3.2056 \(\int \frac{(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^{5/2}} \, dx\)

Optimal. Leaf size=152 \[ -\frac{2 b^4 (d+e x)^{5/2} (b d-a e)}{e^6}+\frac{20 b^3 (d+e x)^{3/2} (b d-a e)^2}{3 e^6}-\frac{20 b^2 \sqrt{d+e x} (b d-a e)^3}{e^6}-\frac{10 b (b d-a e)^4}{e^6 \sqrt{d+e x}}+\frac{2 (b d-a e)^5}{3 e^6 (d+e x)^{3/2}}+\frac{2 b^5 (d+e x)^{7/2}}{7 e^6} \]

[Out]

(2*(b*d - a*e)^5)/(3*e^6*(d + e*x)^(3/2)) - (10*b*(b*d - a*e)^4)/(e^6*Sqrt[d + e
*x]) - (20*b^2*(b*d - a*e)^3*Sqrt[d + e*x])/e^6 + (20*b^3*(b*d - a*e)^2*(d + e*x
)^(3/2))/(3*e^6) - (2*b^4*(b*d - a*e)*(d + e*x)^(5/2))/e^6 + (2*b^5*(d + e*x)^(7
/2))/(7*e^6)

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Rubi [A]  time = 0.135136, antiderivative size = 152, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.061 \[ -\frac{2 b^4 (d+e x)^{5/2} (b d-a e)}{e^6}+\frac{20 b^3 (d+e x)^{3/2} (b d-a e)^2}{3 e^6}-\frac{20 b^2 \sqrt{d+e x} (b d-a e)^3}{e^6}-\frac{10 b (b d-a e)^4}{e^6 \sqrt{d+e x}}+\frac{2 (b d-a e)^5}{3 e^6 (d+e x)^{3/2}}+\frac{2 b^5 (d+e x)^{7/2}}{7 e^6} \]

Antiderivative was successfully verified.

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

[Out]

(2*(b*d - a*e)^5)/(3*e^6*(d + e*x)^(3/2)) - (10*b*(b*d - a*e)^4)/(e^6*Sqrt[d + e
*x]) - (20*b^2*(b*d - a*e)^3*Sqrt[d + e*x])/e^6 + (20*b^3*(b*d - a*e)^2*(d + e*x
)^(3/2))/(3*e^6) - (2*b^4*(b*d - a*e)*(d + e*x)^(5/2))/e^6 + (2*b^5*(d + e*x)^(7
/2))/(7*e^6)

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Rubi in Sympy [A]  time = 71.1484, size = 141, normalized size = 0.93 \[ \frac{2 b^{5} \left (d + e x\right )^{\frac{7}{2}}}{7 e^{6}} + \frac{2 b^{4} \left (d + e x\right )^{\frac{5}{2}} \left (a e - b d\right )}{e^{6}} + \frac{20 b^{3} \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right )^{2}}{3 e^{6}} + \frac{20 b^{2} \sqrt{d + e x} \left (a e - b d\right )^{3}}{e^{6}} - \frac{10 b \left (a e - b d\right )^{4}}{e^{6} \sqrt{d + e x}} - \frac{2 \left (a e - b d\right )^{5}}{3 e^{6} \left (d + e x\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*b**5*(d + e*x)**(7/2)/(7*e**6) + 2*b**4*(d + e*x)**(5/2)*(a*e - b*d)/e**6 + 20
*b**3*(d + e*x)**(3/2)*(a*e - b*d)**2/(3*e**6) + 20*b**2*sqrt(d + e*x)*(a*e - b*
d)**3/e**6 - 10*b*(a*e - b*d)**4/(e**6*sqrt(d + e*x)) - 2*(a*e - b*d)**5/(3*e**6
*(d + e*x)**(3/2))

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Mathematica [A]  time = 0.321366, size = 157, normalized size = 1.03 \[ \frac{2 \sqrt{d+e x} \left (b^3 e x \left (70 a^2 e^2-98 a b d e+37 b^2 d^2\right )+b^2 \left (210 a^3 e^3-560 a^2 b d e^2+511 a b^2 d^2 e-158 b^3 d^3\right )-3 b^4 e^2 x^2 (4 b d-7 a e)-\frac{105 b (b d-a e)^4}{d+e x}+\frac{7 (b d-a e)^5}{(d+e x)^2}+3 b^5 e^3 x^3\right )}{21 e^6} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[d + e*x]*(b^2*(-158*b^3*d^3 + 511*a*b^2*d^2*e - 560*a^2*b*d*e^2 + 210*a^
3*e^3) + b^3*e*(37*b^2*d^2 - 98*a*b*d*e + 70*a^2*e^2)*x - 3*b^4*e^2*(4*b*d - 7*a
*e)*x^2 + 3*b^5*e^3*x^3 + (7*(b*d - a*e)^5)/(d + e*x)^2 - (105*b*(b*d - a*e)^4)/
(d + e*x)))/(21*e^6)

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Maple [B]  time = 0.01, size = 273, normalized size = 1.8 \[ -{\frac{-6\,{x}^{5}{b}^{5}{e}^{5}-42\,{x}^{4}a{b}^{4}{e}^{5}+12\,{x}^{4}{b}^{5}d{e}^{4}-140\,{x}^{3}{a}^{2}{b}^{3}{e}^{5}+112\,{x}^{3}a{b}^{4}d{e}^{4}-32\,{x}^{3}{b}^{5}{d}^{2}{e}^{3}-420\,{x}^{2}{a}^{3}{b}^{2}{e}^{5}+840\,{x}^{2}{a}^{2}{b}^{3}d{e}^{4}-672\,{x}^{2}a{b}^{4}{d}^{2}{e}^{3}+192\,{x}^{2}{b}^{5}{d}^{3}{e}^{2}+210\,x{a}^{4}b{e}^{5}-1680\,x{a}^{3}{b}^{2}d{e}^{4}+3360\,x{a}^{2}{b}^{3}{d}^{2}{e}^{3}-2688\,xa{b}^{4}{d}^{3}{e}^{2}+768\,x{b}^{5}{d}^{4}e+14\,{a}^{5}{e}^{5}+140\,{a}^{4}bd{e}^{4}-1120\,{a}^{3}{b}^{2}{d}^{2}{e}^{3}+2240\,{a}^{2}{b}^{3}{d}^{3}{e}^{2}-1792\,a{b}^{4}{d}^{4}e+512\,{b}^{5}{d}^{5}}{21\,{e}^{6}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/21*(-3*b^5*e^5*x^5-21*a*b^4*e^5*x^4+6*b^5*d*e^4*x^4-70*a^2*b^3*e^5*x^3+56*a*b
^4*d*e^4*x^3-16*b^5*d^2*e^3*x^3-210*a^3*b^2*e^5*x^2+420*a^2*b^3*d*e^4*x^2-336*a*
b^4*d^2*e^3*x^2+96*b^5*d^3*e^2*x^2+105*a^4*b*e^5*x-840*a^3*b^2*d*e^4*x+1680*a^2*
b^3*d^2*e^3*x-1344*a*b^4*d^3*e^2*x+384*b^5*d^4*e*x+7*a^5*e^5+70*a^4*b*d*e^4-560*
a^3*b^2*d^2*e^3+1120*a^2*b^3*d^3*e^2-896*a*b^4*d^4*e+256*b^5*d^5)/(e*x+d)^(3/2)/
e^6

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Maxima [A]  time = 0.714841, size = 358, normalized size = 2.36 \[ \frac{2 \,{\left (\frac{3 \,{\left (e x + d\right )}^{\frac{7}{2}} b^{5} - 21 \,{\left (b^{5} d - a b^{4} e\right )}{\left (e x + d\right )}^{\frac{5}{2}} + 70 \,{\left (b^{5} d^{2} - 2 \, a b^{4} d e + a^{2} b^{3} e^{2}\right )}{\left (e x + d\right )}^{\frac{3}{2}} - 210 \,{\left (b^{5} d^{3} - 3 \, a b^{4} d^{2} e + 3 \, a^{2} b^{3} d e^{2} - a^{3} b^{2} e^{3}\right )} \sqrt{e x + d}}{e^{5}} + \frac{7 \,{\left (b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5} - 15 \,{\left (b^{5} d^{4} - 4 \, a b^{4} d^{3} e + 6 \, a^{2} b^{3} d^{2} e^{2} - 4 \, a^{3} b^{2} d e^{3} + a^{4} b e^{4}\right )}{\left (e x + d\right )}\right )}}{{\left (e x + d\right )}^{\frac{3}{2}} e^{5}}\right )}}{21 \, e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/21*((3*(e*x + d)^(7/2)*b^5 - 21*(b^5*d - a*b^4*e)*(e*x + d)^(5/2) + 70*(b^5*d^
2 - 2*a*b^4*d*e + a^2*b^3*e^2)*(e*x + d)^(3/2) - 210*(b^5*d^3 - 3*a*b^4*d^2*e +
3*a^2*b^3*d*e^2 - a^3*b^2*e^3)*sqrt(e*x + d))/e^5 + 7*(b^5*d^5 - 5*a*b^4*d^4*e +
10*a^2*b^3*d^3*e^2 - 10*a^3*b^2*d^2*e^3 + 5*a^4*b*d*e^4 - a^5*e^5 - 15*(b^5*d^4
- 4*a*b^4*d^3*e + 6*a^2*b^3*d^2*e^2 - 4*a^3*b^2*d*e^3 + a^4*b*e^4)*(e*x + d))/(
(e*x + d)^(3/2)*e^5))/e

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Fricas [A]  time = 0.284265, size = 367, normalized size = 2.41 \[ \frac{2 \,{\left (3 \, b^{5} e^{5} x^{5} - 256 \, b^{5} d^{5} + 896 \, a b^{4} d^{4} e - 1120 \, a^{2} b^{3} d^{3} e^{2} + 560 \, a^{3} b^{2} d^{2} e^{3} - 70 \, a^{4} b d e^{4} - 7 \, a^{5} e^{5} - 3 \,{\left (2 \, b^{5} d e^{4} - 7 \, a b^{4} e^{5}\right )} x^{4} + 2 \,{\left (8 \, b^{5} d^{2} e^{3} - 28 \, a b^{4} d e^{4} + 35 \, a^{2} b^{3} e^{5}\right )} x^{3} - 6 \,{\left (16 \, b^{5} d^{3} e^{2} - 56 \, a b^{4} d^{2} e^{3} + 70 \, a^{2} b^{3} d e^{4} - 35 \, a^{3} b^{2} e^{5}\right )} x^{2} - 3 \,{\left (128 \, b^{5} d^{4} e - 448 \, a b^{4} d^{3} e^{2} + 560 \, a^{2} b^{3} d^{2} e^{3} - 280 \, a^{3} b^{2} d e^{4} + 35 \, a^{4} b e^{5}\right )} x\right )}}{21 \,{\left (e^{7} x + d e^{6}\right )} \sqrt{e x + d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/21*(3*b^5*e^5*x^5 - 256*b^5*d^5 + 896*a*b^4*d^4*e - 1120*a^2*b^3*d^3*e^2 + 560
*a^3*b^2*d^2*e^3 - 70*a^4*b*d*e^4 - 7*a^5*e^5 - 3*(2*b^5*d*e^4 - 7*a*b^4*e^5)*x^
4 + 2*(8*b^5*d^2*e^3 - 28*a*b^4*d*e^4 + 35*a^2*b^3*e^5)*x^3 - 6*(16*b^5*d^3*e^2
- 56*a*b^4*d^2*e^3 + 70*a^2*b^3*d*e^4 - 35*a^3*b^2*e^5)*x^2 - 3*(128*b^5*d^4*e -
448*a*b^4*d^3*e^2 + 560*a^2*b^3*d^2*e^3 - 280*a^3*b^2*d*e^4 + 35*a^4*b*e^5)*x)/
((e^7*x + d*e^6)*sqrt(e*x + d))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (a + b x\right )^{5}}{\left (d + e x\right )^{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((a + b*x)**5/(d + e*x)**(5/2), x)

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GIAC/XCAS [A]  time = 0.299484, size = 451, normalized size = 2.97 \[ \frac{2}{21} \,{\left (3 \,{\left (x e + d\right )}^{\frac{7}{2}} b^{5} e^{36} - 21 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{5} d e^{36} + 70 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{5} d^{2} e^{36} - 210 \, \sqrt{x e + d} b^{5} d^{3} e^{36} + 21 \,{\left (x e + d\right )}^{\frac{5}{2}} a b^{4} e^{37} - 140 \,{\left (x e + d\right )}^{\frac{3}{2}} a b^{4} d e^{37} + 630 \, \sqrt{x e + d} a b^{4} d^{2} e^{37} + 70 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{2} b^{3} e^{38} - 630 \, \sqrt{x e + d} a^{2} b^{3} d e^{38} + 210 \, \sqrt{x e + d} a^{3} b^{2} e^{39}\right )} e^{\left (-42\right )} - \frac{2 \,{\left (15 \,{\left (x e + d\right )} b^{5} d^{4} - b^{5} d^{5} - 60 \,{\left (x e + d\right )} a b^{4} d^{3} e + 5 \, a b^{4} d^{4} e + 90 \,{\left (x e + d\right )} a^{2} b^{3} d^{2} e^{2} - 10 \, a^{2} b^{3} d^{3} e^{2} - 60 \,{\left (x e + d\right )} a^{3} b^{2} d e^{3} + 10 \, a^{3} b^{2} d^{2} e^{3} + 15 \,{\left (x e + d\right )} a^{4} b e^{4} - 5 \, a^{4} b d e^{4} + a^{5} e^{5}\right )} e^{\left (-6\right )}}{3 \,{\left (x e + d\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/21*(3*(x*e + d)^(7/2)*b^5*e^36 - 21*(x*e + d)^(5/2)*b^5*d*e^36 + 70*(x*e + d)^
(3/2)*b^5*d^2*e^36 - 210*sqrt(x*e + d)*b^5*d^3*e^36 + 21*(x*e + d)^(5/2)*a*b^4*e
^37 - 140*(x*e + d)^(3/2)*a*b^4*d*e^37 + 630*sqrt(x*e + d)*a*b^4*d^2*e^37 + 70*(
x*e + d)^(3/2)*a^2*b^3*e^38 - 630*sqrt(x*e + d)*a^2*b^3*d*e^38 + 210*sqrt(x*e +
d)*a^3*b^2*e^39)*e^(-42) - 2/3*(15*(x*e + d)*b^5*d^4 - b^5*d^5 - 60*(x*e + d)*a*
b^4*d^3*e + 5*a*b^4*d^4*e + 90*(x*e + d)*a^2*b^3*d^2*e^2 - 10*a^2*b^3*d^3*e^2 -
60*(x*e + d)*a^3*b^2*d*e^3 + 10*a^3*b^2*d^2*e^3 + 15*(x*e + d)*a^4*b*e^4 - 5*a^4
*b*d*e^4 + a^5*e^5)*e^(-6)/(x*e + d)^(3/2)