3.2104 \(\int \frac{(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx\)

Optimal. Leaf size=258 \[ \frac{4 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^2}{e^5 (a+b x)}-\frac{8 b \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^3}{e^5 (a+b x)}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{e^5 (a+b x) \sqrt{d+e x}}+\frac{2 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{7/2}}{7 e^5 (a+b x)}-\frac{8 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)}{5 e^5 (a+b x)} \]

[Out]

(-2*(b*d - a*e)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^5*(a + b*x)*Sqrt[d + e*x]) -
(8*b*(b*d - a*e)^3*Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^5*(a + b*x))
+ (4*b^2*(b*d - a*e)^2*(d + e*x)^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^5*(a +
b*x)) - (8*b^3*(b*d - a*e)*(d + e*x)^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*e^
5*(a + b*x)) + (2*b^4*(d + e*x)^(7/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*e^5*(a +
b*x))

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Rubi [A]  time = 0.307642, antiderivative size = 258, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.086 \[ \frac{4 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^2}{e^5 (a+b x)}-\frac{8 b \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^3}{e^5 (a+b x)}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{e^5 (a+b x) \sqrt{d+e x}}+\frac{2 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{7/2}}{7 e^5 (a+b x)}-\frac{8 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)}{5 e^5 (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(b*d - a*e)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^5*(a + b*x)*Sqrt[d + e*x]) -
(8*b*(b*d - a*e)^3*Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^5*(a + b*x))
+ (4*b^2*(b*d - a*e)^2*(d + e*x)^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^5*(a +
b*x)) - (8*b^3*(b*d - a*e)*(d + e*x)^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*e^
5*(a + b*x)) + (2*b^4*(d + e*x)^(7/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*e^5*(a +
b*x))

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Rubi in Sympy [A]  time = 38.2836, size = 223, normalized size = 0.86 \[ \frac{16 b \sqrt{d + e x} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{7 e^{2}} + \frac{32 b \left (3 a + 3 b x\right ) \sqrt{d + e x} \left (a e - b d\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{35 e^{3}} + \frac{128 b \sqrt{d + e x} \left (a e - b d\right )^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{35 e^{4}} + \frac{256 b \sqrt{d + e x} \left (a e - b d\right )^{3} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{35 e^{5} \left (a + b x\right )} - \frac{2 \left (a + b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{e \sqrt{d + e x}} \]

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)**(3/2)/(e*x+d)**(3/2),x)

[Out]

16*b*sqrt(d + e*x)*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(7*e**2) + 32*b*(3*a + 3*
b*x)*sqrt(d + e*x)*(a*e - b*d)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(35*e**3) + 128*
b*sqrt(d + e*x)*(a*e - b*d)**2*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(35*e**4) + 256*
b*sqrt(d + e*x)*(a*e - b*d)**3*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(35*e**5*(a + b*
x)) - 2*(a + b*x)*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(e*sqrt(d + e*x))

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Mathematica [A]  time = 0.165275, size = 170, normalized size = 0.66 \[ -\frac{2 \sqrt{(a+b x)^2} \left (35 a^4 e^4-140 a^3 b e^3 (2 d+e x)+70 a^2 b^2 e^2 \left (8 d^2+4 d e x-e^2 x^2\right )-28 a b^3 e \left (16 d^3+8 d^2 e x-2 d e^2 x^2+e^3 x^3\right )+b^4 \left (128 d^4+64 d^3 e x-16 d^2 e^2 x^2+8 d e^3 x^3-5 e^4 x^4\right )\right )}{35 e^5 (a+b x) \sqrt{d+e x}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[(a + b*x)^2]*(35*a^4*e^4 - 140*a^3*b*e^3*(2*d + e*x) + 70*a^2*b^2*e^2*(
8*d^2 + 4*d*e*x - e^2*x^2) - 28*a*b^3*e*(16*d^3 + 8*d^2*e*x - 2*d*e^2*x^2 + e^3*
x^3) + b^4*(128*d^4 + 64*d^3*e*x - 16*d^2*e^2*x^2 + 8*d*e^3*x^3 - 5*e^4*x^4)))/(
35*e^5*(a + b*x)*Sqrt[d + e*x])

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Maple [A]  time = 0.011, size = 202, normalized size = 0.8 \[ -{\frac{-10\,{x}^{4}{b}^{4}{e}^{4}-56\,{x}^{3}a{b}^{3}{e}^{4}+16\,{x}^{3}{b}^{4}d{e}^{3}-140\,{x}^{2}{a}^{2}{b}^{2}{e}^{4}+112\,{x}^{2}a{b}^{3}d{e}^{3}-32\,{x}^{2}{b}^{4}{d}^{2}{e}^{2}-280\,x{a}^{3}b{e}^{4}+560\,x{a}^{2}{b}^{2}d{e}^{3}-448\,xa{b}^{3}{d}^{2}{e}^{2}+128\,x{b}^{4}{d}^{3}e+70\,{a}^{4}{e}^{4}-560\,{a}^{3}bd{e}^{3}+1120\,{a}^{2}{b}^{2}{d}^{2}{e}^{2}-896\,a{b}^{3}{d}^{3}e+256\,{b}^{4}{d}^{4}}{35\, \left ( bx+a \right ) ^{3}{e}^{5}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}{\frac{1}{\sqrt{ex+d}}}} \]

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)^(3/2)/(e*x+d)^(3/2),x)

[Out]

-2/35/(e*x+d)^(1/2)*(-5*b^4*e^4*x^4-28*a*b^3*e^4*x^3+8*b^4*d*e^3*x^3-70*a^2*b^2*
e^4*x^2+56*a*b^3*d*e^3*x^2-16*b^4*d^2*e^2*x^2-140*a^3*b*e^4*x+280*a^2*b^2*d*e^3*
x-224*a*b^3*d^2*e^2*x+64*b^4*d^3*e*x+35*a^4*e^4-280*a^3*b*d*e^3+560*a^2*b^2*d^2*
e^2-448*a*b^3*d^3*e+128*b^4*d^4)*((b*x+a)^2)^(3/2)/e^5/(b*x+a)^3

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Maxima [A]  time = 0.732708, size = 381, normalized size = 1.48 \[ \frac{2 \,{\left (b^{3} e^{3} x^{3} + 16 \, b^{3} d^{3} - 40 \, a b^{2} d^{2} e + 30 \, a^{2} b d e^{2} - 5 \, a^{3} e^{3} -{\left (2 \, b^{3} d e^{2} - 5 \, a b^{2} e^{3}\right )} x^{2} +{\left (8 \, b^{3} d^{2} e - 20 \, a b^{2} d e^{2} + 15 \, a^{2} b e^{3}\right )} x\right )} a}{5 \, \sqrt{e x + d} e^{4}} + \frac{2 \,{\left (5 \, b^{3} e^{4} x^{4} - 128 \, b^{3} d^{4} + 336 \, a b^{2} d^{3} e - 280 \, a^{2} b d^{2} e^{2} + 70 \, a^{3} d e^{3} -{\left (8 \, b^{3} d e^{3} - 21 \, a b^{2} e^{4}\right )} x^{3} +{\left (16 \, b^{3} d^{2} e^{2} - 42 \, a b^{2} d e^{3} + 35 \, a^{2} b e^{4}\right )} x^{2} -{\left (64 \, b^{3} d^{3} e - 168 \, a b^{2} d^{2} e^{2} + 140 \, a^{2} b d e^{3} - 35 \, a^{3} e^{4}\right )} x\right )} b}{35 \, \sqrt{e x + d} e^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/5*(b^3*e^3*x^3 + 16*b^3*d^3 - 40*a*b^2*d^2*e + 30*a^2*b*d*e^2 - 5*a^3*e^3 - (2
*b^3*d*e^2 - 5*a*b^2*e^3)*x^2 + (8*b^3*d^2*e - 20*a*b^2*d*e^2 + 15*a^2*b*e^3)*x)
*a/(sqrt(e*x + d)*e^4) + 2/35*(5*b^3*e^4*x^4 - 128*b^3*d^4 + 336*a*b^2*d^3*e - 2
80*a^2*b*d^2*e^2 + 70*a^3*d*e^3 - (8*b^3*d*e^3 - 21*a*b^2*e^4)*x^3 + (16*b^3*d^2
*e^2 - 42*a*b^2*d*e^3 + 35*a^2*b*e^4)*x^2 - (64*b^3*d^3*e - 168*a*b^2*d^2*e^2 +
140*a^2*b*d*e^3 - 35*a^3*e^4)*x)*b/(sqrt(e*x + d)*e^5)

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Fricas [A]  time = 0.281221, size = 246, normalized size = 0.95 \[ \frac{2 \,{\left (5 \, b^{4} e^{4} x^{4} - 128 \, b^{4} d^{4} + 448 \, a b^{3} d^{3} e - 560 \, a^{2} b^{2} d^{2} e^{2} + 280 \, a^{3} b d e^{3} - 35 \, a^{4} e^{4} - 4 \,{\left (2 \, b^{4} d e^{3} - 7 \, a b^{3} e^{4}\right )} x^{3} + 2 \,{\left (8 \, b^{4} d^{2} e^{2} - 28 \, a b^{3} d e^{3} + 35 \, a^{2} b^{2} e^{4}\right )} x^{2} - 4 \,{\left (16 \, b^{4} d^{3} e - 56 \, a b^{3} d^{2} e^{2} + 70 \, a^{2} b^{2} d e^{3} - 35 \, a^{3} b e^{4}\right )} x\right )}}{35 \, \sqrt{e x + d} e^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (a + b x\right ) \left (\left (a + b x\right )^{2}\right )^{\frac{3}{2}}}{\left (d + e x\right )^{\frac{3}{2}}}\, dx \]

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)**(3/2)/(e*x+d)**(3/2),x)

[Out]

Integral((a + b*x)*((a + b*x)**2)**(3/2)/(d + e*x)**(3/2), x)

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GIAC/XCAS [A]  time = 0.305854, size = 441, normalized size = 1.71 \[ \frac{2}{35} \,{\left (5 \,{\left (x e + d\right )}^{\frac{7}{2}} b^{4} e^{30}{\rm sign}\left (b x + a\right ) - 28 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{4} d e^{30}{\rm sign}\left (b x + a\right ) + 70 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{4} d^{2} e^{30}{\rm sign}\left (b x + a\right ) - 140 \, \sqrt{x e + d} b^{4} d^{3} e^{30}{\rm sign}\left (b x + a\right ) + 28 \,{\left (x e + d\right )}^{\frac{5}{2}} a b^{3} e^{31}{\rm sign}\left (b x + a\right ) - 140 \,{\left (x e + d\right )}^{\frac{3}{2}} a b^{3} d e^{31}{\rm sign}\left (b x + a\right ) + 420 \, \sqrt{x e + d} a b^{3} d^{2} e^{31}{\rm sign}\left (b x + a\right ) + 70 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{2} b^{2} e^{32}{\rm sign}\left (b x + a\right ) - 420 \, \sqrt{x e + d} a^{2} b^{2} d e^{32}{\rm sign}\left (b x + a\right ) + 140 \, \sqrt{x e + d} a^{3} b e^{33}{\rm sign}\left (b x + a\right )\right )} e^{\left (-35\right )} - \frac{2 \,{\left (b^{4} d^{4}{\rm sign}\left (b x + a\right ) - 4 \, a b^{3} d^{3} e{\rm sign}\left (b x + a\right ) + 6 \, a^{2} b^{2} d^{2} e^{2}{\rm sign}\left (b x + a\right ) - 4 \, a^{3} b d e^{3}{\rm sign}\left (b x + a\right ) + a^{4} e^{4}{\rm sign}\left (b x + a\right )\right )} e^{\left (-5\right )}}{\sqrt{x e + d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/35*(5*(x*e + d)^(7/2)*b^4*e^30*sign(b*x + a) - 28*(x*e + d)^(5/2)*b^4*d*e^30*s
ign(b*x + a) + 70*(x*e + d)^(3/2)*b^4*d^2*e^30*sign(b*x + a) - 140*sqrt(x*e + d)
*b^4*d^3*e^30*sign(b*x + a) + 28*(x*e + d)^(5/2)*a*b^3*e^31*sign(b*x + a) - 140*
(x*e + d)^(3/2)*a*b^3*d*e^31*sign(b*x + a) + 420*sqrt(x*e + d)*a*b^3*d^2*e^31*si
gn(b*x + a) + 70*(x*e + d)^(3/2)*a^2*b^2*e^32*sign(b*x + a) - 420*sqrt(x*e + d)*
a^2*b^2*d*e^32*sign(b*x + a) + 140*sqrt(x*e + d)*a^3*b*e^33*sign(b*x + a))*e^(-3
5) - 2*(b^4*d^4*sign(b*x + a) - 4*a*b^3*d^3*e*sign(b*x + a) + 6*a^2*b^2*d^2*e^2*
sign(b*x + a) - 4*a^3*b*d*e^3*sign(b*x + a) + a^4*e^4*sign(b*x + a))*e^(-5)/sqrt
(x*e + d)