3.2018 \(\int (d+e x)^{3/2} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx\)

Optimal. Leaf size=171 \[ \frac{16 \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{105 c^3 d^3 (d+e x)^{3/2}}+\frac{8 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{35 c^2 d^2 \sqrt{d+e x}}+\frac{2 \sqrt{d+e x} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{7 c d} \]

[Out]

(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*(d + e*x)^(3/2)) + (8*(c*d^2 - a*e^2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^
(3/2))/(35*c^2*d^2*Sqrt[d + e*x]) + (2*Sqrt[d + e*x]*(a*d*e + (c*d^2 + a*e^2)*x
+ c*d*e*x^2)^(3/2))/(7*c*d)

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Rubi [A]  time = 0.331083, antiderivative size = 171, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.051 \[ \frac{16 \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{105 c^3 d^3 (d+e x)^{3/2}}+\frac{8 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{35 c^2 d^2 \sqrt{d+e x}}+\frac{2 \sqrt{d+e x} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{7 c d} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^(3/2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2],x]

[Out]

(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*(d + e*x)^(3/2)) + (8*(c*d^2 - a*e^2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^
(3/2))/(35*c^2*d^2*Sqrt[d + e*x]) + (2*Sqrt[d + e*x]*(a*d*e + (c*d^2 + a*e^2)*x
+ c*d*e*x^2)^(3/2))/(7*c*d)

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Rubi in Sympy [A]  time = 52.8861, size = 160, normalized size = 0.94 \[ \frac{2 \sqrt{d + e x} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}}{7 c d} - \frac{8 \left (a e^{2} - c d^{2}\right ) \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}}{35 c^{2} d^{2} \sqrt{d + e x}} + \frac{16 \left (a e^{2} - c d^{2}\right )^{2} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}}{105 c^{3} d^{3} \left (d + e x\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**(3/2)*(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/2),x)

[Out]

2*sqrt(d + e*x)*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))**(3/2)/(7*c*d) - 8*(a
*e**2 - c*d**2)*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))**(3/2)/(35*c**2*d**2*
sqrt(d + e*x)) + 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*(d + e*x)**(3/2))

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Mathematica [A]  time = 0.0996149, size = 118, normalized size = 0.69 \[ \frac{2 \sqrt{(d+e x) (a e+c d x)} \left (8 a^3 e^5-4 a^2 c d e^3 (7 d+e x)+a c^2 d^2 e \left (35 d^2+14 d e x+3 e^2 x^2\right )+c^3 d^3 x \left (35 d^2+42 d e x+15 e^2 x^2\right )\right )}{105 c^3 d^3 \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^(3/2)*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)]*(8*a^3*e^5 - 4*a^2*c*d*e^3*(7*d + e*x) + a*c^2*
d^2*e*(35*d^2 + 14*d*e*x + 3*e^2*x^2) + c^3*d^3*x*(35*d^2 + 42*d*e*x + 15*e^2*x^
2)))/(105*c^3*d^3*Sqrt[d + e*x])

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Maple [A]  time = 0.008, size = 110, normalized size = 0.6 \[{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( 15\,{x}^{2}{c}^{2}{d}^{2}{e}^{2}-12\,xacd{e}^{3}+42\,x{c}^{2}{d}^{3}e+8\,{a}^{2}{e}^{4}-28\,ac{d}^{2}{e}^{2}+35\,{c}^{2}{d}^{4} \right ) }{105\,{c}^{3}{d}^{3}}\sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+aed}{\frac{1}{\sqrt{ex+d}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^(3/2)*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x)

[Out]

2/105*(c*d*x+a*e)*(15*c^2*d^2*e^2*x^2-12*a*c*d*e^3*x+42*c^2*d^3*e*x+8*a^2*e^4-28
*a*c*d^2*e^2+35*c^2*d^4)*(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(1/2)/c^3/d^3/(e*x+d)
^(1/2)

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Maxima [A]  time = 0.756308, size = 189, normalized size = 1.11 \[ \frac{2 \,{\left (15 \, c^{3} d^{3} e^{2} x^{3} + 35 \, a c^{2} d^{4} e - 28 \, a^{2} c d^{2} e^{3} + 8 \, a^{3} e^{5} + 3 \,{\left (14 \, c^{3} d^{4} e + a c^{2} d^{2} e^{3}\right )} x^{2} +{\left (35 \, c^{3} d^{5} + 14 \, a c^{2} d^{3} e^{2} - 4 \, a^{2} c d e^{4}\right )} x\right )} \sqrt{c d x + a e}{\left (e x + d\right )}}{105 \,{\left (c^{3} d^{3} e x + c^{3} d^{4}\right )}} \]

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)^(3/2),x, algorithm="maxima")

[Out]

2/105*(15*c^3*d^3*e^2*x^3 + 35*a*c^2*d^4*e - 28*a^2*c*d^2*e^3 + 8*a^3*e^5 + 3*(1
4*c^3*d^4*e + a*c^2*d^2*e^3)*x^2 + (35*c^3*d^5 + 14*a*c^2*d^3*e^2 - 4*a^2*c*d*e^
4)*x)*sqrt(c*d*x + a*e)*(e*x + d)/(c^3*d^3*e*x + c^3*d^4)

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Fricas [A]  time = 0.251755, size = 347, normalized size = 2.03 \[ \frac{2 \,{\left (15 \, c^{4} d^{4} e^{3} x^{5} + 35 \, a^{2} c^{2} d^{5} e^{2} - 28 \, a^{3} c d^{3} e^{4} + 8 \, a^{4} d e^{6} + 3 \,{\left (19 \, c^{4} d^{5} e^{2} + 6 \, a c^{3} d^{3} e^{4}\right )} x^{4} +{\left (77 \, c^{4} d^{6} e + 74 \, a c^{3} d^{4} e^{3} - a^{2} c^{2} d^{2} e^{5}\right )} x^{3} +{\left (35 \, c^{4} d^{7} + 126 \, a c^{3} d^{5} e^{2} - 15 \, a^{2} c^{2} d^{3} e^{4} + 4 \, a^{3} c d e^{6}\right )} x^{2} +{\left (70 \, a c^{3} d^{6} e + 21 \, a^{2} c^{2} d^{4} e^{3} - 24 \, a^{3} c d^{2} e^{5} + 8 \, a^{4} e^{7}\right )} x\right )}}{105 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{e x + d} c^{3} d^{3}} \]

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)^(3/2),x, algorithm="fricas")

[Out]

2/105*(15*c^4*d^4*e^3*x^5 + 35*a^2*c^2*d^5*e^2 - 28*a^3*c*d^3*e^4 + 8*a^4*d*e^6
+ 3*(19*c^4*d^5*e^2 + 6*a*c^3*d^3*e^4)*x^4 + (77*c^4*d^6*e + 74*a*c^3*d^4*e^3 -
a^2*c^2*d^2*e^5)*x^3 + (35*c^4*d^7 + 126*a*c^3*d^5*e^2 - 15*a^2*c^2*d^3*e^4 + 4*
a^3*c*d*e^6)*x^2 + (70*a*c^3*d^6*e + 21*a^2*c^2*d^4*e^3 - 24*a^3*c*d^2*e^5 + 8*a
^4*e^7)*x)/(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d)*c^3*d^3)

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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)**(3/2)*(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)^(3/2),x, algorithm="giac")

[Out]

Timed out