3.2106 \(\int \frac{(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^{7/2}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.305717, 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 \[ -\frac{12 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{e^5 (a+b x) \sqrt{d+e x}}+\frac{8 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{3 e^5 (a+b x) (d+e x)^{3/2}}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{5 e^5 (a+b x) (d+e x)^{5/2}}+\frac{2 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2}}{3 e^5 (a+b x)}-\frac{8 b^3 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)}{e^5 (a+b x)} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2))/(d + e*x)^(7/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 34.5705, size = 212, normalized size = 0.82 \[ \frac{128 b^{3} \sqrt{d + e x} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{15 e^{4}} + \frac{256 b^{3} \sqrt{d + e x} \left (a e - b d\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{15 e^{5} \left (a + b x\right )} - \frac{32 b^{2} \left (3 a + 3 b x\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{15 e^{3} \sqrt{d + e x}} - \frac{16 b \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{15 e^{2} \left (d + e x\right )^{\frac{3}{2}}} - \frac{2 \left (a + b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{5 e \left (d + e x\right )^{\frac{5}{2}}} \]

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)**(7/2),x)

[Out]

128*b**3*sqrt(d + e*x)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(15*e**4) + 256*b**3*sqr
t(d + e*x)*(a*e - b*d)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(15*e**5*(a + b*x)) - 32
*b**2*(3*a + 3*b*x)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(15*e**3*sqrt(d + e*x)) - 1
6*b*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(15*e**2*(d + e*x)**(3/2)) - 2*(a + b*x)
*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(5*e*(d + e*x)**(5/2))

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Mathematica [A]  time = 0.251409, size = 116, normalized size = 0.45 \[ \frac{2 \sqrt{(a+b x)^2} \sqrt{d+e x} \left (60 a b^3 e-\frac{90 b^2 (b d-a e)^2}{d+e x}+\frac{20 b (b d-a e)^3}{(d+e x)^2}-\frac{3 (b d-a e)^4}{(d+e x)^3}-55 b^4 d+5 b^4 e x\right )}{15 e^5 (a+b x)} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2))/(d + e*x)^(7/2),x]

[Out]

(2*Sqrt[(a + b*x)^2]*Sqrt[d + e*x]*(-55*b^4*d + 60*a*b^3*e + 5*b^4*e*x - (3*(b*d
- a*e)^4)/(d + e*x)^3 + (20*b*(b*d - a*e)^3)/(d + e*x)^2 - (90*b^2*(b*d - a*e)^
2)/(d + e*x)))/(15*e^5*(a + b*x))

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Maple [A]  time = 0.012, size = 202, normalized size = 0.8 \[ -{\frac{-10\,{x}^{4}{b}^{4}{e}^{4}-120\,{x}^{3}a{b}^{3}{e}^{4}+80\,{x}^{3}{b}^{4}d{e}^{3}+180\,{x}^{2}{a}^{2}{b}^{2}{e}^{4}-720\,{x}^{2}a{b}^{3}d{e}^{3}+480\,{x}^{2}{b}^{4}{d}^{2}{e}^{2}+40\,x{a}^{3}b{e}^{4}+240\,x{a}^{2}{b}^{2}d{e}^{3}-960\,xa{b}^{3}{d}^{2}{e}^{2}+640\,x{b}^{4}{d}^{3}e+6\,{a}^{4}{e}^{4}+16\,{a}^{3}bd{e}^{3}+96\,{a}^{2}{b}^{2}{d}^{2}{e}^{2}-384\,a{b}^{3}{d}^{3}e+256\,{b}^{4}{d}^{4}}{15\, \left ( bx+a \right ) ^{3}{e}^{5}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}} \left ( ex+d \right ) ^{-{\frac{5}{2}}}} \]

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)^(7/2),x)

[Out]

-2/15/(e*x+d)^(5/2)*(-5*b^4*e^4*x^4-60*a*b^3*e^4*x^3+40*b^4*d*e^3*x^3+90*a^2*b^2
*e^4*x^2-360*a*b^3*d*e^3*x^2+240*b^4*d^2*e^2*x^2+20*a^3*b*e^4*x+120*a^2*b^2*d*e^
3*x-480*a*b^3*d^2*e^2*x+320*b^4*d^3*e*x+3*a^4*e^4+8*a^3*b*d*e^3+48*a^2*b^2*d^2*e
^2-192*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 = 0.76538, size = 440, normalized size = 1.69 \[ \frac{2 \,{\left (5 \, b^{3} e^{3} x^{3} + 16 \, b^{3} d^{3} - 8 \, a b^{2} d^{2} e - 2 \, a^{2} b d e^{2} - a^{3} e^{3} + 15 \,{\left (2 \, b^{3} d e^{2} - a b^{2} e^{3}\right )} x^{2} + 5 \,{\left (8 \, b^{3} d^{2} e - 4 \, a b^{2} d e^{2} - a^{2} b e^{3}\right )} x\right )} a}{5 \,{\left (e^{6} x^{2} + 2 \, d e^{5} x + d^{2} e^{4}\right )} \sqrt{e x + d}} + \frac{2 \,{\left (5 \, b^{3} e^{4} x^{4} - 128 \, b^{3} d^{4} + 144 \, a b^{2} d^{3} e - 24 \, a^{2} b d^{2} e^{2} - 2 \, a^{3} d e^{3} - 5 \,{\left (8 \, b^{3} d e^{3} - 9 \, a b^{2} e^{4}\right )} x^{3} - 15 \,{\left (16 \, b^{3} d^{2} e^{2} - 18 \, a b^{2} d e^{3} + 3 \, a^{2} b e^{4}\right )} x^{2} - 5 \,{\left (64 \, b^{3} d^{3} e - 72 \, a b^{2} d^{2} e^{2} + 12 \, a^{2} b d e^{3} + a^{3} e^{4}\right )} x\right )} b}{15 \,{\left (e^{7} x^{2} + 2 \, d e^{6} x + d^{2} e^{5}\right )} \sqrt{e x + d}} \]

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)^(7/2),x, algorithm="maxima")

[Out]

2/5*(5*b^3*e^3*x^3 + 16*b^3*d^3 - 8*a*b^2*d^2*e - 2*a^2*b*d*e^2 - a^3*e^3 + 15*(
2*b^3*d*e^2 - a*b^2*e^3)*x^2 + 5*(8*b^3*d^2*e - 4*a*b^2*d*e^2 - a^2*b*e^3)*x)*a/
((e^6*x^2 + 2*d*e^5*x + d^2*e^4)*sqrt(e*x + d)) + 2/15*(5*b^3*e^4*x^4 - 128*b^3*
d^4 + 144*a*b^2*d^3*e - 24*a^2*b*d^2*e^2 - 2*a^3*d*e^3 - 5*(8*b^3*d*e^3 - 9*a*b^
2*e^4)*x^3 - 15*(16*b^3*d^2*e^2 - 18*a*b^2*d*e^3 + 3*a^2*b*e^4)*x^2 - 5*(64*b^3*
d^3*e - 72*a*b^2*d^2*e^2 + 12*a^2*b*d*e^3 + a^3*e^4)*x)*b/((e^7*x^2 + 2*d*e^6*x
+ d^2*e^5)*sqrt(e*x + d))

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Fricas [A]  time = 0.278519, size = 273, normalized size = 1.05 \[ \frac{2 \,{\left (5 \, b^{4} e^{4} x^{4} - 128 \, b^{4} d^{4} + 192 \, a b^{3} d^{3} e - 48 \, a^{2} b^{2} d^{2} e^{2} - 8 \, a^{3} b d e^{3} - 3 \, a^{4} e^{4} - 20 \,{\left (2 \, b^{4} d e^{3} - 3 \, a b^{3} e^{4}\right )} x^{3} - 30 \,{\left (8 \, b^{4} d^{2} e^{2} - 12 \, a b^{3} d e^{3} + 3 \, a^{2} b^{2} e^{4}\right )} x^{2} - 20 \,{\left (16 \, b^{4} d^{3} e - 24 \, a b^{3} d^{2} e^{2} + 6 \, a^{2} b^{2} d e^{3} + a^{3} b e^{4}\right )} x\right )}}{15 \,{\left (e^{7} x^{2} + 2 \, d e^{6} x + d^{2} e^{5}\right )} \sqrt{e x + d}} \]

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)^(7/2),x, algorithm="fricas")

[Out]

2/15*(5*b^4*e^4*x^4 - 128*b^4*d^4 + 192*a*b^3*d^3*e - 48*a^2*b^2*d^2*e^2 - 8*a^3
*b*d*e^3 - 3*a^4*e^4 - 20*(2*b^4*d*e^3 - 3*a*b^3*e^4)*x^3 - 30*(8*b^4*d^2*e^2 -
12*a*b^3*d*e^3 + 3*a^2*b^2*e^4)*x^2 - 20*(16*b^4*d^3*e - 24*a*b^3*d^2*e^2 + 6*a^
2*b^2*d*e^3 + a^3*b*e^4)*x)/((e^7*x^2 + 2*d*e^6*x + d^2*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)**(7/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.329697, size = 427, normalized size = 1.64 \[ \frac{2}{3} \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} b^{4} e^{10}{\rm sign}\left (b x + a\right ) - 12 \, \sqrt{x e + d} b^{4} d e^{10}{\rm sign}\left (b x + a\right ) + 12 \, \sqrt{x e + d} a b^{3} e^{11}{\rm sign}\left (b x + a\right )\right )} e^{\left (-15\right )} - \frac{2 \,{\left (90 \,{\left (x e + d\right )}^{2} b^{4} d^{2}{\rm sign}\left (b x + a\right ) - 20 \,{\left (x e + d\right )} b^{4} d^{3}{\rm sign}\left (b x + a\right ) + 3 \, b^{4} d^{4}{\rm sign}\left (b x + a\right ) - 180 \,{\left (x e + d\right )}^{2} a b^{3} d e{\rm sign}\left (b x + a\right ) + 60 \,{\left (x e + d\right )} a b^{3} d^{2} e{\rm sign}\left (b x + a\right ) - 12 \, a b^{3} d^{3} e{\rm sign}\left (b x + a\right ) + 90 \,{\left (x e + d\right )}^{2} a^{2} b^{2} e^{2}{\rm sign}\left (b x + a\right ) - 60 \,{\left (x e + d\right )} a^{2} b^{2} d e^{2}{\rm sign}\left (b x + a\right ) + 18 \, a^{2} b^{2} d^{2} e^{2}{\rm sign}\left (b x + a\right ) + 20 \,{\left (x e + d\right )} a^{3} b e^{3}{\rm sign}\left (b x + a\right ) - 12 \, a^{3} b d e^{3}{\rm sign}\left (b x + a\right ) + 3 \, a^{4} e^{4}{\rm sign}\left (b x + a\right )\right )} e^{\left (-5\right )}}{15 \,{\left (x e + d\right )}^{\frac{5}{2}}} \]

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)^(7/2),x, algorithm="giac")

[Out]

2/3*((x*e + d)^(3/2)*b^4*e^10*sign(b*x + a) - 12*sqrt(x*e + d)*b^4*d*e^10*sign(b
*x + a) + 12*sqrt(x*e + d)*a*b^3*e^11*sign(b*x + a))*e^(-15) - 2/15*(90*(x*e + d
)^2*b^4*d^2*sign(b*x + a) - 20*(x*e + d)*b^4*d^3*sign(b*x + a) + 3*b^4*d^4*sign(
b*x + a) - 180*(x*e + d)^2*a*b^3*d*e*sign(b*x + a) + 60*(x*e + d)*a*b^3*d^2*e*si
gn(b*x + a) - 12*a*b^3*d^3*e*sign(b*x + a) + 90*(x*e + d)^2*a^2*b^2*e^2*sign(b*x
+ a) - 60*(x*e + d)*a^2*b^2*d*e^2*sign(b*x + a) + 18*a^2*b^2*d^2*e^2*sign(b*x +
a) + 20*(x*e + d)*a^3*b*e^3*sign(b*x + a) - 12*a^3*b*d*e^3*sign(b*x + a) + 3*a^
4*e^4*sign(b*x + a))*e^(-5)/(x*e + d)^(5/2)