3.2043 \(\int (a+b x) (d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right ) \, dx\)

Optimal. Leaf size=100 \[ -\frac{6 b^2 (d+e x)^{11/2} (b d-a e)}{11 e^4}+\frac{2 b (d+e x)^{9/2} (b d-a e)^2}{3 e^4}-\frac{2 (d+e x)^{7/2} (b d-a e)^3}{7 e^4}+\frac{2 b^3 (d+e x)^{13/2}}{13 e^4} \]

[Out]

(-2*(b*d - a*e)^3*(d + e*x)^(7/2))/(7*e^4) + (2*b*(b*d - a*e)^2*(d + e*x)^(9/2))
/(3*e^4) - (6*b^2*(b*d - a*e)*(d + e*x)^(11/2))/(11*e^4) + (2*b^3*(d + e*x)^(13/
2))/(13*e^4)

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Rubi [A]  time = 0.0860453, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065 \[ -\frac{6 b^2 (d+e x)^{11/2} (b d-a e)}{11 e^4}+\frac{2 b (d+e x)^{9/2} (b d-a e)^2}{3 e^4}-\frac{2 (d+e x)^{7/2} (b d-a e)^3}{7 e^4}+\frac{2 b^3 (d+e x)^{13/2}}{13 e^4} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(b*d - a*e)^3*(d + e*x)^(7/2))/(7*e^4) + (2*b*(b*d - a*e)^2*(d + e*x)^(9/2))
/(3*e^4) - (6*b^2*(b*d - a*e)*(d + e*x)^(11/2))/(11*e^4) + (2*b^3*(d + e*x)^(13/
2))/(13*e^4)

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Rubi in Sympy [A]  time = 44.9568, size = 92, normalized size = 0.92 \[ \frac{2 b^{3} \left (d + e x\right )^{\frac{13}{2}}}{13 e^{4}} + \frac{6 b^{2} \left (d + e x\right )^{\frac{11}{2}} \left (a e - b d\right )}{11 e^{4}} + \frac{2 b \left (d + e x\right )^{\frac{9}{2}} \left (a e - b d\right )^{2}}{3 e^{4}} + \frac{2 \left (d + e x\right )^{\frac{7}{2}} \left (a e - b d\right )^{3}}{7 e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.157939, size = 102, normalized size = 1.02 \[ \frac{2 (d+e x)^{7/2} \left (429 a^3 e^3+143 a^2 b e^2 (7 e x-2 d)+13 a b^2 e \left (8 d^2-28 d e x+63 e^2 x^2\right )+b^3 \left (-16 d^3+56 d^2 e x-126 d e^2 x^2+231 e^3 x^3\right )\right )}{3003 e^4} \]

Antiderivative was successfully verified.

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

[Out]

(2*(d + e*x)^(7/2)*(429*a^3*e^3 + 143*a^2*b*e^2*(-2*d + 7*e*x) + 13*a*b^2*e*(8*d
^2 - 28*d*e*x + 63*e^2*x^2) + b^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 = 0.012, size = 116, normalized size = 1.2 \[{\frac{462\,{x}^{3}{b}^{3}{e}^{3}+1638\,{x}^{2}a{b}^{2}{e}^{3}-252\,{x}^{2}{b}^{3}d{e}^{2}+2002\,x{a}^{2}b{e}^{3}-728\,xa{b}^{2}d{e}^{2}+112\,x{b}^{3}{d}^{2}e+858\,{a}^{3}{e}^{3}-572\,{a}^{2}bd{e}^{2}+208\,a{b}^{2}{d}^{2}e-32\,{b}^{3}{d}^{3}}{3003\,{e}^{4}} \left ( ex+d \right ) ^{{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3003*(e*x+d)^(7/2)*(231*b^3*e^3*x^3+819*a*b^2*e^3*x^2-126*b^3*d*e^2*x^2+1001*a
^2*b*e^3*x-364*a*b^2*d*e^2*x+56*b^3*d^2*e*x+429*a^3*e^3-286*a^2*b*d*e^2+104*a*b^
2*d^2*e-16*b^3*d^3)/e^4

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Maxima [A]  time = 0.729957, size = 159, normalized size = 1.59 \[ \frac{2 \,{\left (231 \,{\left (e x + d\right )}^{\frac{13}{2}} b^{3} - 819 \,{\left (b^{3} d - a b^{2} e\right )}{\left (e x + d\right )}^{\frac{11}{2}} + 1001 \,{\left (b^{3} d^{2} - 2 \, a b^{2} d e + a^{2} b e^{2}\right )}{\left (e x + d\right )}^{\frac{9}{2}} - 429 \,{\left (b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}\right )}{\left (e x + d\right )}^{\frac{7}{2}}\right )}}{3003 \, e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3003*(231*(e*x + d)^(13/2)*b^3 - 819*(b^3*d - a*b^2*e)*(e*x + d)^(11/2) + 1001
*(b^3*d^2 - 2*a*b^2*d*e + a^2*b*e^2)*(e*x + d)^(9/2) - 429*(b^3*d^3 - 3*a*b^2*d^
2*e + 3*a^2*b*d*e^2 - a^3*e^3)*(e*x + d)^(7/2))/e^4

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Fricas [A]  time = 0.278486, size = 362, normalized size = 3.62 \[ \frac{2 \,{\left (231 \, b^{3} e^{6} x^{6} - 16 \, b^{3} d^{6} + 104 \, a b^{2} d^{5} e - 286 \, a^{2} b d^{4} e^{2} + 429 \, a^{3} d^{3} e^{3} + 63 \,{\left (9 \, b^{3} d e^{5} + 13 \, a b^{2} e^{6}\right )} x^{5} + 7 \,{\left (53 \, b^{3} d^{2} e^{4} + 299 \, a b^{2} d e^{5} + 143 \, a^{2} b e^{6}\right )} x^{4} +{\left (5 \, b^{3} d^{3} e^{3} + 1469 \, a b^{2} d^{2} e^{4} + 2717 \, a^{2} b d e^{5} + 429 \, a^{3} e^{6}\right )} x^{3} - 3 \,{\left (2 \, b^{3} d^{4} e^{2} - 13 \, a b^{2} d^{3} e^{3} - 715 \, a^{2} b d^{2} e^{4} - 429 \, a^{3} d e^{5}\right )} x^{2} +{\left (8 \, b^{3} d^{5} e - 52 \, a b^{2} d^{4} e^{2} + 143 \, a^{2} b d^{3} e^{3} + 1287 \, a^{3} d^{2} e^{4}\right )} x\right )} \sqrt{e x + d}}{3003 \, e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3003*(231*b^3*e^6*x^6 - 16*b^3*d^6 + 104*a*b^2*d^5*e - 286*a^2*b*d^4*e^2 + 429
*a^3*d^3*e^3 + 63*(9*b^3*d*e^5 + 13*a*b^2*e^6)*x^5 + 7*(53*b^3*d^2*e^4 + 299*a*b
^2*d*e^5 + 143*a^2*b*e^6)*x^4 + (5*b^3*d^3*e^3 + 1469*a*b^2*d^2*e^4 + 2717*a^2*b
*d*e^5 + 429*a^3*e^6)*x^3 - 3*(2*b^3*d^4*e^2 - 13*a*b^2*d^3*e^3 - 715*a^2*b*d^2*
e^4 - 429*a^3*d*e^5)*x^2 + (8*b^3*d^5*e - 52*a*b^2*d^4*e^2 + 143*a^2*b*d^3*e^3 +
1287*a^3*d^2*e^4)*x)*sqrt(e*x + d)/e^4

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Sympy [A]  time = 12.6455, size = 549, normalized size = 5.49 \[ \begin{cases} \frac{2 a^{3} d^{3} \sqrt{d + e x}}{7 e} + \frac{6 a^{3} d^{2} x \sqrt{d + e x}}{7} + \frac{6 a^{3} d e x^{2} \sqrt{d + e x}}{7} + \frac{2 a^{3} e^{2} x^{3} \sqrt{d + e x}}{7} - \frac{4 a^{2} b d^{4} \sqrt{d + e x}}{21 e^{2}} + \frac{2 a^{2} b d^{3} x \sqrt{d + e x}}{21 e} + \frac{10 a^{2} b d^{2} x^{2} \sqrt{d + e x}}{7} + \frac{38 a^{2} b d e x^{3} \sqrt{d + e x}}{21} + \frac{2 a^{2} b e^{2} x^{4} \sqrt{d + e x}}{3} + \frac{16 a b^{2} d^{5} \sqrt{d + e x}}{231 e^{3}} - \frac{8 a b^{2} d^{4} x \sqrt{d + e x}}{231 e^{2}} + \frac{2 a b^{2} d^{3} x^{2} \sqrt{d + e x}}{77 e} + \frac{226 a b^{2} d^{2} x^{3} \sqrt{d + e x}}{231} + \frac{46 a b^{2} d e x^{4} \sqrt{d + e x}}{33} + \frac{6 a b^{2} e^{2} x^{5} \sqrt{d + e x}}{11} - \frac{32 b^{3} d^{6} \sqrt{d + e x}}{3003 e^{4}} + \frac{16 b^{3} d^{5} x \sqrt{d + e x}}{3003 e^{3}} - \frac{4 b^{3} d^{4} x^{2} \sqrt{d + e x}}{1001 e^{2}} + \frac{10 b^{3} d^{3} x^{3} \sqrt{d + e x}}{3003 e} + \frac{106 b^{3} d^{2} x^{4} \sqrt{d + e x}}{429} + \frac{54 b^{3} d e x^{5} \sqrt{d + e x}}{143} + \frac{2 b^{3} e^{2} x^{6} \sqrt{d + e x}}{13} & \text{for}\: e \neq 0 \\d^{\frac{5}{2}} \left (a^{3} x + \frac{3 a^{2} b x^{2}}{2} + a b^{2} x^{3} + \frac{b^{3} x^{4}}{4}\right ) & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((2*a**3*d**3*sqrt(d + e*x)/(7*e) + 6*a**3*d**2*x*sqrt(d + e*x)/7 + 6*a
**3*d*e*x**2*sqrt(d + e*x)/7 + 2*a**3*e**2*x**3*sqrt(d + e*x)/7 - 4*a**2*b*d**4*
sqrt(d + e*x)/(21*e**2) + 2*a**2*b*d**3*x*sqrt(d + e*x)/(21*e) + 10*a**2*b*d**2*
x**2*sqrt(d + e*x)/7 + 38*a**2*b*d*e*x**3*sqrt(d + e*x)/21 + 2*a**2*b*e**2*x**4*
sqrt(d + e*x)/3 + 16*a*b**2*d**5*sqrt(d + e*x)/(231*e**3) - 8*a*b**2*d**4*x*sqrt
(d + e*x)/(231*e**2) + 2*a*b**2*d**3*x**2*sqrt(d + e*x)/(77*e) + 226*a*b**2*d**2
*x**3*sqrt(d + e*x)/231 + 46*a*b**2*d*e*x**4*sqrt(d + e*x)/33 + 6*a*b**2*e**2*x*
*5*sqrt(d + e*x)/11 - 32*b**3*d**6*sqrt(d + e*x)/(3003*e**4) + 16*b**3*d**5*x*sq
rt(d + e*x)/(3003*e**3) - 4*b**3*d**4*x**2*sqrt(d + e*x)/(1001*e**2) + 10*b**3*d
**3*x**3*sqrt(d + e*x)/(3003*e) + 106*b**3*d**2*x**4*sqrt(d + e*x)/429 + 54*b**3
*d*e*x**5*sqrt(d + e*x)/143 + 2*b**3*e**2*x**6*sqrt(d + e*x)/13, Ne(e, 0)), (d**
(5/2)*(a**3*x + 3*a**2*b*x**2/2 + a*b**2*x**3 + b**3*x**4/4), True))

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GIAC/XCAS [A]  time = 0.318348, 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)*(b*x + a)*(e*x + d)^(5/2),x, algorithm="giac")

[Out]

Done