3.2107 \(\int \frac{(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^{9/2}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.309471, 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 \[ -\frac{4 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{e^5 (a+b x) (d+e x)^{3/2}}+\frac{8 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{5 e^5 (a+b x) (d+e x)^{5/2}}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{7 e^5 (a+b x) (d+e x)^{7/2}}+\frac{2 b^4 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x}}{e^5 (a+b x)}+\frac{8 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{e^5 (a+b x) \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2))/(d + e*x)^(9/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 34.9009, size = 212, normalized size = 0.82 \[ \frac{128 b^{3} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{35 e^{4} \sqrt{d + e x}} - \frac{256 b^{3} \left (a e - b d\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{35 e^{5} \left (a + b x\right ) \sqrt{d + e x}} - \frac{32 b^{2} \left (3 a + 3 b x\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{105 e^{3} \left (d + e x\right )^{\frac{3}{2}}} - \frac{16 b \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{35 e^{2} \left (d + e x\right )^{\frac{5}{2}}} - \frac{2 \left (a + b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{7 e \left (d + e x\right )^{\frac{7}{2}}} \]

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)**(3/2)/(e*x+d)**(9/2),x)

[Out]

128*b**3*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(35*e**4*sqrt(d + e*x)) - 256*b**3*(a*
e - b*d)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(35*e**5*(a + b*x)*sqrt(d + e*x)) - 32
*b**2*(3*a + 3*b*x)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(105*e**3*(d + e*x)**(3/2))
- 16*b*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(35*e**2*(d + e*x)**(5/2)) - 2*(a +
b*x)*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(7*e*(d + e*x)**(7/2))

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Mathematica [A]  time = 0.272347, size = 121, normalized size = 0.47 \[ \frac{2 \sqrt{(a+b x)^2} \sqrt{d+e x} \left (\frac{140 b^3 (b d-a e)}{d+e x}-\frac{70 b^2 (b d-a e)^2}{(d+e x)^2}+\frac{28 b (b d-a e)^3}{(d+e x)^3}-\frac{5 (b d-a e)^4}{(d+e x)^4}+35 b^4\right )}{35 e^5 (a+b x)} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2))/(d + e*x)^(9/2),x]

[Out]

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

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Maple [A]  time = 0.012, size = 202, normalized size = 0.8 \[ -{\frac{-70\,{x}^{4}{b}^{4}{e}^{4}+280\,{x}^{3}a{b}^{3}{e}^{4}-560\,{x}^{3}{b}^{4}d{e}^{3}+140\,{x}^{2}{a}^{2}{b}^{2}{e}^{4}+560\,{x}^{2}a{b}^{3}d{e}^{3}-1120\,{x}^{2}{b}^{4}{d}^{2}{e}^{2}+56\,x{a}^{3}b{e}^{4}+112\,x{a}^{2}{b}^{2}d{e}^{3}+448\,xa{b}^{3}{d}^{2}{e}^{2}-896\,x{b}^{4}{d}^{3}e+10\,{a}^{4}{e}^{4}+16\,{a}^{3}bd{e}^{3}+32\,{a}^{2}{b}^{2}{d}^{2}{e}^{2}+128\,a{b}^{3}{d}^{3}e-256\,{b}^{4}{d}^{4}}{35\, \left ( bx+a \right ) ^{3}{e}^{5}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}} \left ( ex+d \right ) ^{-{\frac{7}{2}}}} \]

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)^(3/2)/(e*x+d)^(9/2),x)

[Out]

-2/35/(e*x+d)^(7/2)*(-35*b^4*e^4*x^4+140*a*b^3*e^4*x^3-280*b^4*d*e^3*x^3+70*a^2*
b^2*e^4*x^2+280*a*b^3*d*e^3*x^2-560*b^4*d^2*e^2*x^2+28*a^3*b*e^4*x+56*a^2*b^2*d*
e^3*x+224*a*b^3*d^2*e^2*x-448*b^4*d^3*e*x+5*a^4*e^4+8*a^3*b*d*e^3+16*a^2*b^2*d^2
*e^2+64*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 = 0.737963, size = 470, normalized size = 1.82 \[ -\frac{2 \,{\left (35 \, b^{3} e^{3} x^{3} + 16 \, b^{3} d^{3} + 8 \, a b^{2} d^{2} e + 6 \, a^{2} b d e^{2} + 5 \, a^{3} e^{3} + 35 \,{\left (2 \, b^{3} d e^{2} + a b^{2} e^{3}\right )} x^{2} + 7 \,{\left (8 \, b^{3} d^{2} e + 4 \, a b^{2} d e^{2} + 3 \, a^{2} b e^{3}\right )} x\right )} a}{35 \,{\left (e^{7} x^{3} + 3 \, d e^{6} x^{2} + 3 \, d^{2} e^{5} x + d^{3} e^{4}\right )} \sqrt{e x + d}} + \frac{2 \,{\left (35 \, b^{3} e^{4} x^{4} + 128 \, b^{3} d^{4} - 48 \, a b^{2} d^{3} e - 8 \, a^{2} b d^{2} e^{2} - 2 \, a^{3} d e^{3} + 35 \,{\left (8 \, b^{3} d e^{3} - 3 \, a b^{2} e^{4}\right )} x^{3} + 35 \,{\left (16 \, b^{3} d^{2} e^{2} - 6 \, a b^{2} d e^{3} - a^{2} b e^{4}\right )} x^{2} + 7 \,{\left (64 \, b^{3} d^{3} e - 24 \, a b^{2} d^{2} e^{2} - 4 \, a^{2} b d e^{3} - a^{3} e^{4}\right )} x\right )} b}{35 \,{\left (e^{8} x^{3} + 3 \, d e^{7} x^{2} + 3 \, d^{2} e^{6} x + d^{3} e^{5}\right )} \sqrt{e x + d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*(b*x + a)/(e*x + d)^(9/2),x, algorithm="maxima")

[Out]

-2/35*(35*b^3*e^3*x^3 + 16*b^3*d^3 + 8*a*b^2*d^2*e + 6*a^2*b*d*e^2 + 5*a^3*e^3 +
35*(2*b^3*d*e^2 + a*b^2*e^3)*x^2 + 7*(8*b^3*d^2*e + 4*a*b^2*d*e^2 + 3*a^2*b*e^3
)*x)*a/((e^7*x^3 + 3*d*e^6*x^2 + 3*d^2*e^5*x + d^3*e^4)*sqrt(e*x + d)) + 2/35*(3
5*b^3*e^4*x^4 + 128*b^3*d^4 - 48*a*b^2*d^3*e - 8*a^2*b*d^2*e^2 - 2*a^3*d*e^3 + 3
5*(8*b^3*d*e^3 - 3*a*b^2*e^4)*x^3 + 35*(16*b^3*d^2*e^2 - 6*a*b^2*d*e^3 - a^2*b*e
^4)*x^2 + 7*(64*b^3*d^3*e - 24*a*b^2*d^2*e^2 - 4*a^2*b*d*e^3 - a^3*e^4)*x)*b/((e
^8*x^3 + 3*d*e^7*x^2 + 3*d^2*e^6*x + d^3*e^5)*sqrt(e*x + d))

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Fricas [A]  time = 0.282306, size = 289, normalized size = 1.12 \[ \frac{2 \,{\left (35 \, b^{4} e^{4} x^{4} + 128 \, b^{4} d^{4} - 64 \, a b^{3} d^{3} e - 16 \, a^{2} b^{2} d^{2} e^{2} - 8 \, a^{3} b d e^{3} - 5 \, a^{4} e^{4} + 140 \,{\left (2 \, b^{4} d e^{3} - a b^{3} e^{4}\right )} x^{3} + 70 \,{\left (8 \, b^{4} d^{2} e^{2} - 4 \, a b^{3} d e^{3} - a^{2} b^{2} e^{4}\right )} x^{2} + 28 \,{\left (16 \, b^{4} d^{3} e - 8 \, a b^{3} d^{2} e^{2} - 2 \, a^{2} b^{2} d e^{3} - a^{3} b e^{4}\right )} x\right )}}{35 \,{\left (e^{8} x^{3} + 3 \, d e^{7} x^{2} + 3 \, d^{2} e^{6} x + d^{3} e^{5}\right )} \sqrt{e x + d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*(b*x + a)/(e*x + d)^(9/2),x, algorithm="fricas")

[Out]

2/35*(35*b^4*e^4*x^4 + 128*b^4*d^4 - 64*a*b^3*d^3*e - 16*a^2*b^2*d^2*e^2 - 8*a^3
*b*d*e^3 - 5*a^4*e^4 + 140*(2*b^4*d*e^3 - a*b^3*e^4)*x^3 + 70*(8*b^4*d^2*e^2 - 4
*a*b^3*d*e^3 - a^2*b^2*e^4)*x^2 + 28*(16*b^4*d^3*e - 8*a*b^3*d^2*e^2 - 2*a^2*b^2
*d*e^3 - a^3*b*e^4)*x)/((e^8*x^3 + 3*d*e^7*x^2 + 3*d^2*e^6*x + d^3*e^5)*sqrt(e*x
+ d))

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

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)**(9/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.333767, size = 419, normalized size = 1.62 \[ 2 \, \sqrt{x e + d} b^{4} e^{\left (-5\right )}{\rm sign}\left (b x + a\right ) + \frac{2 \,{\left (140 \,{\left (x e + d\right )}^{3} b^{4} d{\rm sign}\left (b x + a\right ) - 70 \,{\left (x e + d\right )}^{2} b^{4} d^{2}{\rm sign}\left (b x + a\right ) + 28 \,{\left (x e + d\right )} b^{4} d^{3}{\rm sign}\left (b x + a\right ) - 5 \, b^{4} d^{4}{\rm sign}\left (b x + a\right ) - 140 \,{\left (x e + d\right )}^{3} a b^{3} e{\rm sign}\left (b x + a\right ) + 140 \,{\left (x e + d\right )}^{2} a b^{3} d e{\rm sign}\left (b x + a\right ) - 84 \,{\left (x e + d\right )} a b^{3} d^{2} e{\rm sign}\left (b x + a\right ) + 20 \, a b^{3} d^{3} e{\rm sign}\left (b x + a\right ) - 70 \,{\left (x e + d\right )}^{2} a^{2} b^{2} e^{2}{\rm sign}\left (b x + a\right ) + 84 \,{\left (x e + d\right )} a^{2} b^{2} d e^{2}{\rm sign}\left (b x + a\right ) - 30 \, a^{2} b^{2} d^{2} e^{2}{\rm sign}\left (b x + a\right ) - 28 \,{\left (x e + d\right )} a^{3} b e^{3}{\rm sign}\left (b x + a\right ) + 20 \, a^{3} b d e^{3}{\rm sign}\left (b x + a\right ) - 5 \, a^{4} e^{4}{\rm sign}\left (b x + a\right )\right )} e^{\left (-5\right )}}{35 \,{\left (x e + d\right )}^{\frac{7}{2}}} \]

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)^(9/2),x, algorithm="giac")

[Out]

2*sqrt(x*e + d)*b^4*e^(-5)*sign(b*x + a) + 2/35*(140*(x*e + d)^3*b^4*d*sign(b*x
+ a) - 70*(x*e + d)^2*b^4*d^2*sign(b*x + a) + 28*(x*e + d)*b^4*d^3*sign(b*x + a)
- 5*b^4*d^4*sign(b*x + a) - 140*(x*e + d)^3*a*b^3*e*sign(b*x + a) + 140*(x*e +
d)^2*a*b^3*d*e*sign(b*x + a) - 84*(x*e + d)*a*b^3*d^2*e*sign(b*x + a) + 20*a*b^3
*d^3*e*sign(b*x + a) - 70*(x*e + d)^2*a^2*b^2*e^2*sign(b*x + a) + 84*(x*e + d)*a
^2*b^2*d*e^2*sign(b*x + a) - 30*a^2*b^2*d^2*e^2*sign(b*x + a) - 28*(x*e + d)*a^3
*b*e^3*sign(b*x + a) + 20*a^3*b*d*e^3*sign(b*x + a) - 5*a^4*e^4*sign(b*x + a))*e
^(-5)/(x*e + d)^(7/2)