c*d*e*x^2])/(35*c^3*d^3) + (12*(c*d^2 - a*e^2)*(d + e*x)^(3/2)*Sqrt[a*d*e +
(c*d^2 + a*e^2)*x + c*d*e*x^2])/(35*c^2*d^2) + (2*(d + e*x)^(5/2)*Sqrt[a*d*e +
(c*d^2 + a*e^2)*x + c*d*e*x^2])/(7*c*d)

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Rubi in Sympy [A]  time = 77.6819, size = 219, normalized size = 0.94 \[ \frac{2 \left (d + e x\right )^{\frac{5}{2}} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{7 c d} - \frac{12 \left (d + e x\right )^{\frac{3}{2}} \left (a e^{2} - c d^{2}\right ) \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{35 c^{2} d^{2}} + \frac{16 \sqrt{d + e x} \left (a e^{2} - c d^{2}\right )^{2} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{35 c^{3} d^{3}} - \frac{32 \left (a e^{2} - c d^{2}\right )^{3} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{35 c^{4} d^{4} \sqrt{d + e x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**(7/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/2),x)

[Out]

2*(d + e*x)**(5/2)*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))/(7*c*d) - 12*(
d + e*x)**(3/2)*(a*e**2 - c*d**2)*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))
/(35*c**2*d**2) + 16*sqrt(d + e*x)*(a*e**2 - c*d**2)**2*sqrt(a*d*e + c*d*e*x**2
+ x*(a*e**2 + c*d**2))/(35*c**3*d**3) - 32*(a*e**2 - c*d**2)**3*sqrt(a*d*e + c*d
*e*x**2 + x*(a*e**2 + c*d**2))/(35*c**4*d**4*sqrt(d + e*x))

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Mathematica [A]  time = 0.179186, size = 131, normalized size = 0.56 \[ \frac{2 \sqrt{(d+e x) (a e+c d x)} \left (-16 a^3 e^6+8 a^2 c d e^4 (7 d+e x)-2 a c^2 d^2 e^2 \left (35 d^2+14 d e x+3 e^2 x^2\right )+c^3 d^3 \left (35 d^3+35 d^2 e x+21 d e^2 x^2+5 e^3 x^3\right )\right )}{35 c^4 d^4 \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^(7/2)/Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2],x]

[Out]

(2*Sqrt[(a*e + c*d*x)*(d + e*x)]*(-16*a^3*e^6 + 8*a^2*c*d*e^4*(7*d + e*x) - 2*a*
c^2*d^2*e^2*(35*d^2 + 14*d*e*x + 3*e^2*x^2) + c^3*d^3*(35*d^3 + 35*d^2*e*x + 21*
d*e^2*x^2 + 5*e^3*x^3)))/(35*c^4*d^4*Sqrt[d + e*x])

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Maple [A]  time = 0.009, size = 168, normalized size = 0.7 \[ -{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( -5\,{x}^{3}{c}^{3}{d}^{3}{e}^{3}+6\,{x}^{2}a{c}^{2}{d}^{2}{e}^{4}-21\,{x}^{2}{c}^{3}{d}^{4}{e}^{2}-8\,x{a}^{2}cd{e}^{5}+28\,xa{c}^{2}{d}^{3}{e}^{3}-35\,{c}^{3}{d}^{5}ex+16\,{a}^{3}{e}^{6}-56\,{a}^{2}c{d}^{2}{e}^{4}+70\,{c}^{2}{d}^{4}a{e}^{2}-35\,{c}^{3}{d}^{6} \right ) }{35\,{c}^{4}{d}^{4}}\sqrt{ex+d}{\frac{1}{\sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+aed}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^(7/2)/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x)

[Out]

-2/35*(c*d*x+a*e)*(-5*c^3*d^3*e^3*x^3+6*a*c^2*d^2*e^4*x^2-21*c^3*d^4*e^2*x^2-8*a
^2*c*d*e^5*x+28*a*c^2*d^3*e^3*x-35*c^3*d^5*e*x+16*a^3*e^6-56*a^2*c*d^2*e^4+70*a*
c^2*d^4*e^2-35*c^3*d^6)*(e*x+d)^(1/2)/c^4/d^4/(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^
(1/2)

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Maxima [A]  time = 0.819636, size = 259, normalized size = 1.11 \[ \frac{2 \,{\left (5 \, c^{4} d^{4} e^{3} x^{4} + 35 \, a c^{3} d^{6} e - 70 \, a^{2} c^{2} d^{4} e^{3} + 56 \, a^{3} c d^{2} e^{5} - 16 \, a^{4} e^{7} +{\left (21 \, c^{4} d^{5} e^{2} - a c^{3} d^{3} e^{4}\right )} x^{3} +{\left (35 \, c^{4} d^{6} e - 7 \, a c^{3} d^{4} e^{3} + 2 \, a^{2} c^{2} d^{2} e^{5}\right )} x^{2} +{\left (35 \, c^{4} d^{7} - 35 \, a c^{3} d^{5} e^{2} + 28 \, a^{2} c^{2} d^{3} e^{4} - 8 \, a^{3} c d e^{6}\right )} x\right )}}{35 \, \sqrt{c d x + a e} c^{4} d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(7/2)/sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x),x, algorithm="maxima")

[Out]

2/35*(5*c^4*d^4*e^3*x^4 + 35*a*c^3*d^6*e - 70*a^2*c^2*d^4*e^3 + 56*a^3*c*d^2*e^5
- 16*a^4*e^7 + (21*c^4*d^5*e^2 - a*c^3*d^3*e^4)*x^3 + (35*c^4*d^6*e - 7*a*c^3*d
^4*e^3 + 2*a^2*c^2*d^2*e^5)*x^2 + (35*c^4*d^7 - 35*a*c^3*d^5*e^2 + 28*a^2*c^2*d^
3*e^4 - 8*a^3*c*d*e^6)*x)/(sqrt(c*d*x + a*e)*c^4*d^4)

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Fricas [A]  time = 0.216413, size = 362, normalized size = 1.55 \[ \frac{2 \,{\left (5 \, c^{4} d^{4} e^{4} x^{5} + 35 \, a c^{3} d^{7} e - 70 \, a^{2} c^{2} d^{5} e^{3} + 56 \, a^{3} c d^{3} e^{5} - 16 \, a^{4} d e^{7} +{\left (26 \, c^{4} d^{5} e^{3} - a c^{3} d^{3} e^{5}\right )} x^{4} + 2 \,{\left (28 \, c^{4} d^{6} e^{2} - 4 \, a c^{3} d^{4} e^{4} + a^{2} c^{2} d^{2} e^{6}\right )} x^{3} + 2 \,{\left (35 \, c^{4} d^{7} e - 21 \, a c^{3} d^{5} e^{3} + 15 \, a^{2} c^{2} d^{3} e^{5} - 4 \, a^{3} c d e^{7}\right )} x^{2} +{\left (35 \, c^{4} d^{8} - 42 \, a^{2} c^{2} d^{4} e^{4} + 48 \, a^{3} c d^{2} e^{6} - 16 \, a^{4} e^{8}\right )} x\right )}}{35 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{e x + d} c^{4} d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(7/2)/sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x),x, algorithm="fricas")

[Out]

2/35*(5*c^4*d^4*e^4*x^5 + 35*a*c^3*d^7*e - 70*a^2*c^2*d^5*e^3 + 56*a^3*c*d^3*e^5
- 16*a^4*d*e^7 + (26*c^4*d^5*e^3 - a*c^3*d^3*e^5)*x^4 + 2*(28*c^4*d^6*e^2 - 4*a
*c^3*d^4*e^4 + a^2*c^2*d^2*e^6)*x^3 + 2*(35*c^4*d^7*e - 21*a*c^3*d^5*e^3 + 15*a^
2*c^2*d^3*e^5 - 4*a^3*c*d*e^7)*x^2 + (35*c^4*d^8 - 42*a^2*c^2*d^4*e^4 + 48*a^3*c
*d^2*e^6 - 16*a^4*e^8)*x)/(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x
+ d)*c^4*d^4)

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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((e*x+d)**(7/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (e x + d\right )}^{\frac{7}{2}}}{\sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(7/2)/sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x),x, algorithm="giac")

[Out]

integrate((e*x + d)^(7/2)/sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x), x)