3.354 \(\int (d+e x)^{5/2} \left (b x+c x^2\right )^3 \, dx\)

Optimal. Leaf size=248 \[ \frac{2 c (d+e x)^{15/2} \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{5 e^7}-\frac{2 (d+e x)^{13/2} (2 c d-b e) \left (b^2 e^2-10 b c d e+10 c^2 d^2\right )}{13 e^7}+\frac{6 d (d+e x)^{11/2} (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{11 e^7}-\frac{6 c^2 (d+e x)^{17/2} (2 c d-b e)}{17 e^7}+\frac{2 d^3 (d+e x)^{7/2} (c d-b e)^3}{7 e^7}-\frac{2 d^2 (d+e x)^{9/2} (c d-b e)^2 (2 c d-b e)}{3 e^7}+\frac{2 c^3 (d+e x)^{19/2}}{19 e^7} \]

[Out]

(2*d^3*(c*d - b*e)^3*(d + e*x)^(7/2))/(7*e^7) - (2*d^2*(c*d - b*e)^2*(2*c*d - b*
e)*(d + e*x)^(9/2))/(3*e^7) + (6*d*(c*d - b*e)*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2)
*(d + e*x)^(11/2))/(11*e^7) - (2*(2*c*d - b*e)*(10*c^2*d^2 - 10*b*c*d*e + b^2*e^
2)*(d + e*x)^(13/2))/(13*e^7) + (2*c*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2)*(d + e*x)
^(15/2))/(5*e^7) - (6*c^2*(2*c*d - b*e)*(d + e*x)^(17/2))/(17*e^7) + (2*c^3*(d +
e*x)^(19/2))/(19*e^7)

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Rubi [A]  time = 0.316539, antiderivative size = 248, 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 \[ \frac{2 c (d+e x)^{15/2} \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{5 e^7}-\frac{2 (d+e x)^{13/2} (2 c d-b e) \left (b^2 e^2-10 b c d e+10 c^2 d^2\right )}{13 e^7}+\frac{6 d (d+e x)^{11/2} (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{11 e^7}-\frac{6 c^2 (d+e x)^{17/2} (2 c d-b e)}{17 e^7}+\frac{2 d^3 (d+e x)^{7/2} (c d-b e)^3}{7 e^7}-\frac{2 d^2 (d+e x)^{9/2} (c d-b e)^2 (2 c d-b e)}{3 e^7}+\frac{2 c^3 (d+e x)^{19/2}}{19 e^7} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^(5/2)*(b*x + c*x^2)^3,x]

[Out]

(2*d^3*(c*d - b*e)^3*(d + e*x)^(7/2))/(7*e^7) - (2*d^2*(c*d - b*e)^2*(2*c*d - b*
e)*(d + e*x)^(9/2))/(3*e^7) + (6*d*(c*d - b*e)*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2)
*(d + e*x)^(11/2))/(11*e^7) - (2*(2*c*d - b*e)*(10*c^2*d^2 - 10*b*c*d*e + b^2*e^
2)*(d + e*x)^(13/2))/(13*e^7) + (2*c*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2)*(d + e*x)
^(15/2))/(5*e^7) - (6*c^2*(2*c*d - b*e)*(d + e*x)^(17/2))/(17*e^7) + (2*c^3*(d +
e*x)^(19/2))/(19*e^7)

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Rubi in Sympy [A]  time = 58.9065, size = 243, normalized size = 0.98 \[ \frac{2 c^{3} \left (d + e x\right )^{\frac{19}{2}}}{19 e^{7}} + \frac{6 c^{2} \left (d + e x\right )^{\frac{17}{2}} \left (b e - 2 c d\right )}{17 e^{7}} + \frac{2 c \left (d + e x\right )^{\frac{15}{2}} \left (b^{2} e^{2} - 5 b c d e + 5 c^{2} d^{2}\right )}{5 e^{7}} - \frac{2 d^{3} \left (d + e x\right )^{\frac{7}{2}} \left (b e - c d\right )^{3}}{7 e^{7}} + \frac{2 d^{2} \left (d + e x\right )^{\frac{9}{2}} \left (b e - 2 c d\right ) \left (b e - c d\right )^{2}}{3 e^{7}} - \frac{6 d \left (d + e x\right )^{\frac{11}{2}} \left (b e - c d\right ) \left (b^{2} e^{2} - 5 b c d e + 5 c^{2} d^{2}\right )}{11 e^{7}} + \frac{2 \left (d + e x\right )^{\frac{13}{2}} \left (b e - 2 c d\right ) \left (b^{2} e^{2} - 10 b c d e + 10 c^{2} d^{2}\right )}{13 e^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**(5/2)*(c*x**2+b*x)**3,x)

[Out]

2*c**3*(d + e*x)**(19/2)/(19*e**7) + 6*c**2*(d + e*x)**(17/2)*(b*e - 2*c*d)/(17*
e**7) + 2*c*(d + e*x)**(15/2)*(b**2*e**2 - 5*b*c*d*e + 5*c**2*d**2)/(5*e**7) - 2
*d**3*(d + e*x)**(7/2)*(b*e - c*d)**3/(7*e**7) + 2*d**2*(d + e*x)**(9/2)*(b*e -
2*c*d)*(b*e - c*d)**2/(3*e**7) - 6*d*(d + e*x)**(11/2)*(b*e - c*d)*(b**2*e**2 -
5*b*c*d*e + 5*c**2*d**2)/(11*e**7) + 2*(d + e*x)**(13/2)*(b*e - 2*c*d)*(b**2*e**
2 - 10*b*c*d*e + 10*c**2*d**2)/(13*e**7)

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Mathematica [A]  time = 0.221226, size = 232, normalized size = 0.94 \[ \frac{2 (d+e x)^{7/2} \left (1615 b^3 e^3 \left (-16 d^3+56 d^2 e x-126 d e^2 x^2+231 e^3 x^3\right )+323 b^2 c e^2 \left (128 d^4-448 d^3 e x+1008 d^2 e^2 x^2-1848 d e^3 x^3+3003 e^4 x^4\right )+95 b c^2 e \left (-256 d^5+896 d^4 e x-2016 d^3 e^2 x^2+3696 d^2 e^3 x^3-6006 d e^4 x^4+9009 e^5 x^5\right )+5 c^3 \left (1024 d^6-3584 d^5 e x+8064 d^4 e^2 x^2-14784 d^3 e^3 x^3+24024 d^2 e^4 x^4-36036 d e^5 x^5+51051 e^6 x^6\right )\right )}{4849845 e^7} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^(5/2)*(b*x + c*x^2)^3,x]

[Out]

(2*(d + e*x)^(7/2)*(1615*b^3*e^3*(-16*d^3 + 56*d^2*e*x - 126*d*e^2*x^2 + 231*e^3
*x^3) + 323*b^2*c*e^2*(128*d^4 - 448*d^3*e*x + 1008*d^2*e^2*x^2 - 1848*d*e^3*x^3
+ 3003*e^4*x^4) + 95*b*c^2*e*(-256*d^5 + 896*d^4*e*x - 2016*d^3*e^2*x^2 + 3696*
d^2*e^3*x^3 - 6006*d*e^4*x^4 + 9009*e^5*x^5) + 5*c^3*(1024*d^6 - 3584*d^5*e*x +
8064*d^4*e^2*x^2 - 14784*d^3*e^3*x^3 + 24024*d^2*e^4*x^4 - 36036*d*e^5*x^5 + 510
51*e^6*x^6)))/(4849845*e^7)

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Maple [A]  time = 0.012, size = 286, normalized size = 1.2 \[ -{\frac{-510510\,{c}^{3}{x}^{6}{e}^{6}-1711710\,b{c}^{2}{e}^{6}{x}^{5}+360360\,{c}^{3}d{e}^{5}{x}^{5}-1939938\,{b}^{2}c{e}^{6}{x}^{4}+1141140\,b{c}^{2}d{e}^{5}{x}^{4}-240240\,{c}^{3}{d}^{2}{e}^{4}{x}^{4}-746130\,{b}^{3}{e}^{6}{x}^{3}+1193808\,{b}^{2}cd{e}^{5}{x}^{3}-702240\,b{c}^{2}{d}^{2}{e}^{4}{x}^{3}+147840\,{c}^{3}{d}^{3}{e}^{3}{x}^{3}+406980\,{b}^{3}d{e}^{5}{x}^{2}-651168\,{b}^{2}c{d}^{2}{e}^{4}{x}^{2}+383040\,b{c}^{2}{d}^{3}{e}^{3}{x}^{2}-80640\,{c}^{3}{d}^{4}{e}^{2}{x}^{2}-180880\,{b}^{3}{d}^{2}{e}^{4}x+289408\,{b}^{2}c{d}^{3}{e}^{3}x-170240\,b{c}^{2}{d}^{4}{e}^{2}x+35840\,{c}^{3}{d}^{5}ex+51680\,{b}^{3}{d}^{3}{e}^{3}-82688\,{b}^{2}c{d}^{4}{e}^{2}+48640\,b{c}^{2}{d}^{5}e-10240\,{c}^{3}{d}^{6}}{4849845\,{e}^{7}} \left ( ex+d \right ) ^{{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^(5/2)*(c*x^2+b*x)^3,x)

[Out]

-2/4849845*(e*x+d)^(7/2)*(-255255*c^3*e^6*x^6-855855*b*c^2*e^6*x^5+180180*c^3*d*
e^5*x^5-969969*b^2*c*e^6*x^4+570570*b*c^2*d*e^5*x^4-120120*c^3*d^2*e^4*x^4-37306
5*b^3*e^6*x^3+596904*b^2*c*d*e^5*x^3-351120*b*c^2*d^2*e^4*x^3+73920*c^3*d^3*e^3*
x^3+203490*b^3*d*e^5*x^2-325584*b^2*c*d^2*e^4*x^2+191520*b*c^2*d^3*e^3*x^2-40320
*c^3*d^4*e^2*x^2-90440*b^3*d^2*e^4*x+144704*b^2*c*d^3*e^3*x-85120*b*c^2*d^4*e^2*
x+17920*c^3*d^5*e*x+25840*b^3*d^3*e^3-41344*b^2*c*d^4*e^2+24320*b*c^2*d^5*e-5120
*c^3*d^6)/e^7

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Maxima [A]  time = 0.694406, size = 366, normalized size = 1.48 \[ \frac{2 \,{\left (255255 \,{\left (e x + d\right )}^{\frac{19}{2}} c^{3} - 855855 \,{\left (2 \, c^{3} d - b c^{2} e\right )}{\left (e x + d\right )}^{\frac{17}{2}} + 969969 \,{\left (5 \, c^{3} d^{2} - 5 \, b c^{2} d e + b^{2} c e^{2}\right )}{\left (e x + d\right )}^{\frac{15}{2}} - 373065 \,{\left (20 \, c^{3} d^{3} - 30 \, b c^{2} d^{2} e + 12 \, b^{2} c d e^{2} - b^{3} e^{3}\right )}{\left (e x + d\right )}^{\frac{13}{2}} + 1322685 \,{\left (5 \, c^{3} d^{4} - 10 \, b c^{2} d^{3} e + 6 \, b^{2} c d^{2} e^{2} - b^{3} d e^{3}\right )}{\left (e x + d\right )}^{\frac{11}{2}} - 1616615 \,{\left (2 \, c^{3} d^{5} - 5 \, b c^{2} d^{4} e + 4 \, b^{2} c d^{3} e^{2} - b^{3} d^{2} e^{3}\right )}{\left (e x + d\right )}^{\frac{9}{2}} + 692835 \,{\left (c^{3} d^{6} - 3 \, b c^{2} d^{5} e + 3 \, b^{2} c d^{4} e^{2} - b^{3} d^{3} e^{3}\right )}{\left (e x + d\right )}^{\frac{7}{2}}\right )}}{4849845 \, e^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^3*(e*x + d)^(5/2),x, algorithm="maxima")

[Out]

2/4849845*(255255*(e*x + d)^(19/2)*c^3 - 855855*(2*c^3*d - b*c^2*e)*(e*x + d)^(1
7/2) + 969969*(5*c^3*d^2 - 5*b*c^2*d*e + b^2*c*e^2)*(e*x + d)^(15/2) - 373065*(2
0*c^3*d^3 - 30*b*c^2*d^2*e + 12*b^2*c*d*e^2 - b^3*e^3)*(e*x + d)^(13/2) + 132268
5*(5*c^3*d^4 - 10*b*c^2*d^3*e + 6*b^2*c*d^2*e^2 - b^3*d*e^3)*(e*x + d)^(11/2) -
1616615*(2*c^3*d^5 - 5*b*c^2*d^4*e + 4*b^2*c*d^3*e^2 - b^3*d^2*e^3)*(e*x + d)^(9
/2) + 692835*(c^3*d^6 - 3*b*c^2*d^5*e + 3*b^2*c*d^4*e^2 - b^3*d^3*e^3)*(e*x + d)
^(7/2))/e^7

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Fricas [A]  time = 0.216948, size = 575, normalized size = 2.32 \[ \frac{2 \,{\left (255255 \, c^{3} e^{9} x^{9} + 5120 \, c^{3} d^{9} - 24320 \, b c^{2} d^{8} e + 41344 \, b^{2} c d^{7} e^{2} - 25840 \, b^{3} d^{6} e^{3} + 45045 \,{\left (13 \, c^{3} d e^{8} + 19 \, b c^{2} e^{9}\right )} x^{8} + 3003 \,{\left (115 \, c^{3} d^{2} e^{7} + 665 \, b c^{2} d e^{8} + 323 \, b^{2} c e^{9}\right )} x^{7} + 231 \,{\left (5 \, c^{3} d^{3} e^{6} + 5225 \, b c^{2} d^{2} e^{7} + 10013 \, b^{2} c d e^{8} + 1615 \, b^{3} e^{9}\right )} x^{6} - 63 \,{\left (20 \, c^{3} d^{4} e^{5} - 95 \, b c^{2} d^{3} e^{6} - 22933 \, b^{2} c d^{2} e^{7} - 14535 \, b^{3} d e^{8}\right )} x^{5} + 35 \,{\left (40 \, c^{3} d^{5} e^{4} - 190 \, b c^{2} d^{4} e^{5} + 323 \, b^{2} c d^{3} e^{6} + 17119 \, b^{3} d^{2} e^{7}\right )} x^{4} - 5 \,{\left (320 \, c^{3} d^{6} e^{3} - 1520 \, b c^{2} d^{5} e^{4} + 2584 \, b^{2} c d^{4} e^{5} - 1615 \, b^{3} d^{3} e^{6}\right )} x^{3} + 6 \,{\left (320 \, c^{3} d^{7} e^{2} - 1520 \, b c^{2} d^{6} e^{3} + 2584 \, b^{2} c d^{5} e^{4} - 1615 \, b^{3} d^{4} e^{5}\right )} x^{2} - 8 \,{\left (320 \, c^{3} d^{8} e - 1520 \, b c^{2} d^{7} e^{2} + 2584 \, b^{2} c d^{6} e^{3} - 1615 \, b^{3} d^{5} e^{4}\right )} x\right )} \sqrt{e x + d}}{4849845 \, e^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^3*(e*x + d)^(5/2),x, algorithm="fricas")

[Out]

2/4849845*(255255*c^3*e^9*x^9 + 5120*c^3*d^9 - 24320*b*c^2*d^8*e + 41344*b^2*c*d
^7*e^2 - 25840*b^3*d^6*e^3 + 45045*(13*c^3*d*e^8 + 19*b*c^2*e^9)*x^8 + 3003*(115
*c^3*d^2*e^7 + 665*b*c^2*d*e^8 + 323*b^2*c*e^9)*x^7 + 231*(5*c^3*d^3*e^6 + 5225*
b*c^2*d^2*e^7 + 10013*b^2*c*d*e^8 + 1615*b^3*e^9)*x^6 - 63*(20*c^3*d^4*e^5 - 95*
b*c^2*d^3*e^6 - 22933*b^2*c*d^2*e^7 - 14535*b^3*d*e^8)*x^5 + 35*(40*c^3*d^5*e^4
- 190*b*c^2*d^4*e^5 + 323*b^2*c*d^3*e^6 + 17119*b^3*d^2*e^7)*x^4 - 5*(320*c^3*d^
6*e^3 - 1520*b*c^2*d^5*e^4 + 2584*b^2*c*d^4*e^5 - 1615*b^3*d^3*e^6)*x^3 + 6*(320
*c^3*d^7*e^2 - 1520*b*c^2*d^6*e^3 + 2584*b^2*c*d^5*e^4 - 1615*b^3*d^4*e^5)*x^2 -
8*(320*c^3*d^8*e - 1520*b*c^2*d^7*e^2 + 2584*b^2*c*d^6*e^3 - 1615*b^3*d^5*e^4)*
x)*sqrt(e*x + d)/e^7

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Sympy [A]  time = 13.1906, size = 1207, normalized size = 4.87 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**(5/2)*(c*x**2+b*x)**3,x)

[Out]

2*b**3*d**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**4 + 4*b**3*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**4 + 2*b**3*(-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)**(1
1/2)/11 + (d + e*x)**(13/2)/13)/e**4 + 6*b**2*c*d**2*(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 + 12*b**2*c*d*(-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 + 6*b**2*c*(d**6*(d + e*x)**(3/2)/
3 - 6*d**5*(d + e*x)**(5/2)/5 + 15*d**4*(d + e*x)**(7/2)/7 - 20*d**3*(d + e*x)**
(9/2)/9 + 15*d**2*(d + e*x)**(11/2)/11 - 6*d*(d + e*x)**(13/2)/13 + (d + e*x)**(
15/2)/15)/e**5 + 6*b*c**2*d**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**6 + 12*b*c**2*d*(d**6*(d + e*x)**(3/2)/3 - 6*d*
*5*(d + e*x)**(5/2)/5 + 15*d**4*(d + e*x)**(7/2)/7 - 20*d**3*(d + e*x)**(9/2)/9
+ 15*d**2*(d + e*x)**(11/2)/11 - 6*d*(d + e*x)**(13/2)/13 + (d + e*x)**(15/2)/15
)/e**6 + 6*b*c**2*(-d**7*(d + e*x)**(3/2)/3 + 7*d**6*(d + e*x)**(5/2)/5 - 3*d**5
*(d + e*x)**(7/2) + 35*d**4*(d + e*x)**(9/2)/9 - 35*d**3*(d + e*x)**(11/2)/11 +
21*d**2*(d + e*x)**(13/2)/13 - 7*d*(d + e*x)**(15/2)/15 + (d + e*x)**(17/2)/17)/
e**6 + 2*c**3*d**2*(d**6*(d + e*x)**(3/2)/3 - 6*d**5*(d + e*x)**(5/2)/5 + 15*d**
4*(d + e*x)**(7/2)/7 - 20*d**3*(d + e*x)**(9/2)/9 + 15*d**2*(d + e*x)**(11/2)/11
- 6*d*(d + e*x)**(13/2)/13 + (d + e*x)**(15/2)/15)/e**7 + 4*c**3*d*(-d**7*(d +
e*x)**(3/2)/3 + 7*d**6*(d + e*x)**(5/2)/5 - 3*d**5*(d + e*x)**(7/2) + 35*d**4*(d
+ e*x)**(9/2)/9 - 35*d**3*(d + e*x)**(11/2)/11 + 21*d**2*(d + e*x)**(13/2)/13 -
7*d*(d + e*x)**(15/2)/15 + (d + e*x)**(17/2)/17)/e**7 + 2*c**3*(d**8*(d + e*x)*
*(3/2)/3 - 8*d**7*(d + e*x)**(5/2)/5 + 4*d**6*(d + e*x)**(7/2) - 56*d**5*(d + e*
x)**(9/2)/9 + 70*d**4*(d + e*x)**(11/2)/11 - 56*d**3*(d + e*x)**(13/2)/13 + 28*d
**2*(d + e*x)**(15/2)/15 - 8*d*(d + e*x)**(17/2)/17 + (d + e*x)**(19/2)/19)/e**7

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.229398, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^3*(e*x + d)^(5/2),x, algorithm="giac")

[Out]

Done