3.2016 \(\int (d+e x)^{7/2} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx\)

Optimal. Leaf size=295 \[ \frac{256 \left (c d^2-a e^2\right )^4 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3465 c^5 d^5 (d+e x)^{3/2}}+\frac{128 \left (c d^2-a e^2\right )^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{1155 c^4 d^4 \sqrt{d+e x}}+\frac{32 \sqrt{d+e x} \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{231 c^3 d^3}+\frac{16 (d+e x)^{3/2} \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{99 c^2 d^2}+\frac{2 (d+e x)^{5/2} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{11 c d} \]

[Out]

(256*(c*d^2 - a*e^2)^4*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(3465*c^5*
d^5*(d + e*x)^(3/2)) + (128*(c*d^2 - a*e^2)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e
*x^2)^(3/2))/(1155*c^4*d^4*Sqrt[d + e*x]) + (32*(c*d^2 - a*e^2)^2*Sqrt[d + e*x]*
(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(231*c^3*d^3) + (16*(c*d^2 - a*e^
2)*(d + e*x)^(3/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(99*c^2*d^2) +
(2*(d + e*x)^(5/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(11*c*d)

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Rubi [A]  time = 0.71276, antiderivative size = 295, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.051 \[ \frac{256 \left (c d^2-a e^2\right )^4 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3465 c^5 d^5 (d+e x)^{3/2}}+\frac{128 \left (c d^2-a e^2\right )^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{1155 c^4 d^4 \sqrt{d+e x}}+\frac{32 \sqrt{d+e x} \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{231 c^3 d^3}+\frac{16 (d+e x)^{3/2} \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{99 c^2 d^2}+\frac{2 (d+e x)^{5/2} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{11 c d} \]

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]

(256*(c*d^2 - a*e^2)^4*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(3465*c^5*
d^5*(d + e*x)^(3/2)) + (128*(c*d^2 - a*e^2)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e
*x^2)^(3/2))/(1155*c^4*d^4*Sqrt[d + e*x]) + (32*(c*d^2 - a*e^2)^2*Sqrt[d + e*x]*
(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(231*c^3*d^3) + (16*(c*d^2 - a*e^
2)*(d + e*x)^(3/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(99*c^2*d^2) +
(2*(d + e*x)^(5/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(11*c*d)

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Rubi in Sympy [A]  time = 102.566, size = 279, normalized size = 0.95 \[ \frac{2 \left (d + e x\right )^{\frac{5}{2}} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}}{11 c d} - \frac{16 \left (d + e x\right )^{\frac{3}{2}} \left (a e^{2} - c d^{2}\right ) \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}}{99 c^{2} d^{2}} + \frac{32 \sqrt{d + e x} \left (a e^{2} - c d^{2}\right )^{2} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}}{231 c^{3} d^{3}} - \frac{128 \left (a e^{2} - c d^{2}\right )^{3} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}}{1155 c^{4} d^{4} \sqrt{d + e x}} + \frac{256 \left (a e^{2} - c d^{2}\right )^{4} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}}{3465 c^{5} d^{5} \left (d + e x\right )^{\frac{3}{2}}} \]

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)*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))**(3/2)/(11*c*d) -
16*(d + e*x)**(3/2)*(a*e**2 - c*d**2)*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))
**(3/2)/(99*c**2*d**2) + 32*sqrt(d + e*x)*(a*e**2 - c*d**2)**2*(a*d*e + c*d*e*x*
*2 + x*(a*e**2 + c*d**2))**(3/2)/(231*c**3*d**3) - 128*(a*e**2 - c*d**2)**3*(a*d
*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))**(3/2)/(1155*c**4*d**4*sqrt(d + e*x)) + 2
56*(a*e**2 - c*d**2)**4*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))**(3/2)/(3465*
c**5*d**5*(d + e*x)**(3/2))

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Mathematica [A]  time = 0.198712, size = 239, normalized size = 0.81 \[ \frac{2 \sqrt{(d+e x) (a e+c d x)} \left (128 a^5 e^9-64 a^4 c d e^7 (11 d+e x)+16 a^3 c^2 d^2 e^5 \left (99 d^2+22 d e x+3 e^2 x^2\right )-8 a^2 c^3 d^3 e^3 \left (231 d^3+99 d^2 e x+33 d e^2 x^2+5 e^3 x^3\right )+a c^4 d^4 e \left (1155 d^4+924 d^3 e x+594 d^2 e^2 x^2+220 d e^3 x^3+35 e^4 x^4\right )+c^5 d^5 x \left (1155 d^4+2772 d^3 e x+2970 d^2 e^2 x^2+1540 d e^3 x^3+315 e^4 x^4\right )\right )}{3465 c^5 d^5 \sqrt{d+e x}} \]

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)]*(128*a^5*e^9 - 64*a^4*c*d*e^7*(11*d + e*x) + 16
*a^3*c^2*d^2*e^5*(99*d^2 + 22*d*e*x + 3*e^2*x^2) - 8*a^2*c^3*d^3*e^3*(231*d^3 +
99*d^2*e*x + 33*d*e^2*x^2 + 5*e^3*x^3) + a*c^4*d^4*e*(1155*d^4 + 924*d^3*e*x + 5
94*d^2*e^2*x^2 + 220*d*e^3*x^3 + 35*e^4*x^4) + c^5*d^5*x*(1155*d^4 + 2772*d^3*e*
x + 2970*d^2*e^2*x^2 + 1540*d*e^3*x^3 + 315*e^4*x^4)))/(3465*c^5*d^5*Sqrt[d + e*
x])

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Maple [A]  time = 0.01, size = 243, normalized size = 0.8 \[{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( 315\,{x}^{4}{c}^{4}{d}^{4}{e}^{4}-280\,{x}^{3}a{c}^{3}{d}^{3}{e}^{5}+1540\,{x}^{3}{c}^{4}{d}^{5}{e}^{3}+240\,{x}^{2}{a}^{2}{c}^{2}{d}^{2}{e}^{6}-1320\,{x}^{2}a{c}^{3}{d}^{4}{e}^{4}+2970\,{x}^{2}{c}^{4}{d}^{6}{e}^{2}-192\,x{a}^{3}cd{e}^{7}+1056\,x{a}^{2}{c}^{2}{d}^{3}{e}^{5}-2376\,xa{c}^{3}{d}^{5}{e}^{3}+2772\,{c}^{4}{d}^{7}ex+128\,{a}^{4}{e}^{8}-704\,{a}^{3}c{d}^{2}{e}^{6}+1584\,{a}^{2}{c}^{2}{d}^{4}{e}^{4}-1848\,a{c}^{3}{d}^{6}{e}^{2}+1155\,{c}^{4}{d}^{8} \right ) }{3465\,{c}^{5}{d}^{5}}\sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+aed}{\frac{1}{\sqrt{ex+d}}}} \]

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]

2/3465*(c*d*x+a*e)*(315*c^4*d^4*e^4*x^4-280*a*c^3*d^3*e^5*x^3+1540*c^4*d^5*e^3*x
^3+240*a^2*c^2*d^2*e^6*x^2-1320*a*c^3*d^4*e^4*x^2+2970*c^4*d^6*e^2*x^2-192*a^3*c
*d*e^7*x+1056*a^2*c^2*d^3*e^5*x-2376*a*c^3*d^5*e^3*x+2772*c^4*d^7*e*x+128*a^4*e^
8-704*a^3*c*d^2*e^6+1584*a^2*c^2*d^4*e^4-1848*a*c^3*d^6*e^2+1155*c^4*d^8)*(c*d*e
*x^2+a*e^2*x+c*d^2*x+a*d*e)^(1/2)/c^5/d^5/(e*x+d)^(1/2)

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Maxima [A]  time = 0.737935, size = 400, normalized size = 1.36 \[ \frac{2 \,{\left (315 \, c^{5} d^{5} e^{4} x^{5} + 1155 \, a c^{4} d^{8} e - 1848 \, a^{2} c^{3} d^{6} e^{3} + 1584 \, a^{3} c^{2} d^{4} e^{5} - 704 \, a^{4} c d^{2} e^{7} + 128 \, a^{5} e^{9} + 35 \,{\left (44 \, c^{5} d^{6} e^{3} + a c^{4} d^{4} e^{5}\right )} x^{4} + 10 \,{\left (297 \, c^{5} d^{7} e^{2} + 22 \, a c^{4} d^{5} e^{4} - 4 \, a^{2} c^{3} d^{3} e^{6}\right )} x^{3} + 6 \,{\left (462 \, c^{5} d^{8} e + 99 \, a c^{4} d^{6} e^{3} - 44 \, a^{2} c^{3} d^{4} e^{5} + 8 \, a^{3} c^{2} d^{2} e^{7}\right )} x^{2} +{\left (1155 \, c^{5} d^{9} + 924 \, a c^{4} d^{7} e^{2} - 792 \, a^{2} c^{3} d^{5} e^{4} + 352 \, a^{3} c^{2} d^{3} e^{6} - 64 \, a^{4} c d e^{8}\right )} x\right )} \sqrt{c d x + a e}{\left (e x + d\right )}}{3465 \,{\left (c^{5} d^{5} e x + c^{5} d^{6}\right )}} \]

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)^(7/2),x, algorithm="maxima")

[Out]

2/3465*(315*c^5*d^5*e^4*x^5 + 1155*a*c^4*d^8*e - 1848*a^2*c^3*d^6*e^3 + 1584*a^3
*c^2*d^4*e^5 - 704*a^4*c*d^2*e^7 + 128*a^5*e^9 + 35*(44*c^5*d^6*e^3 + a*c^4*d^4*
e^5)*x^4 + 10*(297*c^5*d^7*e^2 + 22*a*c^4*d^5*e^4 - 4*a^2*c^3*d^3*e^6)*x^3 + 6*(
462*c^5*d^8*e + 99*a*c^4*d^6*e^3 - 44*a^2*c^3*d^4*e^5 + 8*a^3*c^2*d^2*e^7)*x^2 +
(1155*c^5*d^9 + 924*a*c^4*d^7*e^2 - 792*a^2*c^3*d^5*e^4 + 352*a^3*c^2*d^3*e^6 -
64*a^4*c*d*e^8)*x)*sqrt(c*d*x + a*e)*(e*x + d)/(c^5*d^5*e*x + c^5*d^6)

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Fricas [A]  time = 0.241849, size = 632, normalized size = 2.14 \[ \frac{2 \,{\left (315 \, c^{6} d^{6} e^{5} x^{7} + 1155 \, a^{2} c^{4} d^{9} e^{2} - 1848 \, a^{3} c^{3} d^{7} e^{4} + 1584 \, a^{4} c^{2} d^{5} e^{6} - 704 \, a^{5} c d^{3} e^{8} + 128 \, a^{6} d e^{10} + 35 \,{\left (53 \, c^{6} d^{7} e^{4} + 10 \, a c^{5} d^{5} e^{6}\right )} x^{6} + 5 \,{\left (902 \, c^{6} d^{8} e^{3} + 422 \, a c^{5} d^{6} e^{5} - a^{2} c^{4} d^{4} e^{7}\right )} x^{5} +{\left (5742 \, c^{6} d^{9} e^{2} + 5324 \, a c^{5} d^{7} e^{4} - 49 \, a^{2} c^{4} d^{5} e^{6} + 8 \, a^{3} c^{3} d^{3} e^{8}\right )} x^{4} +{\left (3927 \, c^{6} d^{10} e + 7260 \, a c^{5} d^{8} e^{3} - 242 \, a^{2} c^{4} d^{6} e^{5} + 96 \, a^{3} c^{3} d^{4} e^{7} - 16 \, a^{4} c^{2} d^{2} e^{9}\right )} x^{3} +{\left (1155 \, c^{6} d^{11} + 6006 \, a c^{5} d^{9} e^{2} - 1122 \, a^{2} c^{4} d^{7} e^{4} + 880 \, a^{3} c^{3} d^{5} e^{6} - 368 \, a^{4} c^{2} d^{3} e^{8} + 64 \, a^{5} c d e^{10}\right )} x^{2} +{\left (2310 \, a c^{5} d^{10} e + 231 \, a^{2} c^{4} d^{8} e^{3} - 1056 \, a^{3} c^{3} d^{6} e^{5} + 1232 \, a^{4} c^{2} d^{4} e^{7} - 640 \, a^{5} c d^{2} e^{9} + 128 \, a^{6} e^{11}\right )} x\right )}}{3465 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{e x + d} c^{5} d^{5}} \]

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)^(7/2),x, algorithm="fricas")

[Out]

2/3465*(315*c^6*d^6*e^5*x^7 + 1155*a^2*c^4*d^9*e^2 - 1848*a^3*c^3*d^7*e^4 + 1584
*a^4*c^2*d^5*e^6 - 704*a^5*c*d^3*e^8 + 128*a^6*d*e^10 + 35*(53*c^6*d^7*e^4 + 10*
a*c^5*d^5*e^6)*x^6 + 5*(902*c^6*d^8*e^3 + 422*a*c^5*d^6*e^5 - a^2*c^4*d^4*e^7)*x
^5 + (5742*c^6*d^9*e^2 + 5324*a*c^5*d^7*e^4 - 49*a^2*c^4*d^5*e^6 + 8*a^3*c^3*d^3
*e^8)*x^4 + (3927*c^6*d^10*e + 7260*a*c^5*d^8*e^3 - 242*a^2*c^4*d^6*e^5 + 96*a^3
*c^3*d^4*e^7 - 16*a^4*c^2*d^2*e^9)*x^3 + (1155*c^6*d^11 + 6006*a*c^5*d^9*e^2 - 1
122*a^2*c^4*d^7*e^4 + 880*a^3*c^3*d^5*e^6 - 368*a^4*c^2*d^3*e^8 + 64*a^5*c*d*e^1
0)*x^2 + (2310*a*c^5*d^10*e + 231*a^2*c^4*d^8*e^3 - 1056*a^3*c^3*d^6*e^5 + 1232*
a^4*c^2*d^4*e^7 - 640*a^5*c*d^2*e^9 + 128*a^6*e^11)*x)/(sqrt(c*d*e*x^2 + a*d*e +
(c*d^2 + a*e^2)*x)*sqrt(e*x + d)*c^5*d^5)

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

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)^(7/2),x, algorithm="giac")

[Out]

Timed out