3.2017 \(\int (d+e x)^{5/2} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx\)

Optimal. Leaf size=233 \[ \frac{32 \left (c d^2-a e^2\right )^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{315 c^4 d^4 (d+e x)^{3/2}}+\frac{16 \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{105 c^3 d^3 \sqrt{d+e x}}+\frac{4 \sqrt{d+e x} \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{21 c^2 d^2}+\frac{2 (d+e x)^{3/2} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{9 c d} \]

[Out]

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

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Rubi [A]  time = 0.497592, 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 \[ \frac{32 \left (c d^2-a e^2\right )^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{315 c^4 d^4 (d+e x)^{3/2}}+\frac{16 \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{105 c^3 d^3 \sqrt{d+e x}}+\frac{4 \sqrt{d+e x} \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{21 c^2 d^2}+\frac{2 (d+e x)^{3/2} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{9 c d} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^(5/2)*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*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(315*c^4*d^
4*(d + e*x)^(3/2)) + (16*(c*d^2 - a*e^2)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^
2)^(3/2))/(105*c^3*d^3*Sqrt[d + e*x]) + (4*(c*d^2 - a*e^2)*Sqrt[d + e*x]*(a*d*e
+ (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(21*c^2*d^2) + (2*(d + e*x)^(3/2)*(a*d*e
+ (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(9*c*d)

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Rubi in Sympy [A]  time = 76.824, size = 219, normalized size = 0.94 \[ \frac{2 \left (d + e x\right )^{\frac{3}{2}} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}}{9 c d} - \frac{4 \sqrt{d + e x} \left (a e^{2} - c d^{2}\right ) \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}}{21 c^{2} d^{2}} + \frac{16 \left (a e^{2} - c d^{2}\right )^{2} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}}{105 c^{3} d^{3} \sqrt{d + e x}} - \frac{32 \left (a e^{2} - c d^{2}\right )^{3} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}}{315 c^{4} d^{4} \left (d + e x\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**(5/2)*(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.162723, size = 172, normalized size = 0.74 \[ \frac{2 \sqrt{(d+e x) (a e+c d x)} \left (-16 a^4 e^7+8 a^3 c d e^5 (9 d+e x)-6 a^2 c^2 d^2 e^3 \left (21 d^2+6 d e x+e^2 x^2\right )+a c^3 d^3 e \left (105 d^3+63 d^2 e x+27 d e^2 x^2+5 e^3 x^3\right )+c^4 d^4 x \left (105 d^3+189 d^2 e x+135 d e^2 x^2+35 e^3 x^3\right )\right )}{315 c^4 d^4 \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^(5/2)*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^4*e^7 + 8*a^3*c*d*e^5*(9*d + e*x) - 6*a^
2*c^2*d^2*e^3*(21*d^2 + 6*d*e*x + e^2*x^2) + a*c^3*d^3*e*(105*d^3 + 63*d^2*e*x +
27*d*e^2*x^2 + 5*e^3*x^3) + c^4*d^4*x*(105*d^3 + 189*d^2*e*x + 135*d*e^2*x^2 +
35*e^3*x^3)))/(315*c^4*d^4*Sqrt[d + e*x])

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Maple [A]  time = 0.01, size = 168, normalized size = 0.7 \[ -{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( -35\,{x}^{3}{c}^{3}{d}^{3}{e}^{3}+30\,{x}^{2}a{c}^{2}{d}^{2}{e}^{4}-135\,{x}^{2}{c}^{3}{d}^{4}{e}^{2}-24\,x{a}^{2}cd{e}^{5}+108\,xa{c}^{2}{d}^{3}{e}^{3}-189\,{c}^{3}{d}^{5}ex+16\,{a}^{3}{e}^{6}-72\,{a}^{2}c{d}^{2}{e}^{4}+126\,{c}^{2}{d}^{4}a{e}^{2}-105\,{c}^{3}{d}^{6} \right ) }{315\,{c}^{4}{d}^{4}}\sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+aed}{\frac{1}{\sqrt{ex+d}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^(5/2)*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x)

[Out]

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

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Maxima [A]  time = 0.752082, size = 285, normalized size = 1.22 \[ \frac{2 \,{\left (35 \, c^{4} d^{4} e^{3} x^{4} + 105 \, a c^{3} d^{6} e - 126 \, a^{2} c^{2} d^{4} e^{3} + 72 \, a^{3} c d^{2} e^{5} - 16 \, a^{4} e^{7} + 5 \,{\left (27 \, c^{4} d^{5} e^{2} + a c^{3} d^{3} e^{4}\right )} x^{3} + 3 \,{\left (63 \, c^{4} d^{6} e + 9 \, a c^{3} d^{4} e^{3} - 2 \, a^{2} c^{2} d^{2} e^{5}\right )} x^{2} +{\left (105 \, c^{4} d^{7} + 63 \, a c^{3} d^{5} e^{2} - 36 \, a^{2} c^{2} d^{3} e^{4} + 8 \, a^{3} c d e^{6}\right )} x\right )} \sqrt{c d x + a e}{\left (e x + d\right )}}{315 \,{\left (c^{4} d^{4} e x + c^{4} d^{5}\right )}} \]

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)^(5/2),x, algorithm="maxima")

[Out]

2/315*(35*c^4*d^4*e^3*x^4 + 105*a*c^3*d^6*e - 126*a^2*c^2*d^4*e^3 + 72*a^3*c*d^2
*e^5 - 16*a^4*e^7 + 5*(27*c^4*d^5*e^2 + a*c^3*d^3*e^4)*x^3 + 3*(63*c^4*d^6*e + 9
*a*c^3*d^4*e^3 - 2*a^2*c^2*d^2*e^5)*x^2 + (105*c^4*d^7 + 63*a*c^3*d^5*e^2 - 36*a
^2*c^2*d^3*e^4 + 8*a^3*c*d*e^6)*x)*sqrt(c*d*x + a*e)*(e*x + d)/(c^4*d^4*e*x + c^
4*d^5)

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Fricas [A]  time = 0.255885, size = 481, normalized size = 2.06 \[ \frac{2 \,{\left (35 \, c^{5} d^{5} e^{4} x^{6} + 105 \, a^{2} c^{3} d^{7} e^{2} - 126 \, a^{3} c^{2} d^{5} e^{4} + 72 \, a^{4} c d^{3} e^{6} - 16 \, a^{5} d e^{8} + 10 \,{\left (17 \, c^{5} d^{6} e^{3} + 4 \, a c^{4} d^{4} e^{5}\right )} x^{5} +{\left (324 \, c^{5} d^{7} e^{2} + 202 \, a c^{4} d^{5} e^{4} - a^{2} c^{3} d^{3} e^{6}\right )} x^{4} + 2 \,{\left (147 \, c^{5} d^{8} e + 207 \, a c^{4} d^{6} e^{3} - 5 \, a^{2} c^{3} d^{4} e^{5} + a^{3} c^{2} d^{2} e^{7}\right )} x^{3} +{\left (105 \, c^{5} d^{9} + 462 \, a c^{4} d^{7} e^{2} - 72 \, a^{2} c^{3} d^{5} e^{4} + 38 \, a^{3} c^{2} d^{3} e^{6} - 8 \, a^{4} c d e^{8}\right )} x^{2} + 2 \,{\left (105 \, a c^{4} d^{8} e + 21 \, a^{2} c^{3} d^{6} e^{3} - 45 \, a^{3} c^{2} d^{4} e^{5} + 32 \, a^{4} c d^{2} e^{7} - 8 \, a^{5} e^{9}\right )} x\right )}}{315 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{e x + d} c^{4} d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(e*x + d)^(5/2),x, algorithm="fricas")

[Out]

2/315*(35*c^5*d^5*e^4*x^6 + 105*a^2*c^3*d^7*e^2 - 126*a^3*c^2*d^5*e^4 + 72*a^4*c
*d^3*e^6 - 16*a^5*d*e^8 + 10*(17*c^5*d^6*e^3 + 4*a*c^4*d^4*e^5)*x^5 + (324*c^5*d
^7*e^2 + 202*a*c^4*d^5*e^4 - a^2*c^3*d^3*e^6)*x^4 + 2*(147*c^5*d^8*e + 207*a*c^4
*d^6*e^3 - 5*a^2*c^3*d^4*e^5 + a^3*c^2*d^2*e^7)*x^3 + (105*c^5*d^9 + 462*a*c^4*d
^7*e^2 - 72*a^2*c^3*d^5*e^4 + 38*a^3*c^2*d^3*e^6 - 8*a^4*c*d*e^8)*x^2 + 2*(105*a
*c^4*d^8*e + 21*a^2*c^3*d^6*e^3 - 45*a^3*c^2*d^4*e^5 + 32*a^4*c*d^2*e^7 - 8*a^5*
e^9)*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)**(5/2)*(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)^(5/2),x, algorithm="giac")

[Out]

Timed out