3.347 \(\int (d+e x)^{3/2} \left (b x+c x^2\right )^2 \, dx\)

Optimal. Leaf size=147 \[ \frac{2 (d+e x)^{9/2} \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )}{9 e^5}+\frac{2 d^2 (d+e x)^{5/2} (c d-b e)^2}{5 e^5}-\frac{4 c (d+e x)^{11/2} (2 c d-b e)}{11 e^5}-\frac{4 d (d+e x)^{7/2} (c d-b e) (2 c d-b e)}{7 e^5}+\frac{2 c^2 (d+e x)^{13/2}}{13 e^5} \]

[Out]

(2*d^2*(c*d - b*e)^2*(d + e*x)^(5/2))/(5*e^5) - (4*d*(c*d - b*e)*(2*c*d - b*e)*(
d + e*x)^(7/2))/(7*e^5) + (2*(6*c^2*d^2 - 6*b*c*d*e + b^2*e^2)*(d + e*x)^(9/2))/
(9*e^5) - (4*c*(2*c*d - b*e)*(d + e*x)^(11/2))/(11*e^5) + (2*c^2*(d + e*x)^(13/2
))/(13*e^5)

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Rubi [A]  time = 0.197829, antiderivative size = 147, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095 \[ \frac{2 (d+e x)^{9/2} \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )}{9 e^5}+\frac{2 d^2 (d+e x)^{5/2} (c d-b e)^2}{5 e^5}-\frac{4 c (d+e x)^{11/2} (2 c d-b e)}{11 e^5}-\frac{4 d (d+e x)^{7/2} (c d-b e) (2 c d-b e)}{7 e^5}+\frac{2 c^2 (d+e x)^{13/2}}{13 e^5} \]

Antiderivative was successfully verified.

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

[Out]

(2*d^2*(c*d - b*e)^2*(d + e*x)^(5/2))/(5*e^5) - (4*d*(c*d - b*e)*(2*c*d - b*e)*(
d + e*x)^(7/2))/(7*e^5) + (2*(6*c^2*d^2 - 6*b*c*d*e + b^2*e^2)*(d + e*x)^(9/2))/
(9*e^5) - (4*c*(2*c*d - b*e)*(d + e*x)^(11/2))/(11*e^5) + (2*c^2*(d + e*x)^(13/2
))/(13*e^5)

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Rubi in Sympy [A]  time = 33.4962, size = 141, normalized size = 0.96 \[ \frac{2 c^{2} \left (d + e x\right )^{\frac{13}{2}}}{13 e^{5}} + \frac{4 c \left (d + e x\right )^{\frac{11}{2}} \left (b e - 2 c d\right )}{11 e^{5}} + \frac{2 d^{2} \left (d + e x\right )^{\frac{5}{2}} \left (b e - c d\right )^{2}}{5 e^{5}} - \frac{4 d \left (d + e x\right )^{\frac{7}{2}} \left (b e - 2 c d\right ) \left (b e - c d\right )}{7 e^{5}} + \frac{2 \left (d + e x\right )^{\frac{9}{2}} \left (b^{2} e^{2} - 6 b c d e + 6 c^{2} d^{2}\right )}{9 e^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*c**2*(d + e*x)**(13/2)/(13*e**5) + 4*c*(d + e*x)**(11/2)*(b*e - 2*c*d)/(11*e**
5) + 2*d**2*(d + e*x)**(5/2)*(b*e - c*d)**2/(5*e**5) - 4*d*(d + e*x)**(7/2)*(b*e
- 2*c*d)*(b*e - c*d)/(7*e**5) + 2*(d + e*x)**(9/2)*(b**2*e**2 - 6*b*c*d*e + 6*c
**2*d**2)/(9*e**5)

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Mathematica [A]  time = 0.129082, size = 125, normalized size = 0.85 \[ \frac{2 (d+e x)^{5/2} \left (143 b^2 e^2 \left (8 d^2-20 d e x+35 e^2 x^2\right )+78 b c e \left (-16 d^3+40 d^2 e x-70 d e^2 x^2+105 e^3 x^3\right )+3 c^2 \left (128 d^4-320 d^3 e x+560 d^2 e^2 x^2-840 d e^3 x^3+1155 e^4 x^4\right )\right )}{45045 e^5} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^(3/2)*(b*x + c*x^2)^2,x]

[Out]

(2*(d + e*x)^(5/2)*(143*b^2*e^2*(8*d^2 - 20*d*e*x + 35*e^2*x^2) + 78*b*c*e*(-16*
d^3 + 40*d^2*e*x - 70*d*e^2*x^2 + 105*e^3*x^3) + 3*c^2*(128*d^4 - 320*d^3*e*x +
560*d^2*e^2*x^2 - 840*d*e^3*x^3 + 1155*e^4*x^4)))/(45045*e^5)

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Maple [A]  time = 0.008, size = 141, normalized size = 1. \[{\frac{6930\,{c}^{2}{x}^{4}{e}^{4}+16380\,bc{e}^{4}{x}^{3}-5040\,{c}^{2}d{e}^{3}{x}^{3}+10010\,{b}^{2}{e}^{4}{x}^{2}-10920\,bcd{e}^{3}{x}^{2}+3360\,{c}^{2}{d}^{2}{e}^{2}{x}^{2}-5720\,{b}^{2}d{e}^{3}x+6240\,bc{d}^{2}{e}^{2}x-1920\,{c}^{2}{d}^{3}ex+2288\,{b}^{2}{d}^{2}{e}^{2}-2496\,bc{d}^{3}e+768\,{c}^{2}{d}^{4}}{45045\,{e}^{5}} \left ( ex+d \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/45045*(e*x+d)^(5/2)*(3465*c^2*e^4*x^4+8190*b*c*e^4*x^3-2520*c^2*d*e^3*x^3+5005
*b^2*e^4*x^2-5460*b*c*d*e^3*x^2+1680*c^2*d^2*e^2*x^2-2860*b^2*d*e^3*x+3120*b*c*d
^2*e^2*x-960*c^2*d^3*e*x+1144*b^2*d^2*e^2-1248*b*c*d^3*e+384*c^2*d^4)/e^5

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Maxima [A]  time = 0.69511, size = 188, normalized size = 1.28 \[ \frac{2 \,{\left (3465 \,{\left (e x + d\right )}^{\frac{13}{2}} c^{2} - 8190 \,{\left (2 \, c^{2} d - b c e\right )}{\left (e x + d\right )}^{\frac{11}{2}} + 5005 \,{\left (6 \, c^{2} d^{2} - 6 \, b c d e + b^{2} e^{2}\right )}{\left (e x + d\right )}^{\frac{9}{2}} - 12870 \,{\left (2 \, c^{2} d^{3} - 3 \, b c d^{2} e + b^{2} d e^{2}\right )}{\left (e x + d\right )}^{\frac{7}{2}} + 9009 \,{\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2}\right )}{\left (e x + d\right )}^{\frac{5}{2}}\right )}}{45045 \, e^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^2*(e*x + d)^(3/2),x, algorithm="maxima")

[Out]

2/45045*(3465*(e*x + d)^(13/2)*c^2 - 8190*(2*c^2*d - b*c*e)*(e*x + d)^(11/2) + 5
005*(6*c^2*d^2 - 6*b*c*d*e + b^2*e^2)*(e*x + d)^(9/2) - 12870*(2*c^2*d^3 - 3*b*c
*d^2*e + b^2*d*e^2)*(e*x + d)^(7/2) + 9009*(c^2*d^4 - 2*b*c*d^3*e + b^2*d^2*e^2)
*(e*x + d)^(5/2))/e^5

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Fricas [A]  time = 0.209469, size = 289, normalized size = 1.97 \[ \frac{2 \,{\left (3465 \, c^{2} e^{6} x^{6} + 384 \, c^{2} d^{6} - 1248 \, b c d^{5} e + 1144 \, b^{2} d^{4} e^{2} + 630 \,{\left (7 \, c^{2} d e^{5} + 13 \, b c e^{6}\right )} x^{5} + 35 \,{\left (3 \, c^{2} d^{2} e^{4} + 312 \, b c d e^{5} + 143 \, b^{2} e^{6}\right )} x^{4} - 10 \,{\left (12 \, c^{2} d^{3} e^{3} - 39 \, b c d^{2} e^{4} - 715 \, b^{2} d e^{5}\right )} x^{3} + 3 \,{\left (48 \, c^{2} d^{4} e^{2} - 156 \, b c d^{3} e^{3} + 143 \, b^{2} d^{2} e^{4}\right )} x^{2} - 4 \,{\left (48 \, c^{2} d^{5} e - 156 \, b c d^{4} e^{2} + 143 \, b^{2} d^{3} e^{3}\right )} x\right )} \sqrt{e x + d}}{45045 \, e^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^2*(e*x + d)^(3/2),x, algorithm="fricas")

[Out]

2/45045*(3465*c^2*e^6*x^6 + 384*c^2*d^6 - 1248*b*c*d^5*e + 1144*b^2*d^4*e^2 + 63
0*(7*c^2*d*e^5 + 13*b*c*e^6)*x^5 + 35*(3*c^2*d^2*e^4 + 312*b*c*d*e^5 + 143*b^2*e
^6)*x^4 - 10*(12*c^2*d^3*e^3 - 39*b*c*d^2*e^4 - 715*b^2*d*e^5)*x^3 + 3*(48*c^2*d
^4*e^2 - 156*b*c*d^3*e^3 + 143*b^2*d^2*e^4)*x^2 - 4*(48*c^2*d^5*e - 156*b*c*d^4*
e^2 + 143*b^2*d^3*e^3)*x)*sqrt(e*x + d)/e^5

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Sympy [A]  time = 6.2724, size = 413, normalized size = 2.81 \[ \frac{2 b^{2} d \left (\frac{d^{2} \left (d + e x\right )^{\frac{3}{2}}}{3} - \frac{2 d \left (d + e x\right )^{\frac{5}{2}}}{5} + \frac{\left (d + e x\right )^{\frac{7}{2}}}{7}\right )}{e^{3}} + \frac{2 b^{2} \left (- \frac{d^{3} \left (d + e x\right )^{\frac{3}{2}}}{3} + \frac{3 d^{2} \left (d + e x\right )^{\frac{5}{2}}}{5} - \frac{3 d \left (d + e x\right )^{\frac{7}{2}}}{7} + \frac{\left (d + e x\right )^{\frac{9}{2}}}{9}\right )}{e^{3}} + \frac{4 b c d \left (- \frac{d^{3} \left (d + e x\right )^{\frac{3}{2}}}{3} + \frac{3 d^{2} \left (d + e x\right )^{\frac{5}{2}}}{5} - \frac{3 d \left (d + e x\right )^{\frac{7}{2}}}{7} + \frac{\left (d + e x\right )^{\frac{9}{2}}}{9}\right )}{e^{4}} + \frac{4 b c \left (\frac{d^{4} \left (d + e x\right )^{\frac{3}{2}}}{3} - \frac{4 d^{3} \left (d + e x\right )^{\frac{5}{2}}}{5} + \frac{6 d^{2} \left (d + e x\right )^{\frac{7}{2}}}{7} - \frac{4 d \left (d + e x\right )^{\frac{9}{2}}}{9} + \frac{\left (d + e x\right )^{\frac{11}{2}}}{11}\right )}{e^{4}} + \frac{2 c^{2} d \left (\frac{d^{4} \left (d + e x\right )^{\frac{3}{2}}}{3} - \frac{4 d^{3} \left (d + e x\right )^{\frac{5}{2}}}{5} + \frac{6 d^{2} \left (d + e x\right )^{\frac{7}{2}}}{7} - \frac{4 d \left (d + e x\right )^{\frac{9}{2}}}{9} + \frac{\left (d + e x\right )^{\frac{11}{2}}}{11}\right )}{e^{5}} + \frac{2 c^{2} \left (- \frac{d^{5} \left (d + e x\right )^{\frac{3}{2}}}{3} + d^{4} \left (d + e x\right )^{\frac{5}{2}} - \frac{10 d^{3} \left (d + e x\right )^{\frac{7}{2}}}{7} + \frac{10 d^{2} \left (d + e x\right )^{\frac{9}{2}}}{9} - \frac{5 d \left (d + e x\right )^{\frac{11}{2}}}{11} + \frac{\left (d + e x\right )^{\frac{13}{2}}}{13}\right )}{e^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**(3/2)*(c*x**2+b*x)**2,x)

[Out]

2*b**2*d*(d**2*(d + e*x)**(3/2)/3 - 2*d*(d + e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)
/e**3 + 2*b**2*(-d**3*(d + e*x)**(3/2)/3 + 3*d**2*(d + e*x)**(5/2)/5 - 3*d*(d +
e*x)**(7/2)/7 + (d + e*x)**(9/2)/9)/e**3 + 4*b*c*d*(-d**3*(d + e*x)**(3/2)/3 + 3
*d**2*(d + e*x)**(5/2)/5 - 3*d*(d + e*x)**(7/2)/7 + (d + e*x)**(9/2)/9)/e**4 + 4
*b*c*(d**4*(d + e*x)**(3/2)/3 - 4*d**3*(d + e*x)**(5/2)/5 + 6*d**2*(d + e*x)**(7
/2)/7 - 4*d*(d + e*x)**(9/2)/9 + (d + e*x)**(11/2)/11)/e**4 + 2*c**2*d*(d**4*(d
+ e*x)**(3/2)/3 - 4*d**3*(d + e*x)**(5/2)/5 + 6*d**2*(d + e*x)**(7/2)/7 - 4*d*(d
+ e*x)**(9/2)/9 + (d + e*x)**(11/2)/11)/e**5 + 2*c**2*(-d**5*(d + e*x)**(3/2)/3
+ d**4*(d + e*x)**(5/2) - 10*d**3*(d + e*x)**(7/2)/7 + 10*d**2*(d + e*x)**(9/2)
/9 - 5*d*(d + e*x)**(11/2)/11 + (d + e*x)**(13/2)/13)/e**5

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GIAC/XCAS [A]  time = 0.214547, size = 579, normalized size = 3.94 \[ \frac{2}{45045} \,{\left (429 \,{\left (15 \,{\left (x e + d\right )}^{\frac{7}{2}} e^{12} - 42 \,{\left (x e + d\right )}^{\frac{5}{2}} d e^{12} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2} e^{12}\right )} b^{2} d e^{\left (-14\right )} + 286 \,{\left (35 \,{\left (x e + d\right )}^{\frac{9}{2}} e^{24} - 135 \,{\left (x e + d\right )}^{\frac{7}{2}} d e^{24} + 189 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{2} e^{24} - 105 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{3} e^{24}\right )} b c d e^{\left (-27\right )} + 13 \,{\left (315 \,{\left (x e + d\right )}^{\frac{11}{2}} e^{40} - 1540 \,{\left (x e + d\right )}^{\frac{9}{2}} d e^{40} + 2970 \,{\left (x e + d\right )}^{\frac{7}{2}} d^{2} e^{40} - 2772 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{3} e^{40} + 1155 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{4} e^{40}\right )} c^{2} d e^{\left (-44\right )} + 143 \,{\left (35 \,{\left (x e + d\right )}^{\frac{9}{2}} e^{24} - 135 \,{\left (x e + d\right )}^{\frac{7}{2}} d e^{24} + 189 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{2} e^{24} - 105 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{3} e^{24}\right )} b^{2} e^{\left (-26\right )} + 26 \,{\left (315 \,{\left (x e + d\right )}^{\frac{11}{2}} e^{40} - 1540 \,{\left (x e + d\right )}^{\frac{9}{2}} d e^{40} + 2970 \,{\left (x e + d\right )}^{\frac{7}{2}} d^{2} e^{40} - 2772 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{3} e^{40} + 1155 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{4} e^{40}\right )} b c e^{\left (-43\right )} + 5 \,{\left (693 \,{\left (x e + d\right )}^{\frac{13}{2}} e^{60} - 4095 \,{\left (x e + d\right )}^{\frac{11}{2}} d e^{60} + 10010 \,{\left (x e + d\right )}^{\frac{9}{2}} d^{2} e^{60} - 12870 \,{\left (x e + d\right )}^{\frac{7}{2}} d^{3} e^{60} + 9009 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{4} e^{60} - 3003 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{5} e^{60}\right )} c^{2} e^{\left (-64\right )}\right )} e^{\left (-1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^2*(e*x + d)^(3/2),x, algorithm="giac")

[Out]

2/45045*(429*(15*(x*e + d)^(7/2)*e^12 - 42*(x*e + d)^(5/2)*d*e^12 + 35*(x*e + d)
^(3/2)*d^2*e^12)*b^2*d*e^(-14) + 286*(35*(x*e + d)^(9/2)*e^24 - 135*(x*e + d)^(7
/2)*d*e^24 + 189*(x*e + d)^(5/2)*d^2*e^24 - 105*(x*e + d)^(3/2)*d^3*e^24)*b*c*d*
e^(-27) + 13*(315*(x*e + d)^(11/2)*e^40 - 1540*(x*e + d)^(9/2)*d*e^40 + 2970*(x*
e + d)^(7/2)*d^2*e^40 - 2772*(x*e + d)^(5/2)*d^3*e^40 + 1155*(x*e + d)^(3/2)*d^4
*e^40)*c^2*d*e^(-44) + 143*(35*(x*e + d)^(9/2)*e^24 - 135*(x*e + d)^(7/2)*d*e^24
+ 189*(x*e + d)^(5/2)*d^2*e^24 - 105*(x*e + d)^(3/2)*d^3*e^24)*b^2*e^(-26) + 26
*(315*(x*e + d)^(11/2)*e^40 - 1540*(x*e + d)^(9/2)*d*e^40 + 2970*(x*e + d)^(7/2)
*d^2*e^40 - 2772*(x*e + d)^(5/2)*d^3*e^40 + 1155*(x*e + d)^(3/2)*d^4*e^40)*b*c*e
^(-43) + 5*(693*(x*e + d)^(13/2)*e^60 - 4095*(x*e + d)^(11/2)*d*e^60 + 10010*(x*
e + d)^(9/2)*d^2*e^60 - 12870*(x*e + d)^(7/2)*d^3*e^60 + 9009*(x*e + d)^(5/2)*d^
4*e^60 - 3003*(x*e + d)^(3/2)*d^5*e^60)*c^2*e^(-64))*e^(-1)