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

Optimal. Leaf size=147 \[ \frac{2 (d+e x)^{11/2} \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )}{11 e^5}+\frac{2 d^2 (d+e x)^{7/2} (c d-b e)^2}{7 e^5}-\frac{4 c (d+e x)^{13/2} (2 c d-b e)}{13 e^5}-\frac{4 d (d+e x)^{9/2} (c d-b e) (2 c d-b e)}{9 e^5}+\frac{2 c^2 (d+e x)^{15/2}}{15 e^5} \]

[Out]

(2*d^2*(c*d - b*e)^2*(d + e*x)^(7/2))/(7*e^5) - (4*d*(c*d - b*e)*(2*c*d - b*e)*(
d + e*x)^(9/2))/(9*e^5) + (2*(6*c^2*d^2 - 6*b*c*d*e + b^2*e^2)*(d + e*x)^(11/2))
/(11*e^5) - (4*c*(2*c*d - b*e)*(d + e*x)^(13/2))/(13*e^5) + (2*c^2*(d + e*x)^(15
/2))/(15*e^5)

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Rubi [A]  time = 0.204066, 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 \[ \frac{2 (d+e x)^{11/2} \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )}{11 e^5}+\frac{2 d^2 (d+e x)^{7/2} (c d-b e)^2}{7 e^5}-\frac{4 c (d+e x)^{13/2} (2 c d-b e)}{13 e^5}-\frac{4 d (d+e x)^{9/2} (c d-b e) (2 c d-b e)}{9 e^5}+\frac{2 c^2 (d+e x)^{15/2}}{15 e^5} \]

Antiderivative was successfully verified.

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

[Out]

(2*d^2*(c*d - b*e)^2*(d + e*x)^(7/2))/(7*e^5) - (4*d*(c*d - b*e)*(2*c*d - b*e)*(
d + e*x)^(9/2))/(9*e^5) + (2*(6*c^2*d^2 - 6*b*c*d*e + b^2*e^2)*(d + e*x)^(11/2))
/(11*e^5) - (4*c*(2*c*d - b*e)*(d + e*x)^(13/2))/(13*e^5) + (2*c^2*(d + e*x)^(15
/2))/(15*e^5)

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Rubi in Sympy [A]  time = 34.5772, size = 141, normalized size = 0.96 \[ \frac{2 c^{2} \left (d + e x\right )^{\frac{15}{2}}}{15 e^{5}} + \frac{4 c \left (d + e x\right )^{\frac{13}{2}} \left (b e - 2 c d\right )}{13 e^{5}} + \frac{2 d^{2} \left (d + e x\right )^{\frac{7}{2}} \left (b e - c d\right )^{2}}{7 e^{5}} - \frac{4 d \left (d + e x\right )^{\frac{9}{2}} \left (b e - 2 c d\right ) \left (b e - c d\right )}{9 e^{5}} + \frac{2 \left (d + e x\right )^{\frac{11}{2}} \left (b^{2} e^{2} - 6 b c d e + 6 c^{2} d^{2}\right )}{11 e^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*c**2*(d + e*x)**(15/2)/(15*e**5) + 4*c*(d + e*x)**(13/2)*(b*e - 2*c*d)/(13*e**
5) + 2*d**2*(d + e*x)**(7/2)*(b*e - c*d)**2/(7*e**5) - 4*d*(d + e*x)**(9/2)*(b*e
- 2*c*d)*(b*e - c*d)/(9*e**5) + 2*(d + e*x)**(11/2)*(b**2*e**2 - 6*b*c*d*e + 6*
c**2*d**2)/(11*e**5)

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Mathematica [A]  time = 0.132676, size = 124, normalized size = 0.84 \[ \frac{2 (d+e x)^{7/2} \left (65 b^2 e^2 \left (8 d^2-28 d e x+63 e^2 x^2\right )+30 b c e \left (-16 d^3+56 d^2 e x-126 d e^2 x^2+231 e^3 x^3\right )+c^2 \left (128 d^4-448 d^3 e x+1008 d^2 e^2 x^2-1848 d e^3 x^3+3003 e^4 x^4\right )\right )}{45045 e^5} \]

Antiderivative was successfully verified.

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

[Out]

(2*(d + e*x)^(7/2)*(65*b^2*e^2*(8*d^2 - 28*d*e*x + 63*e^2*x^2) + 30*b*c*e*(-16*d
^3 + 56*d^2*e*x - 126*d*e^2*x^2 + 231*e^3*x^3) + c^2*(128*d^4 - 448*d^3*e*x + 10
08*d^2*e^2*x^2 - 1848*d*e^3*x^3 + 3003*e^4*x^4)))/(45045*e^5)

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Maple [A]  time = 0.008, size = 141, normalized size = 1. \[{\frac{6006\,{c}^{2}{x}^{4}{e}^{4}+13860\,bc{e}^{4}{x}^{3}-3696\,{c}^{2}d{e}^{3}{x}^{3}+8190\,{b}^{2}{e}^{4}{x}^{2}-7560\,bcd{e}^{3}{x}^{2}+2016\,{c}^{2}{d}^{2}{e}^{2}{x}^{2}-3640\,{b}^{2}d{e}^{3}x+3360\,bc{d}^{2}{e}^{2}x-896\,{c}^{2}{d}^{3}ex+1040\,{b}^{2}{d}^{2}{e}^{2}-960\,bc{d}^{3}e+256\,{c}^{2}{d}^{4}}{45045\,{e}^{5}} \left ( ex+d \right ) ^{{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/45045*(e*x+d)^(7/2)*(3003*c^2*e^4*x^4+6930*b*c*e^4*x^3-1848*c^2*d*e^3*x^3+4095
*b^2*e^4*x^2-3780*b*c*d*e^3*x^2+1008*c^2*d^2*e^2*x^2-1820*b^2*d*e^3*x+1680*b*c*d
^2*e^2*x-448*c^2*d^3*e*x+520*b^2*d^2*e^2-480*b*c*d^3*e+128*c^2*d^4)/e^5

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Maxima [A]  time = 0.698526, size = 188, normalized size = 1.28 \[ \frac{2 \,{\left (3003 \,{\left (e x + d\right )}^{\frac{15}{2}} c^{2} - 6930 \,{\left (2 \, c^{2} d - b c e\right )}{\left (e x + d\right )}^{\frac{13}{2}} + 4095 \,{\left (6 \, c^{2} d^{2} - 6 \, b c d e + b^{2} e^{2}\right )}{\left (e x + d\right )}^{\frac{11}{2}} - 10010 \,{\left (2 \, c^{2} d^{3} - 3 \, b c d^{2} e + b^{2} d e^{2}\right )}{\left (e x + d\right )}^{\frac{9}{2}} + 6435 \,{\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2}\right )}{\left (e x + d\right )}^{\frac{7}{2}}\right )}}{45045 \, e^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/45045*(3003*(e*x + d)^(15/2)*c^2 - 6930*(2*c^2*d - b*c*e)*(e*x + d)^(13/2) + 4
095*(6*c^2*d^2 - 6*b*c*d*e + b^2*e^2)*(e*x + d)^(11/2) - 10010*(2*c^2*d^3 - 3*b*
c*d^2*e + b^2*d*e^2)*(e*x + d)^(9/2) + 6435*(c^2*d^4 - 2*b*c*d^3*e + b^2*d^2*e^2
)*(e*x + d)^(7/2))/e^5

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Fricas [A]  time = 0.214587, size = 339, normalized size = 2.31 \[ \frac{2 \,{\left (3003 \, c^{2} e^{7} x^{7} + 128 \, c^{2} d^{7} - 480 \, b c d^{6} e + 520 \, b^{2} d^{5} e^{2} + 231 \,{\left (31 \, c^{2} d e^{6} + 30 \, b c e^{7}\right )} x^{6} + 63 \,{\left (71 \, c^{2} d^{2} e^{5} + 270 \, b c d e^{6} + 65 \, b^{2} e^{7}\right )} x^{5} + 35 \,{\left (c^{2} d^{3} e^{4} + 318 \, b c d^{2} e^{5} + 299 \, b^{2} d e^{6}\right )} x^{4} - 5 \,{\left (8 \, c^{2} d^{4} e^{3} - 30 \, b c d^{3} e^{4} - 1469 \, b^{2} d^{2} e^{5}\right )} x^{3} + 3 \,{\left (16 \, c^{2} d^{5} e^{2} - 60 \, b c d^{4} e^{3} + 65 \, b^{2} d^{3} e^{4}\right )} x^{2} - 4 \,{\left (16 \, c^{2} d^{6} e - 60 \, b c d^{5} e^{2} + 65 \, b^{2} d^{4} e^{3}\right )} x\right )} \sqrt{e x + d}}{45045 \, e^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/45045*(3003*c^2*e^7*x^7 + 128*c^2*d^7 - 480*b*c*d^6*e + 520*b^2*d^5*e^2 + 231*
(31*c^2*d*e^6 + 30*b*c*e^7)*x^6 + 63*(71*c^2*d^2*e^5 + 270*b*c*d*e^6 + 65*b^2*e^
7)*x^5 + 35*(c^2*d^3*e^4 + 318*b*c*d^2*e^5 + 299*b^2*d*e^6)*x^4 - 5*(8*c^2*d^4*e
^3 - 30*b*c*d^3*e^4 - 1469*b^2*d^2*e^5)*x^3 + 3*(16*c^2*d^5*e^2 - 60*b*c*d^4*e^3
+ 65*b^2*d^3*e^4)*x^2 - 4*(16*c^2*d^6*e - 60*b*c*d^5*e^2 + 65*b^2*d^4*e^3)*x)*s
qrt(e*x + d)/e^5

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Sympy [A]  time = 9.7793, size = 695, normalized size = 4.73 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*b**2*d**2*(d**2*(d + e*x)**(3/2)/3 - 2*d*(d + e*x)**(5/2)/5 + (d + e*x)**(7/2)
/7)/e**3 + 4*b**2*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**3 + 2*b**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**3 + 4*b*c*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 + 8*b*c*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 + 4*b*c*(-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**4 + 2*c**2*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 + 4*c**2*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 + 2*c**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)**(1
3/2)/13 + (d + e*x)**(15/2)/15)/e**5

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GIAC/XCAS [A]  time = 0.217735, size = 1, normalized size = 0.01 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done