3.2105 \(\int \frac{(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^{5/2}} \, dx\)

Optimal. Leaf size=260 \[ \frac{12 b^2 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^2}{e^5 (a+b x)}+\frac{8 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{e^5 (a+b x) \sqrt{d+e x}}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{3 e^5 (a+b x) (d+e x)^{3/2}}+\frac{2 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2}}{5 e^5 (a+b x)}-\frac{8 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)}{3 e^5 (a+b x)} \]

[Out]

(-2*(b*d - a*e)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^5*(a + b*x)*(d + e*x)^(3/2
)) + (8*b*(b*d - a*e)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^5*(a + b*x)*Sqrt[d + e
*x]) + (12*b^2*(b*d - a*e)^2*Sqrt[d + e*x]*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)^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3
*e^5*(a + b*x)) + (2*b^4*(d + e*x)^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*e^5*(
a + b*x))

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Rubi [A]  time = 0.301441, antiderivative size = 260, 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 \[ \frac{12 b^2 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^2}{e^5 (a+b x)}+\frac{8 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{e^5 (a+b x) \sqrt{d+e x}}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{3 e^5 (a+b x) (d+e x)^{3/2}}+\frac{2 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2}}{5 e^5 (a+b x)}-\frac{8 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)}{3 e^5 (a+b x)} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2))/(d + e*x)^(5/2),x]

[Out]

(-2*(b*d - a*e)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^5*(a + b*x)*(d + e*x)^(3/2
)) + (8*b*(b*d - a*e)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^5*(a + b*x)*Sqrt[d + e
*x]) + (12*b^2*(b*d - a*e)^2*Sqrt[d + e*x]*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)^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3
*e^5*(a + b*x)) + (2*b^4*(d + e*x)^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*e^5*(
a + b*x))

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Rubi in Sympy [A]  time = 36.4898, size = 221, normalized size = 0.85 \[ \frac{32 b^{2} \left (3 a + 3 b x\right ) \sqrt{d + e x} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{15 e^{3}} + \frac{128 b^{2} \sqrt{d + e x} \left (a e - b d\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{15 e^{4}} + \frac{256 b^{2} \sqrt{d + e x} \left (a e - b d\right )^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{15 e^{5} \left (a + b x\right )} - \frac{16 b \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{3 e^{2} \sqrt{d + e x}} - \frac{2 \left (a + b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{3 e \left (d + e x\right )^{\frac{3}{2}}} \]

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)**(3/2)/(e*x+d)**(5/2),x)

[Out]

32*b**2*(3*a + 3*b*x)*sqrt(d + e*x)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(15*e**3) +
128*b**2*sqrt(d + e*x)*(a*e - b*d)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(15*e**4) +
256*b**2*sqrt(d + e*x)*(a*e - b*d)**2*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(15*e**5
*(a + b*x)) - 16*b*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(3*e**2*sqrt(d + e*x)) -
2*(a + b*x)*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(3*e*(d + e*x)**(3/2))

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Mathematica [A]  time = 0.342545, size = 128, normalized size = 0.49 \[ \frac{2 \sqrt{(a+b x)^2} \sqrt{d+e x} \left (b^2 \left (90 a^2 e^2-160 a b d e+73 b^2 d^2\right )-2 b^3 e x (7 b d-10 a e)+\frac{60 b (b d-a e)^3}{d+e x}-\frac{5 (b d-a e)^4}{(d+e x)^2}+3 b^4 e^2 x^2\right )}{15 e^5 (a+b x)} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2))/(d + e*x)^(5/2),x]

[Out]

(2*Sqrt[(a + b*x)^2]*Sqrt[d + e*x]*(b^2*(73*b^2*d^2 - 160*a*b*d*e + 90*a^2*e^2)
- 2*b^3*e*(7*b*d - 10*a*e)*x + 3*b^4*e^2*x^2 - (5*(b*d - a*e)^4)/(d + e*x)^2 + (
60*b*(b*d - a*e)^3)/(d + e*x)))/(15*e^5*(a + b*x))

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Maple [A]  time = 0.011, size = 202, normalized size = 0.8 \[ -{\frac{-6\,{x}^{4}{b}^{4}{e}^{4}-40\,{x}^{3}a{b}^{3}{e}^{4}+16\,{x}^{3}{b}^{4}d{e}^{3}-180\,{x}^{2}{a}^{2}{b}^{2}{e}^{4}+240\,{x}^{2}a{b}^{3}d{e}^{3}-96\,{x}^{2}{b}^{4}{d}^{2}{e}^{2}+120\,x{a}^{3}b{e}^{4}-720\,x{a}^{2}{b}^{2}d{e}^{3}+960\,xa{b}^{3}{d}^{2}{e}^{2}-384\,x{b}^{4}{d}^{3}e+10\,{a}^{4}{e}^{4}+80\,{a}^{3}bd{e}^{3}-480\,{a}^{2}{b}^{2}{d}^{2}{e}^{2}+640\,a{b}^{3}{d}^{3}e-256\,{b}^{4}{d}^{4}}{15\, \left ( bx+a \right ) ^{3}{e}^{5}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}} \]

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)^(3/2)/(e*x+d)^(5/2),x)

[Out]

-2/15/(e*x+d)^(3/2)*(-3*b^4*e^4*x^4-20*a*b^3*e^4*x^3+8*b^4*d*e^3*x^3-90*a^2*b^2*
e^4*x^2+120*a*b^3*d*e^3*x^2-48*b^4*d^2*e^2*x^2+60*a^3*b*e^4*x-360*a^2*b^2*d*e^3*
x+480*a*b^3*d^2*e^2*x-192*b^4*d^3*e*x+5*a^4*e^4+40*a^3*b*d*e^3-240*a^2*b^2*d^2*e
^2+320*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 = 0.726026, size = 410, normalized size = 1.58 \[ \frac{2 \,{\left (b^{3} e^{3} x^{3} - 16 \, b^{3} d^{3} + 24 \, a b^{2} d^{2} e - 6 \, a^{2} b d e^{2} - a^{3} e^{3} - 3 \,{\left (2 \, b^{3} d e^{2} - 3 \, a b^{2} e^{3}\right )} x^{2} - 3 \,{\left (8 \, b^{3} d^{2} e - 12 \, a b^{2} d e^{2} + 3 \, a^{2} b e^{3}\right )} x\right )} a}{3 \,{\left (e^{5} x + d e^{4}\right )} \sqrt{e x + d}} + \frac{2 \,{\left (3 \, b^{3} e^{4} x^{4} + 128 \, b^{3} d^{4} - 240 \, a b^{2} d^{3} e + 120 \, a^{2} b d^{2} e^{2} - 10 \, a^{3} d e^{3} -{\left (8 \, b^{3} d e^{3} - 15 \, a b^{2} e^{4}\right )} x^{3} + 3 \,{\left (16 \, b^{3} d^{2} e^{2} - 30 \, a b^{2} d e^{3} + 15 \, a^{2} b e^{4}\right )} x^{2} + 3 \,{\left (64 \, b^{3} d^{3} e - 120 \, a b^{2} d^{2} e^{2} + 60 \, a^{2} b d e^{3} - 5 \, a^{3} e^{4}\right )} x\right )} b}{15 \,{\left (e^{6} x + d e^{5}\right )} \sqrt{e x + d}} \]

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)^(5/2),x, algorithm="maxima")

[Out]

2/3*(b^3*e^3*x^3 - 16*b^3*d^3 + 24*a*b^2*d^2*e - 6*a^2*b*d*e^2 - a^3*e^3 - 3*(2*
b^3*d*e^2 - 3*a*b^2*e^3)*x^2 - 3*(8*b^3*d^2*e - 12*a*b^2*d*e^2 + 3*a^2*b*e^3)*x)
*a/((e^5*x + d*e^4)*sqrt(e*x + d)) + 2/15*(3*b^3*e^4*x^4 + 128*b^3*d^4 - 240*a*b
^2*d^3*e + 120*a^2*b*d^2*e^2 - 10*a^3*d*e^3 - (8*b^3*d*e^3 - 15*a*b^2*e^4)*x^3 +
3*(16*b^3*d^2*e^2 - 30*a*b^2*d*e^3 + 15*a^2*b*e^4)*x^2 + 3*(64*b^3*d^3*e - 120*
a*b^2*d^2*e^2 + 60*a^2*b*d*e^3 - 5*a^3*e^4)*x)*b/((e^6*x + d*e^5)*sqrt(e*x + d))

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Fricas [A]  time = 0.277812, size = 259, normalized size = 1. \[ \frac{2 \,{\left (3 \, b^{4} e^{4} x^{4} + 128 \, b^{4} d^{4} - 320 \, a b^{3} d^{3} e + 240 \, a^{2} b^{2} d^{2} e^{2} - 40 \, a^{3} b d e^{3} - 5 \, a^{4} e^{4} - 4 \,{\left (2 \, b^{4} d e^{3} - 5 \, a b^{3} e^{4}\right )} x^{3} + 6 \,{\left (8 \, b^{4} d^{2} e^{2} - 20 \, a b^{3} d e^{3} + 15 \, a^{2} b^{2} e^{4}\right )} x^{2} + 12 \,{\left (16 \, b^{4} d^{3} e - 40 \, a b^{3} d^{2} e^{2} + 30 \, a^{2} b^{2} d e^{3} - 5 \, a^{3} b e^{4}\right )} x\right )}}{15 \,{\left (e^{6} x + d e^{5}\right )} \sqrt{e x + d}} \]

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)^(5/2),x, algorithm="fricas")

[Out]

2/15*(3*b^4*e^4*x^4 + 128*b^4*d^4 - 320*a*b^3*d^3*e + 240*a^2*b^2*d^2*e^2 - 40*a
^3*b*d*e^3 - 5*a^4*e^4 - 4*(2*b^4*d*e^3 - 5*a*b^3*e^4)*x^3 + 6*(8*b^4*d^2*e^2 -
20*a*b^3*d*e^3 + 15*a^2*b^2*e^4)*x^2 + 12*(16*b^4*d^3*e - 40*a*b^3*d^2*e^2 + 30*
a^2*b^2*d*e^3 - 5*a^3*b*e^4)*x)/((e^6*x + d*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)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.290771, size = 431, normalized size = 1.66 \[ \frac{2}{15} \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{4} e^{20}{\rm sign}\left (b x + a\right ) - 20 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{4} d e^{20}{\rm sign}\left (b x + a\right ) + 90 \, \sqrt{x e + d} b^{4} d^{2} e^{20}{\rm sign}\left (b x + a\right ) + 20 \,{\left (x e + d\right )}^{\frac{3}{2}} a b^{3} e^{21}{\rm sign}\left (b x + a\right ) - 180 \, \sqrt{x e + d} a b^{3} d e^{21}{\rm sign}\left (b x + a\right ) + 90 \, \sqrt{x e + d} a^{2} b^{2} e^{22}{\rm sign}\left (b x + a\right )\right )} e^{\left (-25\right )} + \frac{2 \,{\left (12 \,{\left (x e + d\right )} b^{4} d^{3}{\rm sign}\left (b x + a\right ) - b^{4} d^{4}{\rm sign}\left (b x + a\right ) - 36 \,{\left (x e + d\right )} a b^{3} d^{2} e{\rm sign}\left (b x + a\right ) + 4 \, a b^{3} d^{3} e{\rm sign}\left (b x + a\right ) + 36 \,{\left (x e + d\right )} a^{2} b^{2} d e^{2}{\rm sign}\left (b x + a\right ) - 6 \, a^{2} b^{2} d^{2} e^{2}{\rm sign}\left (b x + a\right ) - 12 \,{\left (x e + d\right )} a^{3} b e^{3}{\rm sign}\left (b x + a\right ) + 4 \, a^{3} b d e^{3}{\rm sign}\left (b x + a\right ) - a^{4} e^{4}{\rm sign}\left (b x + a\right )\right )} e^{\left (-5\right )}}{3 \,{\left (x e + d\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*(b*x + a)/(e*x + d)^(5/2),x, algorithm="giac")

[Out]

2/15*(3*(x*e + d)^(5/2)*b^4*e^20*sign(b*x + a) - 20*(x*e + d)^(3/2)*b^4*d*e^20*s
ign(b*x + a) + 90*sqrt(x*e + d)*b^4*d^2*e^20*sign(b*x + a) + 20*(x*e + d)^(3/2)*
a*b^3*e^21*sign(b*x + a) - 180*sqrt(x*e + d)*a*b^3*d*e^21*sign(b*x + a) + 90*sqr
t(x*e + d)*a^2*b^2*e^22*sign(b*x + a))*e^(-25) + 2/3*(12*(x*e + d)*b^4*d^3*sign(
b*x + a) - b^4*d^4*sign(b*x + a) - 36*(x*e + d)*a*b^3*d^2*e*sign(b*x + a) + 4*a*
b^3*d^3*e*sign(b*x + a) + 36*(x*e + d)*a^2*b^2*d*e^2*sign(b*x + a) - 6*a^2*b^2*d
^2*e^2*sign(b*x + a) - 12*(x*e + d)*a^3*b*e^3*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)/(x*e + d)^(3/2)