3.1848 \(\int \frac{(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx\)

Optimal. Leaf size=302 \[ -\frac{2 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (-3 a B e-A b e+4 b B d)}{5 e^5 (a+b x)}+\frac{2 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e) (-a B e-A b e+2 b B d)}{e^5 (a+b x)}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{e^5 (a+b x)}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3 (B d-A e)}{e^5 (a+b x) \sqrt{d+e x}}+\frac{2 b^3 B \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{7/2}}{7 e^5 (a+b x)} \]

[Out]

(-2*(b*d - a*e)^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*d - a*e)^2*(4*b*B*d - 3*A*b*e - a*B*e)*Sqrt[d + e*x]*Sqrt[a^2
+ 2*a*b*x + b^2*x^2])/(e^5*(a + b*x)) + (2*b*(b*d - a*e)*(2*b*B*d - A*b*e - a*B
*e)*(d + e*x)^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^5*(a + b*x)) - (2*b^2*(4*b
*B*d - A*b*e - 3*a*B*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^3*B*(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 = 0.469652, antiderivative size = 302, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.057 \[ -\frac{2 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (-3 a B e-A b e+4 b B d)}{5 e^5 (a+b x)}+\frac{2 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e) (-a B e-A b e+2 b B d)}{e^5 (a+b x)}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{e^5 (a+b x)}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3 (B d-A e)}{e^5 (a+b x) \sqrt{d+e x}}+\frac{2 b^3 B \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{7/2}}{7 e^5 (a+b x)} \]

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)^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*d - a*e)^2*(4*b*B*d - 3*A*b*e - a*B*e)*Sqrt[d + e*x]*Sqrt[a^2
+ 2*a*b*x + b^2*x^2])/(e^5*(a + b*x)) + (2*b*(b*d - a*e)*(2*b*B*d - A*b*e - a*B
*e)*(d + e*x)^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^5*(a + b*x)) - (2*b^2*(4*b
*B*d - A*b*e - 3*a*B*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^3*B*(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 = 55.6727, size = 304, normalized size = 1.01 \[ - \frac{\left (2 a + 2 b x\right ) \left (A e - B d\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{e \sqrt{d + e x} \left (a e - b d\right )} + \frac{2 \sqrt{d + e x} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}} \left (7 A b e + B a e - 8 B b d\right )}{7 e^{2} \left (a e - b d\right )} + \frac{4 \left (3 a + 3 b x\right ) \sqrt{d + e x} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \left (7 A b e + B a e - 8 B b d\right )}{35 e^{3}} + \frac{16 \sqrt{d + e x} \left (a e - b d\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \left (7 A b e + B a e - 8 B b d\right )}{35 e^{4}} + \frac{32 \sqrt{d + e x} \left (a e - b d\right )^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \left (7 A b e + B a e - 8 B b d\right )}{35 e^{5} \left (a + b x\right )} \]

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]

-(2*a + 2*b*x)*(A*e - B*d)*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(e*sqrt(d + e*x)*
(a*e - b*d)) + 2*sqrt(d + e*x)*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)*(7*A*b*e + B*
a*e - 8*B*b*d)/(7*e**2*(a*e - b*d)) + 4*(3*a + 3*b*x)*sqrt(d + e*x)*sqrt(a**2 +
2*a*b*x + b**2*x**2)*(7*A*b*e + B*a*e - 8*B*b*d)/(35*e**3) + 16*sqrt(d + e*x)*(a
*e - b*d)*sqrt(a**2 + 2*a*b*x + b**2*x**2)*(7*A*b*e + B*a*e - 8*B*b*d)/(35*e**4)
+ 32*sqrt(d + e*x)*(a*e - b*d)**2*sqrt(a**2 + 2*a*b*x + b**2*x**2)*(7*A*b*e + B
*a*e - 8*B*b*d)/(35*e**5*(a + b*x))

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Mathematica [A]  time = 0.281193, size = 240, normalized size = 0.79 \[ \frac{2 \sqrt{(a+b x)^2} \left (35 a^3 e^3 (-A e+2 B d+B e x)+35 a^2 b e^2 \left (3 A e (2 d+e x)+B \left (-8 d^2-4 d e x+e^2 x^2\right )\right )+7 a b^2 e \left (5 A e \left (-8 d^2-4 d e x+e^2 x^2\right )+3 B \left (16 d^3+8 d^2 e x-2 d e^2 x^2+e^3 x^3\right )\right )+b^3 \left (7 A e \left (16 d^3+8 d^2 e x-2 d e^2 x^2+e^3 x^3\right )+B \left (-128 d^4-64 d^3 e x+16 d^2 e^2 x^2-8 d e^3 x^3+5 e^4 x^4\right )\right )\right )}{35 e^5 (a+b x) \sqrt{d+e x}} \]

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^3*e^3*(2*B*d - A*e + B*e*x) + 35*a^2*b*e^2*(3*A*e*(2*
d + e*x) + B*(-8*d^2 - 4*d*e*x + e^2*x^2)) + 7*a*b^2*e*(5*A*e*(-8*d^2 - 4*d*e*x
+ e^2*x^2) + 3*B*(16*d^3 + 8*d^2*e*x - 2*d*e^2*x^2 + e^3*x^3)) + b^3*(7*A*e*(16*
d^3 + 8*d^2*e*x - 2*d*e^2*x^2 + e^3*x^3) + B*(-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 = 0.013, size = 317, normalized size = 1.1 \[ -{\frac{-10\,B{x}^{4}{b}^{3}{e}^{4}-14\,A{x}^{3}{b}^{3}{e}^{4}-42\,B{x}^{3}a{b}^{2}{e}^{4}+16\,B{x}^{3}{b}^{3}d{e}^{3}-70\,A{x}^{2}a{b}^{2}{e}^{4}+28\,A{x}^{2}{b}^{3}d{e}^{3}-70\,B{x}^{2}{a}^{2}b{e}^{4}+84\,B{x}^{2}a{b}^{2}d{e}^{3}-32\,B{x}^{2}{b}^{3}{d}^{2}{e}^{2}-210\,Ax{a}^{2}b{e}^{4}+280\,Axa{b}^{2}d{e}^{3}-112\,Ax{b}^{3}{d}^{2}{e}^{2}-70\,Bx{a}^{3}{e}^{4}+280\,Bx{a}^{2}bd{e}^{3}-336\,Bxa{b}^{2}{d}^{2}{e}^{2}+128\,Bx{b}^{3}{d}^{3}e+70\,A{a}^{3}{e}^{4}-420\,Ad{e}^{3}{a}^{2}b+560\,Aa{b}^{2}{d}^{2}{e}^{2}-224\,A{b}^{3}{d}^{3}e-140\,Bd{e}^{3}{a}^{3}+560\,B{a}^{2}b{d}^{2}{e}^{2}-672\,Ba{b}^{2}{d}^{3}e+256\,B{b}^{3}{d}^{4}}{35\, \left ( bx+a \right ) ^{3}{e}^{5}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}{\frac{1}{\sqrt{ex+d}}}} \]

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*b^3*e^4*x^4-7*A*b^3*e^4*x^3-21*B*a*b^2*e^4*x^3+8*B*b^3
*d*e^3*x^3-35*A*a*b^2*e^4*x^2+14*A*b^3*d*e^3*x^2-35*B*a^2*b*e^4*x^2+42*B*a*b^2*d
*e^3*x^2-16*B*b^3*d^2*e^2*x^2-105*A*a^2*b*e^4*x+140*A*a*b^2*d*e^3*x-56*A*b^3*d^2
*e^2*x-35*B*a^3*e^4*x+140*B*a^2*b*d*e^3*x-168*B*a*b^2*d^2*e^2*x+64*B*b^3*d^3*e*x
+35*A*a^3*e^4-210*A*a^2*b*d*e^3+280*A*a*b^2*d^2*e^2-112*A*b^3*d^3*e-70*B*a^3*d*e
^3+280*B*a^2*b*d^2*e^2-336*B*a*b^2*d^3*e+128*B*b^3*d^4)*((b*x+a)^2)^(3/2)/e^5/(b
*x+a)^3

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Maxima [A]  time = 0.737434, size = 381, normalized size = 1.26 \[ \frac{2 \,{\left (b^{3} e^{3} x^{3} + 16 \, b^{3} d^{3} - 40 \, a b^{2} d^{2} e + 30 \, a^{2} b d e^{2} - 5 \, a^{3} e^{3} -{\left (2 \, b^{3} d e^{2} - 5 \, a b^{2} e^{3}\right )} x^{2} +{\left (8 \, b^{3} d^{2} e - 20 \, a b^{2} d e^{2} + 15 \, a^{2} b e^{3}\right )} x\right )} A}{5 \, \sqrt{e x + d} e^{4}} + \frac{2 \,{\left (5 \, b^{3} e^{4} x^{4} - 128 \, b^{3} d^{4} + 336 \, a b^{2} d^{3} e - 280 \, a^{2} b d^{2} e^{2} + 70 \, a^{3} d e^{3} -{\left (8 \, b^{3} d e^{3} - 21 \, a b^{2} e^{4}\right )} x^{3} +{\left (16 \, b^{3} d^{2} e^{2} - 42 \, a b^{2} d e^{3} + 35 \, a^{2} b e^{4}\right )} x^{2} -{\left (64 \, b^{3} d^{3} e - 168 \, a b^{2} d^{2} e^{2} + 140 \, a^{2} b d e^{3} - 35 \, a^{3} e^{4}\right )} x\right )} B}{35 \, \sqrt{e x + d} e^{5}} \]

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 = 0.277826, size = 354, normalized size = 1.17 \[ \frac{2 \,{\left (5 \, B b^{3} e^{4} x^{4} - 128 \, B b^{3} d^{4} - 35 \, A a^{3} e^{4} + 112 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e - 280 \,{\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} + 70 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3} -{\left (8 \, B b^{3} d e^{3} - 7 \,{\left (3 \, B a b^{2} + A b^{3}\right )} e^{4}\right )} x^{3} +{\left (16 \, B b^{3} d^{2} e^{2} - 14 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d e^{3} + 35 \,{\left (B a^{2} b + A a b^{2}\right )} e^{4}\right )} x^{2} -{\left (64 \, B b^{3} d^{3} e - 56 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{2} + 140 \,{\left (B a^{2} b + A a b^{2}\right )} d e^{3} - 35 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} e^{4}\right )} x\right )}}{35 \, \sqrt{e x + d} e^{5}} \]

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*b^3*e^4*x^4 - 128*B*b^3*d^4 - 35*A*a^3*e^4 + 112*(3*B*a*b^2 + A*b^3)*d
^3*e - 280*(B*a^2*b + A*a*b^2)*d^2*e^2 + 70*(B*a^3 + 3*A*a^2*b)*d*e^3 - (8*B*b^3
*d*e^3 - 7*(3*B*a*b^2 + A*b^3)*e^4)*x^3 + (16*B*b^3*d^2*e^2 - 14*(3*B*a*b^2 + A*
b^3)*d*e^3 + 35*(B*a^2*b + A*a*b^2)*e^4)*x^2 - (64*B*b^3*d^3*e - 56*(3*B*a*b^2 +
A*b^3)*d^2*e^2 + 140*(B*a^2*b + A*a*b^2)*d*e^3 - 35*(B*a^3 + 3*A*a^2*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 = 0.302134, size = 709, normalized size = 2.35 \[ \frac{2}{35} \,{\left (5 \,{\left (x e + d\right )}^{\frac{7}{2}} B b^{3} e^{30}{\rm sign}\left (b x + a\right ) - 28 \,{\left (x e + d\right )}^{\frac{5}{2}} B b^{3} d e^{30}{\rm sign}\left (b x + a\right ) + 70 \,{\left (x e + d\right )}^{\frac{3}{2}} B b^{3} d^{2} e^{30}{\rm sign}\left (b x + a\right ) - 140 \, \sqrt{x e + d} B b^{3} d^{3} e^{30}{\rm sign}\left (b x + a\right ) + 21 \,{\left (x e + d\right )}^{\frac{5}{2}} B a b^{2} e^{31}{\rm sign}\left (b x + a\right ) + 7 \,{\left (x e + d\right )}^{\frac{5}{2}} A b^{3} e^{31}{\rm sign}\left (b x + a\right ) - 105 \,{\left (x e + d\right )}^{\frac{3}{2}} B a b^{2} d e^{31}{\rm sign}\left (b x + a\right ) - 35 \,{\left (x e + d\right )}^{\frac{3}{2}} A b^{3} d e^{31}{\rm sign}\left (b x + a\right ) + 315 \, \sqrt{x e + d} B a b^{2} d^{2} e^{31}{\rm sign}\left (b x + a\right ) + 105 \, \sqrt{x e + d} A b^{3} d^{2} e^{31}{\rm sign}\left (b x + a\right ) + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} B a^{2} b e^{32}{\rm sign}\left (b x + a\right ) + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} A a b^{2} e^{32}{\rm sign}\left (b x + a\right ) - 210 \, \sqrt{x e + d} B a^{2} b d e^{32}{\rm sign}\left (b x + a\right ) - 210 \, \sqrt{x e + d} A a b^{2} d e^{32}{\rm sign}\left (b x + a\right ) + 35 \, \sqrt{x e + d} B a^{3} e^{33}{\rm sign}\left (b x + a\right ) + 105 \, \sqrt{x e + d} A a^{2} b e^{33}{\rm sign}\left (b x + a\right )\right )} e^{\left (-35\right )} - \frac{2 \,{\left (B b^{3} d^{4}{\rm sign}\left (b x + a\right ) - 3 \, B a b^{2} d^{3} e{\rm sign}\left (b x + a\right ) - A b^{3} d^{3} e{\rm sign}\left (b x + a\right ) + 3 \, B a^{2} b d^{2} e^{2}{\rm sign}\left (b x + a\right ) + 3 \, A a b^{2} d^{2} e^{2}{\rm sign}\left (b x + a\right ) - B a^{3} d e^{3}{\rm sign}\left (b x + a\right ) - 3 \, A a^{2} b d e^{3}{\rm sign}\left (b x + a\right ) + A a^{3} e^{4}{\rm sign}\left (b x + a\right )\right )} e^{\left (-5\right )}}{\sqrt{x e + d}} \]

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*b^3*e^30*sign(b*x + a) - 28*(x*e + d)^(5/2)*B*b^3*d*e^
30*sign(b*x + a) + 70*(x*e + d)^(3/2)*B*b^3*d^2*e^30*sign(b*x + a) - 140*sqrt(x*
e + d)*B*b^3*d^3*e^30*sign(b*x + a) + 21*(x*e + d)^(5/2)*B*a*b^2*e^31*sign(b*x +
a) + 7*(x*e + d)^(5/2)*A*b^3*e^31*sign(b*x + a) - 105*(x*e + d)^(3/2)*B*a*b^2*d
*e^31*sign(b*x + a) - 35*(x*e + d)^(3/2)*A*b^3*d*e^31*sign(b*x + a) + 315*sqrt(x
*e + d)*B*a*b^2*d^2*e^31*sign(b*x + a) + 105*sqrt(x*e + d)*A*b^3*d^2*e^31*sign(b
*x + a) + 35*(x*e + d)^(3/2)*B*a^2*b*e^32*sign(b*x + a) + 35*(x*e + d)^(3/2)*A*a
*b^2*e^32*sign(b*x + a) - 210*sqrt(x*e + d)*B*a^2*b*d*e^32*sign(b*x + a) - 210*s
qrt(x*e + d)*A*a*b^2*d*e^32*sign(b*x + a) + 35*sqrt(x*e + d)*B*a^3*e^33*sign(b*x
+ a) + 105*sqrt(x*e + d)*A*a^2*b*e^33*sign(b*x + a))*e^(-35) - 2*(B*b^3*d^4*sig
n(b*x + a) - 3*B*a*b^2*d^3*e*sign(b*x + a) - A*b^3*d^3*e*sign(b*x + a) + 3*B*a^2
*b*d^2*e^2*sign(b*x + a) + 3*A*a*b^2*d^2*e^2*sign(b*x + a) - B*a^3*d*e^3*sign(b*
x + a) - 3*A*a^2*b*d*e^3*sign(b*x + a) + A*a^3*e^4*sign(b*x + a))*e^(-5)/sqrt(x*
e + d)