3.1982 \(\int \frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{\sqrt{d+e x}} \, dx\)
Optimal. Leaf size=119 \[ -\frac{6 c^2 d^2 (d+e x)^{11/2} \left (c d^2-a e^2\right )}{11 e^4}+\frac{2 c d (d+e x)^{9/2} \left (c d^2-a e^2\right )^2}{3 e^4}-\frac{2 (d+e x)^{7/2} \left (c d^2-a e^2\right )^3}{7 e^4}+\frac{2 c^3 d^3 (d+e x)^{13/2}}{13 e^4} \]
[Out]
(-2*(c*d^2 - a*e^2)^3*(d + e*x)^(7/2))/(7*e^4) + (2*c*d*(c*d^2 - a*e^2)^2*(d + e
*x)^(9/2))/(3*e^4) - (6*c^2*d^2*(c*d^2 - a*e^2)*(d + e*x)^(11/2))/(11*e^4) + (2*
c^3*d^3*(d + e*x)^(13/2))/(13*e^4)
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Rubi [A] time = 0.188628, antiderivative size = 119, normalized size of antiderivative = 1.,
number of steps used = 3, number of rules used = 2, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.054
\[ -\frac{6 c^2 d^2 (d+e x)^{11/2} \left (c d^2-a e^2\right )}{11 e^4}+\frac{2 c d (d+e x)^{9/2} \left (c d^2-a e^2\right )^2}{3 e^4}-\frac{2 (d+e x)^{7/2} \left (c d^2-a e^2\right )^3}{7 e^4}+\frac{2 c^3 d^3 (d+e x)^{13/2}}{13 e^4} \]
Antiderivative was successfully verified.
[In] Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3/Sqrt[d + e*x],x]
[Out]
(-2*(c*d^2 - a*e^2)^3*(d + e*x)^(7/2))/(7*e^4) + (2*c*d*(c*d^2 - a*e^2)^2*(d + e
*x)^(9/2))/(3*e^4) - (6*c^2*d^2*(c*d^2 - a*e^2)*(d + e*x)^(11/2))/(11*e^4) + (2*
c^3*d^3*(d + e*x)^(13/2))/(13*e^4)
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Rubi in Sympy [A] time = 47.063, size = 110, normalized size = 0.92 \[ \frac{2 c^{3} d^{3} \left (d + e x\right )^{\frac{13}{2}}}{13 e^{4}} + \frac{6 c^{2} d^{2} \left (d + e x\right )^{\frac{11}{2}} \left (a e^{2} - c d^{2}\right )}{11 e^{4}} + \frac{2 c d \left (d + e x\right )^{\frac{9}{2}} \left (a e^{2} - c d^{2}\right )^{2}}{3 e^{4}} + \frac{2 \left (d + e x\right )^{\frac{7}{2}} \left (a e^{2} - c d^{2}\right )^{3}}{7 e^{4}} \]
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)**3/(e*x+d)**(1/2),x)
[Out]
2*c**3*d**3*(d + e*x)**(13/2)/(13*e**4) + 6*c**2*d**2*(d + e*x)**(11/2)*(a*e**2
- c*d**2)/(11*e**4) + 2*c*d*(d + e*x)**(9/2)*(a*e**2 - c*d**2)**2/(3*e**4) + 2*(
d + e*x)**(7/2)*(a*e**2 - c*d**2)**3/(7*e**4)
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Mathematica [A] time = 0.189075, size = 111, normalized size = 0.93 \[ \frac{2 (d+e x)^{7/2} \left (429 a^3 e^6-143 a^2 c d e^4 (2 d-7 e x)+13 a c^2 d^2 e^2 \left (8 d^2-28 d e x+63 e^2 x^2\right )+c^3 d^3 \left (-16 d^3+56 d^2 e x-126 d e^2 x^2+231 e^3 x^3\right )\right )}{3003 e^4} \]
Antiderivative was successfully verified.
[In] Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3/Sqrt[d + e*x],x]
[Out]
(2*(d + e*x)^(7/2)*(429*a^3*e^6 - 143*a^2*c*d*e^4*(2*d - 7*e*x) + 13*a*c^2*d^2*e
^2*(8*d^2 - 28*d*e*x + 63*e^2*x^2) + c^3*d^3*(-16*d^3 + 56*d^2*e*x - 126*d*e^2*x
^2 + 231*e^3*x^3)))/(3003*e^4)
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Maple [A] time = 0.01, size = 131, normalized size = 1.1 \[{\frac{462\,{x}^{3}{c}^{3}{d}^{3}{e}^{3}+1638\,{x}^{2}a{c}^{2}{d}^{2}{e}^{4}-252\,{x}^{2}{c}^{3}{d}^{4}{e}^{2}+2002\,x{a}^{2}cd{e}^{5}-728\,xa{c}^{2}{d}^{3}{e}^{3}+112\,{c}^{3}{d}^{5}ex+858\,{a}^{3}{e}^{6}-572\,{a}^{2}c{d}^{2}{e}^{4}+208\,{c}^{2}{d}^{4}a{e}^{2}-32\,{c}^{3}{d}^{6}}{3003\,{e}^{4}} \left ( ex+d \right ) ^{{\frac{7}{2}}}} \]
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)^3/(e*x+d)^(1/2),x)
[Out]
2/3003*(e*x+d)^(7/2)*(231*c^3*d^3*e^3*x^3+819*a*c^2*d^2*e^4*x^2-126*c^3*d^4*e^2*
x^2+1001*a^2*c*d*e^5*x-364*a*c^2*d^3*e^3*x+56*c^3*d^5*e*x+429*a^3*e^6-286*a^2*c*
d^2*e^4+104*a*c^2*d^4*e^2-16*c^3*d^6)/e^4
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Maxima [A] time = 0.756086, size = 825, normalized size = 6.93 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^3/sqrt(e*x + d),x, algorithm="maxima")
[Out]
2/15015*(15015*sqrt(e*x + d)*a^3*d^3*e^3 + 3003*((3*(e*x + d)^(5/2) - 10*(e*x +
d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*c*d/e + 5*(c*d^2 + a*e^2)*((e*x + d)^(3/2) -
3*sqrt(e*x + d)*d)/e)*a^2*d^2*e^2 + 5*(231*(e*x + d)^(13/2) - 1638*(e*x + d)^(11
/2)*d + 5005*(e*x + d)^(9/2)*d^2 - 8580*(e*x + d)^(7/2)*d^3 + 9009*(e*x + d)^(5/
2)*d^4 - 6006*(e*x + d)^(3/2)*d^5 + 3003*sqrt(e*x + d)*d^6)*c^3*d^3/e^3 + 143*((
35*(e*x + d)^(9/2) - 180*(e*x + d)^(7/2)*d + 378*(e*x + d)^(5/2)*d^2 - 420*(e*x
+ d)^(3/2)*d^3 + 315*sqrt(e*x + d)*d^4)*c^2*d^2/e^2 + 18*(5*(e*x + d)^(7/2) - 21
*(e*x + d)^(5/2)*d + 35*(e*x + d)^(3/2)*d^2 - 35*sqrt(e*x + d)*d^3)*(c*d^2 + a*e
^2)*c*d/e^2 + 21*(c*d^2 + a*e^2)^2*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 1
5*sqrt(e*x + d)*d^2)/e^2)*a*d*e + 65*(63*(e*x + d)^(11/2) - 385*(e*x + d)^(9/2)*
d + 990*(e*x + d)^(7/2)*d^2 - 1386*(e*x + d)^(5/2)*d^3 + 1155*(e*x + d)^(3/2)*d^
4 - 693*sqrt(e*x + d)*d^5)*(c*d^2 + a*e^2)*c^2*d^2/e^3 + 143*(35*(e*x + d)^(9/2)
- 180*(e*x + d)^(7/2)*d + 378*(e*x + d)^(5/2)*d^2 - 420*(e*x + d)^(3/2)*d^3 + 3
15*sqrt(e*x + d)*d^4)*(c*d^2 + a*e^2)^2*c*d/e^3 + 429*(5*(e*x + d)^(7/2) - 21*(e
*x + d)^(5/2)*d + 35*(e*x + d)^(3/2)*d^2 - 35*sqrt(e*x + d)*d^3)*(c*d^2 + a*e^2)
^3/e^3)/e
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Fricas [A] time = 0.224763, size = 382, normalized size = 3.21 \[ \frac{2 \,{\left (231 \, c^{3} d^{3} e^{6} x^{6} - 16 \, c^{3} d^{9} + 104 \, a c^{2} d^{7} e^{2} - 286 \, a^{2} c d^{5} e^{4} + 429 \, a^{3} d^{3} e^{6} + 63 \,{\left (9 \, c^{3} d^{4} e^{5} + 13 \, a c^{2} d^{2} e^{7}\right )} x^{5} + 7 \,{\left (53 \, c^{3} d^{5} e^{4} + 299 \, a c^{2} d^{3} e^{6} + 143 \, a^{2} c d e^{8}\right )} x^{4} +{\left (5 \, c^{3} d^{6} e^{3} + 1469 \, a c^{2} d^{4} e^{5} + 2717 \, a^{2} c d^{2} e^{7} + 429 \, a^{3} e^{9}\right )} x^{3} - 3 \,{\left (2 \, c^{3} d^{7} e^{2} - 13 \, a c^{2} d^{5} e^{4} - 715 \, a^{2} c d^{3} e^{6} - 429 \, a^{3} d e^{8}\right )} x^{2} +{\left (8 \, c^{3} d^{8} e - 52 \, a c^{2} d^{6} e^{3} + 143 \, a^{2} c d^{4} e^{5} + 1287 \, a^{3} d^{2} e^{7}\right )} x\right )} \sqrt{e x + d}}{3003 \, e^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^3/sqrt(e*x + d),x, algorithm="fricas")
[Out]
2/3003*(231*c^3*d^3*e^6*x^6 - 16*c^3*d^9 + 104*a*c^2*d^7*e^2 - 286*a^2*c*d^5*e^4
+ 429*a^3*d^3*e^6 + 63*(9*c^3*d^4*e^5 + 13*a*c^2*d^2*e^7)*x^5 + 7*(53*c^3*d^5*e
^4 + 299*a*c^2*d^3*e^6 + 143*a^2*c*d*e^8)*x^4 + (5*c^3*d^6*e^3 + 1469*a*c^2*d^4*
e^5 + 2717*a^2*c*d^2*e^7 + 429*a^3*e^9)*x^3 - 3*(2*c^3*d^7*e^2 - 13*a*c^2*d^5*e^
4 - 715*a^2*c*d^3*e^6 - 429*a^3*d*e^8)*x^2 + (8*c^3*d^8*e - 52*a*c^2*d^6*e^3 + 1
43*a^2*c*d^4*e^5 + 1287*a^3*d^2*e^7)*x)*sqrt(e*x + d)/e^4
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Sympy [A] time = 164.25, size = 1357, normalized size = 11.4 \[ \text{result too large to display} \]
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)**3/(e*x+d)**(1/2),x)
[Out]
Piecewise((-(2*a**3*d**4*e**3/sqrt(d + e*x) + 8*a**3*d**3*e**3*(-d/sqrt(d + e*x)
- sqrt(d + e*x)) + 12*a**3*d**2*e**3*(d**2/sqrt(d + e*x) + 2*d*sqrt(d + e*x) -
(d + e*x)**(3/2)/3) + 8*a**3*d*e**3*(-d**3/sqrt(d + e*x) - 3*d**2*sqrt(d + e*x)
+ d*(d + e*x)**(3/2) - (d + e*x)**(5/2)/5) + 2*a**3*e**3*(d**4/sqrt(d + e*x) + 4
*d**3*sqrt(d + e*x) - 2*d**2*(d + e*x)**(3/2) + 4*d*(d + e*x)**(5/2)/5 - (d + e*
x)**(7/2)/7) + 6*a**2*c*d**5*e*(-d/sqrt(d + e*x) - sqrt(d + e*x)) + 24*a**2*c*d*
*4*e*(d**2/sqrt(d + e*x) + 2*d*sqrt(d + e*x) - (d + e*x)**(3/2)/3) + 36*a**2*c*d
**3*e*(-d**3/sqrt(d + e*x) - 3*d**2*sqrt(d + e*x) + d*(d + e*x)**(3/2) - (d + e*
x)**(5/2)/5) + 24*a**2*c*d**2*e*(d**4/sqrt(d + e*x) + 4*d**3*sqrt(d + e*x) - 2*d
**2*(d + e*x)**(3/2) + 4*d*(d + e*x)**(5/2)/5 - (d + e*x)**(7/2)/7) + 6*a**2*c*d
*e*(-d**5/sqrt(d + e*x) - 5*d**4*sqrt(d + e*x) + 10*d**3*(d + e*x)**(3/2)/3 - 2*
d**2*(d + e*x)**(5/2) + 5*d*(d + e*x)**(7/2)/7 - (d + e*x)**(9/2)/9) + 6*a*c**2*
d**6*(d**2/sqrt(d + e*x) + 2*d*sqrt(d + e*x) - (d + e*x)**(3/2)/3)/e + 24*a*c**2
*d**5*(-d**3/sqrt(d + e*x) - 3*d**2*sqrt(d + e*x) + d*(d + e*x)**(3/2) - (d + e*
x)**(5/2)/5)/e + 36*a*c**2*d**4*(d**4/sqrt(d + e*x) + 4*d**3*sqrt(d + e*x) - 2*d
**2*(d + e*x)**(3/2) + 4*d*(d + e*x)**(5/2)/5 - (d + e*x)**(7/2)/7)/e + 24*a*c**
2*d**3*(-d**5/sqrt(d + e*x) - 5*d**4*sqrt(d + e*x) + 10*d**3*(d + e*x)**(3/2)/3
- 2*d**2*(d + e*x)**(5/2) + 5*d*(d + e*x)**(7/2)/7 - (d + e*x)**(9/2)/9)/e + 6*a
*c**2*d**2*(d**6/sqrt(d + e*x) + 6*d**5*sqrt(d + e*x) - 5*d**4*(d + e*x)**(3/2)
+ 4*d**3*(d + e*x)**(5/2) - 15*d**2*(d + e*x)**(7/2)/7 + 2*d*(d + e*x)**(9/2)/3
- (d + e*x)**(11/2)/11)/e + 2*c**3*d**7*(-d**3/sqrt(d + e*x) - 3*d**2*sqrt(d + e
*x) + d*(d + e*x)**(3/2) - (d + e*x)**(5/2)/5)/e**3 + 8*c**3*d**6*(d**4/sqrt(d +
e*x) + 4*d**3*sqrt(d + e*x) - 2*d**2*(d + e*x)**(3/2) + 4*d*(d + e*x)**(5/2)/5
- (d + e*x)**(7/2)/7)/e**3 + 12*c**3*d**5*(-d**5/sqrt(d + e*x) - 5*d**4*sqrt(d +
e*x) + 10*d**3*(d + e*x)**(3/2)/3 - 2*d**2*(d + e*x)**(5/2) + 5*d*(d + e*x)**(7
/2)/7 - (d + e*x)**(9/2)/9)/e**3 + 8*c**3*d**4*(d**6/sqrt(d + e*x) + 6*d**5*sqrt
(d + e*x) - 5*d**4*(d + e*x)**(3/2) + 4*d**3*(d + e*x)**(5/2) - 15*d**2*(d + e*x
)**(7/2)/7 + 2*d*(d + e*x)**(9/2)/3 - (d + e*x)**(11/2)/11)/e**3 + 2*c**3*d**3*(
-d**7/sqrt(d + e*x) - 7*d**6*sqrt(d + e*x) + 7*d**5*(d + e*x)**(3/2) - 7*d**4*(d
+ e*x)**(5/2) + 5*d**3*(d + e*x)**(7/2) - 7*d**2*(d + e*x)**(9/2)/3 + 7*d*(d +
e*x)**(11/2)/11 - (d + e*x)**(13/2)/13)/e**3)/e, Ne(e, 0)), (c**3*d**(11/2)*x**4
/4, True))
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GIAC/XCAS [A] time = 0.214888, size = 1, normalized size = 0.01 \[ \mathit{Done} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^3/sqrt(e*x + d),x, algorithm="giac")
[Out]
Done