3.1618 \(\int (d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx\)

Optimal. Leaf size=129 \[ -\frac{8 b^3 (d+e x)^{15/2} (b d-a e)}{15 e^5}+\frac{12 b^2 (d+e x)^{13/2} (b d-a e)^2}{13 e^5}-\frac{8 b (d+e x)^{11/2} (b d-a e)^3}{11 e^5}+\frac{2 (d+e x)^{9/2} (b d-a e)^4}{9 e^5}+\frac{2 b^4 (d+e x)^{17/2}}{17 e^5} \]

[Out]

(2*(b*d - a*e)^4*(d + e*x)^(9/2))/(9*e^5) - (8*b*(b*d - a*e)^3*(d + e*x)^(11/2))
/(11*e^5) + (12*b^2*(b*d - a*e)^2*(d + e*x)^(13/2))/(13*e^5) - (8*b^3*(b*d - a*e
)*(d + e*x)^(15/2))/(15*e^5) + (2*b^4*(d + e*x)^(17/2))/(17*e^5)

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Rubi [A]  time = 0.157147, antiderivative size = 129, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071 \[ -\frac{8 b^3 (d+e x)^{15/2} (b d-a e)}{15 e^5}+\frac{12 b^2 (d+e x)^{13/2} (b d-a e)^2}{13 e^5}-\frac{8 b (d+e x)^{11/2} (b d-a e)^3}{11 e^5}+\frac{2 (d+e x)^{9/2} (b d-a e)^4}{9 e^5}+\frac{2 b^4 (d+e x)^{17/2}}{17 e^5} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^(7/2)*(a^2 + 2*a*b*x + b^2*x^2)^2,x]

[Out]

(2*(b*d - a*e)^4*(d + e*x)^(9/2))/(9*e^5) - (8*b*(b*d - a*e)^3*(d + e*x)^(11/2))
/(11*e^5) + (12*b^2*(b*d - a*e)^2*(d + e*x)^(13/2))/(13*e^5) - (8*b^3*(b*d - a*e
)*(d + e*x)^(15/2))/(15*e^5) + (2*b^4*(d + e*x)^(17/2))/(17*e^5)

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Rubi in Sympy [A]  time = 56.9613, size = 119, normalized size = 0.92 \[ \frac{2 b^{4} \left (d + e x\right )^{\frac{17}{2}}}{17 e^{5}} + \frac{8 b^{3} \left (d + e x\right )^{\frac{15}{2}} \left (a e - b d\right )}{15 e^{5}} + \frac{12 b^{2} \left (d + e x\right )^{\frac{13}{2}} \left (a e - b d\right )^{2}}{13 e^{5}} + \frac{8 b \left (d + e x\right )^{\frac{11}{2}} \left (a e - b d\right )^{3}}{11 e^{5}} + \frac{2 \left (d + e x\right )^{\frac{9}{2}} \left (a e - b d\right )^{4}}{9 e^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**(7/2)*(b**2*x**2+2*a*b*x+a**2)**2,x)

[Out]

2*b**4*(d + e*x)**(17/2)/(17*e**5) + 8*b**3*(d + e*x)**(15/2)*(a*e - b*d)/(15*e*
*5) + 12*b**2*(d + e*x)**(13/2)*(a*e - b*d)**2/(13*e**5) + 8*b*(d + e*x)**(11/2)
*(a*e - b*d)**3/(11*e**5) + 2*(d + e*x)**(9/2)*(a*e - b*d)**4/(9*e**5)

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Mathematica [A]  time = 0.240141, size = 154, normalized size = 1.19 \[ \frac{2 (d+e x)^{9/2} \left (12155 a^4 e^4+4420 a^3 b e^3 (9 e x-2 d)+510 a^2 b^2 e^2 \left (8 d^2-36 d e x+99 e^2 x^2\right )+68 a b^3 e \left (-16 d^3+72 d^2 e x-198 d e^2 x^2+429 e^3 x^3\right )+b^4 \left (128 d^4-576 d^3 e x+1584 d^2 e^2 x^2-3432 d e^3 x^3+6435 e^4 x^4\right )\right )}{109395 e^5} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^(7/2)*(a^2 + 2*a*b*x + b^2*x^2)^2,x]

[Out]

(2*(d + e*x)^(9/2)*(12155*a^4*e^4 + 4420*a^3*b*e^3*(-2*d + 9*e*x) + 510*a^2*b^2*
e^2*(8*d^2 - 36*d*e*x + 99*e^2*x^2) + 68*a*b^3*e*(-16*d^3 + 72*d^2*e*x - 198*d*e
^2*x^2 + 429*e^3*x^3) + b^4*(128*d^4 - 576*d^3*e*x + 1584*d^2*e^2*x^2 - 3432*d*e
^3*x^3 + 6435*e^4*x^4)))/(109395*e^5)

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Maple [A]  time = 0.011, size = 186, normalized size = 1.4 \[{\frac{12870\,{x}^{4}{b}^{4}{e}^{4}+58344\,{x}^{3}a{b}^{3}{e}^{4}-6864\,{x}^{3}{b}^{4}d{e}^{3}+100980\,{x}^{2}{a}^{2}{b}^{2}{e}^{4}-26928\,{x}^{2}a{b}^{3}d{e}^{3}+3168\,{x}^{2}{b}^{4}{d}^{2}{e}^{2}+79560\,x{a}^{3}b{e}^{4}-36720\,x{a}^{2}{b}^{2}d{e}^{3}+9792\,xa{b}^{3}{d}^{2}{e}^{2}-1152\,x{b}^{4}{d}^{3}e+24310\,{a}^{4}{e}^{4}-17680\,{a}^{3}bd{e}^{3}+8160\,{a}^{2}{b}^{2}{d}^{2}{e}^{2}-2176\,a{b}^{3}{d}^{3}e+256\,{b}^{4}{d}^{4}}{109395\,{e}^{5}} \left ( ex+d \right ) ^{{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^(7/2)*(b^2*x^2+2*a*b*x+a^2)^2,x)

[Out]

2/109395*(e*x+d)^(9/2)*(6435*b^4*e^4*x^4+29172*a*b^3*e^4*x^3-3432*b^4*d*e^3*x^3+
50490*a^2*b^2*e^4*x^2-13464*a*b^3*d*e^3*x^2+1584*b^4*d^2*e^2*x^2+39780*a^3*b*e^4
*x-18360*a^2*b^2*d*e^3*x+4896*a*b^3*d^2*e^2*x-576*b^4*d^3*e*x+12155*a^4*e^4-8840
*a^3*b*d*e^3+4080*a^2*b^2*d^2*e^2-1088*a*b^3*d^3*e+128*b^4*d^4)/e^5

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Maxima [A]  time = 0.733522, size = 244, normalized size = 1.89 \[ \frac{2 \,{\left (6435 \,{\left (e x + d\right )}^{\frac{17}{2}} b^{4} - 29172 \,{\left (b^{4} d - a b^{3} e\right )}{\left (e x + d\right )}^{\frac{15}{2}} + 50490 \,{\left (b^{4} d^{2} - 2 \, a b^{3} d e + a^{2} b^{2} e^{2}\right )}{\left (e x + d\right )}^{\frac{13}{2}} - 39780 \,{\left (b^{4} d^{3} - 3 \, a b^{3} d^{2} e + 3 \, a^{2} b^{2} d e^{2} - a^{3} b e^{3}\right )}{\left (e x + d\right )}^{\frac{11}{2}} + 12155 \,{\left (b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}\right )}{\left (e x + d\right )}^{\frac{9}{2}}\right )}}{109395 \, e^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^2*(e*x + d)^(7/2),x, algorithm="maxima")

[Out]

2/109395*(6435*(e*x + d)^(17/2)*b^4 - 29172*(b^4*d - a*b^3*e)*(e*x + d)^(15/2) +
50490*(b^4*d^2 - 2*a*b^3*d*e + a^2*b^2*e^2)*(e*x + d)^(13/2) - 39780*(b^4*d^3 -
3*a*b^3*d^2*e + 3*a^2*b^2*d*e^2 - a^3*b*e^3)*(e*x + d)^(11/2) + 12155*(b^4*d^4
- 4*a*b^3*d^3*e + 6*a^2*b^2*d^2*e^2 - 4*a^3*b*d*e^3 + a^4*e^4)*(e*x + d)^(9/2))/
e^5

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Fricas [A]  time = 0.207897, size = 599, normalized size = 4.64 \[ \frac{2 \,{\left (6435 \, b^{4} e^{8} x^{8} + 128 \, b^{4} d^{8} - 1088 \, a b^{3} d^{7} e + 4080 \, a^{2} b^{2} d^{6} e^{2} - 8840 \, a^{3} b d^{5} e^{3} + 12155 \, a^{4} d^{4} e^{4} + 1716 \,{\left (13 \, b^{4} d e^{7} + 17 \, a b^{3} e^{8}\right )} x^{7} + 66 \,{\left (401 \, b^{4} d^{2} e^{6} + 1564 \, a b^{3} d e^{7} + 765 \, a^{2} b^{2} e^{8}\right )} x^{6} + 36 \,{\left (303 \, b^{4} d^{3} e^{5} + 3502 \, a b^{3} d^{2} e^{6} + 5100 \, a^{2} b^{2} d e^{7} + 1105 \, a^{3} b e^{8}\right )} x^{5} + 5 \,{\left (7 \, b^{4} d^{4} e^{4} + 10880 \, a b^{3} d^{3} e^{5} + 46716 \, a^{2} b^{2} d^{2} e^{6} + 30056 \, a^{3} b d e^{7} + 2431 \, a^{4} e^{8}\right )} x^{4} - 20 \,{\left (2 \, b^{4} d^{5} e^{3} - 17 \, a b^{3} d^{4} e^{4} - 5406 \, a^{2} b^{2} d^{3} e^{5} - 10166 \, a^{3} b d^{2} e^{6} - 2431 \, a^{4} d e^{7}\right )} x^{3} + 6 \,{\left (8 \, b^{4} d^{6} e^{2} - 68 \, a b^{3} d^{5} e^{3} + 255 \, a^{2} b^{2} d^{4} e^{4} + 17680 \, a^{3} b d^{3} e^{5} + 12155 \, a^{4} d^{2} e^{6}\right )} x^{2} - 4 \,{\left (16 \, b^{4} d^{7} e - 136 \, a b^{3} d^{6} e^{2} + 510 \, a^{2} b^{2} d^{5} e^{3} - 1105 \, a^{3} b d^{4} e^{4} - 12155 \, a^{4} d^{3} e^{5}\right )} x\right )} \sqrt{e x + d}}{109395 \, e^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^2*(e*x + d)^(7/2),x, algorithm="fricas")

[Out]

2/109395*(6435*b^4*e^8*x^8 + 128*b^4*d^8 - 1088*a*b^3*d^7*e + 4080*a^2*b^2*d^6*e
^2 - 8840*a^3*b*d^5*e^3 + 12155*a^4*d^4*e^4 + 1716*(13*b^4*d*e^7 + 17*a*b^3*e^8)
*x^7 + 66*(401*b^4*d^2*e^6 + 1564*a*b^3*d*e^7 + 765*a^2*b^2*e^8)*x^6 + 36*(303*b
^4*d^3*e^5 + 3502*a*b^3*d^2*e^6 + 5100*a^2*b^2*d*e^7 + 1105*a^3*b*e^8)*x^5 + 5*(
7*b^4*d^4*e^4 + 10880*a*b^3*d^3*e^5 + 46716*a^2*b^2*d^2*e^6 + 30056*a^3*b*d*e^7
+ 2431*a^4*e^8)*x^4 - 20*(2*b^4*d^5*e^3 - 17*a*b^3*d^4*e^4 - 5406*a^2*b^2*d^3*e^
5 - 10166*a^3*b*d^2*e^6 - 2431*a^4*d*e^7)*x^3 + 6*(8*b^4*d^6*e^2 - 68*a*b^3*d^5*
e^3 + 255*a^2*b^2*d^4*e^4 + 17680*a^3*b*d^3*e^5 + 12155*a^4*d^2*e^6)*x^2 - 4*(16
*b^4*d^7*e - 136*a*b^3*d^6*e^2 + 510*a^2*b^2*d^5*e^3 - 1105*a^3*b*d^4*e^4 - 1215
5*a^4*d^3*e^5)*x)*sqrt(e*x + d)/e^5

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Sympy [A]  time = 37.7509, size = 903, normalized size = 7. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**(7/2)*(b**2*x**2+2*a*b*x+a**2)**2,x)

[Out]

Piecewise((2*a**4*d**4*sqrt(d + e*x)/(9*e) + 8*a**4*d**3*x*sqrt(d + e*x)/9 + 4*a
**4*d**2*e*x**2*sqrt(d + e*x)/3 + 8*a**4*d*e**2*x**3*sqrt(d + e*x)/9 + 2*a**4*e*
*3*x**4*sqrt(d + e*x)/9 - 16*a**3*b*d**5*sqrt(d + e*x)/(99*e**2) + 8*a**3*b*d**4
*x*sqrt(d + e*x)/(99*e) + 64*a**3*b*d**3*x**2*sqrt(d + e*x)/33 + 368*a**3*b*d**2
*e*x**3*sqrt(d + e*x)/99 + 272*a**3*b*d*e**2*x**4*sqrt(d + e*x)/99 + 8*a**3*b*e*
*3*x**5*sqrt(d + e*x)/11 + 32*a**2*b**2*d**6*sqrt(d + e*x)/(429*e**3) - 16*a**2*
b**2*d**5*x*sqrt(d + e*x)/(429*e**2) + 4*a**2*b**2*d**4*x**2*sqrt(d + e*x)/(143*
e) + 848*a**2*b**2*d**3*x**3*sqrt(d + e*x)/429 + 1832*a**2*b**2*d**2*e*x**4*sqrt
(d + e*x)/429 + 480*a**2*b**2*d*e**2*x**5*sqrt(d + e*x)/143 + 12*a**2*b**2*e**3*
x**6*sqrt(d + e*x)/13 - 128*a*b**3*d**7*sqrt(d + e*x)/(6435*e**4) + 64*a*b**3*d*
*6*x*sqrt(d + e*x)/(6435*e**3) - 16*a*b**3*d**5*x**2*sqrt(d + e*x)/(2145*e**2) +
8*a*b**3*d**4*x**3*sqrt(d + e*x)/(1287*e) + 1280*a*b**3*d**3*x**4*sqrt(d + e*x)
/1287 + 1648*a*b**3*d**2*e*x**5*sqrt(d + e*x)/715 + 368*a*b**3*d*e**2*x**6*sqrt(
d + e*x)/195 + 8*a*b**3*e**3*x**7*sqrt(d + e*x)/15 + 256*b**4*d**8*sqrt(d + e*x)
/(109395*e**5) - 128*b**4*d**7*x*sqrt(d + e*x)/(109395*e**4) + 32*b**4*d**6*x**2
*sqrt(d + e*x)/(36465*e**3) - 16*b**4*d**5*x**3*sqrt(d + e*x)/(21879*e**2) + 14*
b**4*d**4*x**4*sqrt(d + e*x)/(21879*e) + 2424*b**4*d**3*x**5*sqrt(d + e*x)/12155
+ 1604*b**4*d**2*e*x**6*sqrt(d + e*x)/3315 + 104*b**4*d*e**2*x**7*sqrt(d + e*x)
/255 + 2*b**4*e**3*x**8*sqrt(d + e*x)/17, Ne(e, 0)), (d**(7/2)*(a**4*x + 2*a**3*
b*x**2 + 2*a**2*b**2*x**3 + a*b**3*x**4 + b**4*x**5/5), True))

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GIAC/XCAS [A]  time = 0.243372, size = 1, normalized size = 0.01 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^2*(e*x + d)^(7/2),x, algorithm="giac")

[Out]

Done