3.1849 \(\int \frac{(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^{5/2}} \, dx\)

Optimal. Leaf size=304 \[ -\frac{2 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (-3 a B e-A b e+4 b B d)}{3 e^5 (a+b x)}+\frac{6 b \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e) (-a B e-A b e+2 b B d)}{e^5 (a+b x)}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{e^5 (a+b x) \sqrt{d+e x}}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3 (B d-A e)}{3 e^5 (a+b x) (d+e x)^{3/2}}+\frac{2 b^3 B \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2}}{5 e^5 (a+b x)} \]

[Out]

(-2*(b*d - a*e)^3*(B*d - A*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^5*(a + b*x)*(d
+ e*x)^(3/2)) + (2*(b*d - a*e)^2*(4*b*B*d - 3*A*b*e - a*B*e)*Sqrt[a^2 + 2*a*b*x
+ b^2*x^2])/(e^5*(a + b*x)*Sqrt[d + e*x]) + (6*b*(b*d - a*e)*(2*b*B*d - 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^2*(4
*b*B*d - A*b*e - 3*a*B*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^3*B*(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 = 0.47206, antiderivative size = 304, 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 \[ -\frac{2 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (-3 a B e-A b e+4 b B d)}{3 e^5 (a+b x)}+\frac{6 b \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e) (-a B e-A b e+2 b B d)}{e^5 (a+b x)}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{e^5 (a+b x) \sqrt{d+e x}}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3 (B d-A e)}{3 e^5 (a+b x) (d+e x)^{3/2}}+\frac{2 b^3 B \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2}}{5 e^5 (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(b*d - a*e)^3*(B*d - A*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^5*(a + b*x)*(d
+ e*x)^(3/2)) + (2*(b*d - a*e)^2*(4*b*B*d - 3*A*b*e - a*B*e)*Sqrt[a^2 + 2*a*b*x
+ b^2*x^2])/(e^5*(a + b*x)*Sqrt[d + e*x]) + (6*b*(b*d - a*e)*(2*b*B*d - 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^2*(4
*b*B*d - A*b*e - 3*a*B*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^3*B*(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 = 54.0805, size = 316, normalized size = 1.04 \[ \frac{4 b \left (3 a + 3 b x\right ) \sqrt{d + e x} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \left (5 A b e + 3 B a e - 8 B b d\right )}{15 e^{3} \left (a e - b d\right )} + \frac{16 b \sqrt{d + e x} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \left (5 A b e + 3 B a e - 8 B b d\right )}{15 e^{4}} + \frac{32 b \sqrt{d + e x} \left (a e - b d\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \left (5 A b e + 3 B a e - 8 B b d\right )}{15 e^{5} \left (a + b x\right )} - \frac{\left (2 a + 2 b x\right ) \left (A e - B d\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{3 e \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right )} - \frac{2 \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}} \left (5 A b e + 3 B a e - 8 B b d\right )}{3 e^{2} \sqrt{d + e x} \left (a e - b d\right )} \]

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)**(5/2),x)

[Out]

4*b*(3*a + 3*b*x)*sqrt(d + e*x)*sqrt(a**2 + 2*a*b*x + b**2*x**2)*(5*A*b*e + 3*B*
a*e - 8*B*b*d)/(15*e**3*(a*e - b*d)) + 16*b*sqrt(d + e*x)*sqrt(a**2 + 2*a*b*x +
b**2*x**2)*(5*A*b*e + 3*B*a*e - 8*B*b*d)/(15*e**4) + 32*b*sqrt(d + e*x)*(a*e - b
*d)*sqrt(a**2 + 2*a*b*x + b**2*x**2)*(5*A*b*e + 3*B*a*e - 8*B*b*d)/(15*e**5*(a +
b*x)) - (2*a + 2*b*x)*(A*e - B*d)*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(3*e*(d +
e*x)**(3/2)*(a*e - b*d)) - 2*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)*(5*A*b*e + 3*B
*a*e - 8*B*b*d)/(3*e**2*sqrt(d + e*x)*(a*e - b*d))

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Mathematica [A]  time = 0.524279, size = 170, normalized size = 0.56 \[ \frac{2 \left ((a+b x)^2\right )^{3/2} \sqrt{d+e x} \left (b \left (45 a^2 B e^2+15 a b e (3 A e-8 B d)+b^2 d (73 B d-40 A e)\right )+b^2 e x (15 a B e+5 A b e-14 b B d)-\frac{15 (b d-a e)^2 (a B e+3 A b e-4 b B d)}{d+e x}-\frac{5 (b d-a e)^3 (B d-A e)}{(d+e x)^2}+3 b^3 B e^2 x^2\right )}{15 e^5 (a+b x)^3} \]

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*((a + b*x)^2)^(3/2)*Sqrt[d + e*x]*(b*(45*a^2*B*e^2 + b^2*d*(73*B*d - 40*A*e)
+ 15*a*b*e*(-8*B*d + 3*A*e)) + b^2*e*(-14*b*B*d + 5*A*b*e + 15*a*B*e)*x + 3*b^3*
B*e^2*x^2 - (5*(b*d - a*e)^3*(B*d - A*e))/(d + e*x)^2 - (15*(b*d - a*e)^2*(-4*b*
B*d + 3*A*b*e + a*B*e))/(d + e*x)))/(15*e^5*(a + b*x)^3)

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Maple [A]  time = 0.013, size = 317, normalized size = 1. \[ -{\frac{-6\,B{x}^{4}{b}^{3}{e}^{4}-10\,A{x}^{3}{b}^{3}{e}^{4}-30\,B{x}^{3}a{b}^{2}{e}^{4}+16\,B{x}^{3}{b}^{3}d{e}^{3}-90\,A{x}^{2}a{b}^{2}{e}^{4}+60\,A{x}^{2}{b}^{3}d{e}^{3}-90\,B{x}^{2}{a}^{2}b{e}^{4}+180\,B{x}^{2}a{b}^{2}d{e}^{3}-96\,B{x}^{2}{b}^{3}{d}^{2}{e}^{2}+90\,Ax{a}^{2}b{e}^{4}-360\,Axa{b}^{2}d{e}^{3}+240\,Ax{b}^{3}{d}^{2}{e}^{2}+30\,Bx{a}^{3}{e}^{4}-360\,Bx{a}^{2}bd{e}^{3}+720\,Bxa{b}^{2}{d}^{2}{e}^{2}-384\,Bx{b}^{3}{d}^{3}e+10\,A{a}^{3}{e}^{4}+60\,Ad{e}^{3}{a}^{2}b-240\,Aa{b}^{2}{d}^{2}{e}^{2}+160\,A{b}^{3}{d}^{3}e+20\,Bd{e}^{3}{a}^{3}-240\,B{a}^{2}b{d}^{2}{e}^{2}+480\,Ba{b}^{2}{d}^{3}e-256\,B{b}^{3}{d}^{4}}{15\, \left ( bx+a \right ) ^{3}{e}^{5}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}} \]

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*b^3*e^4*x^4-5*A*b^3*e^4*x^3-15*B*a*b^2*e^4*x^3+8*B*b^3
*d*e^3*x^3-45*A*a*b^2*e^4*x^2+30*A*b^3*d*e^3*x^2-45*B*a^2*b*e^4*x^2+90*B*a*b^2*d
*e^3*x^2-48*B*b^3*d^2*e^2*x^2+45*A*a^2*b*e^4*x-180*A*a*b^2*d*e^3*x+120*A*b^3*d^2
*e^2*x+15*B*a^3*e^4*x-180*B*a^2*b*d*e^3*x+360*B*a*b^2*d^2*e^2*x-192*B*b^3*d^3*e*
x+5*A*a^3*e^4+30*A*a^2*b*d*e^3-120*A*a*b^2*d^2*e^2+80*A*b^3*d^3*e+10*B*a^3*d*e^3
-120*B*a^2*b*d^2*e^2+240*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 = 0.749701, size = 410, normalized size = 1.35 \[ \frac{2 \,{\left (b^{3} e^{3} x^{3} - 16 \, b^{3} d^{3} + 24 \, a b^{2} d^{2} e - 6 \, a^{2} b d e^{2} - a^{3} e^{3} - 3 \,{\left (2 \, b^{3} d e^{2} - 3 \, a b^{2} e^{3}\right )} x^{2} - 3 \,{\left (8 \, b^{3} d^{2} e - 12 \, a b^{2} d e^{2} + 3 \, a^{2} b e^{3}\right )} x\right )} A}{3 \,{\left (e^{5} x + d e^{4}\right )} \sqrt{e x + d}} + \frac{2 \,{\left (3 \, b^{3} e^{4} x^{4} + 128 \, b^{3} d^{4} - 240 \, a b^{2} d^{3} e + 120 \, a^{2} b d^{2} e^{2} - 10 \, a^{3} d e^{3} -{\left (8 \, b^{3} d e^{3} - 15 \, a b^{2} e^{4}\right )} x^{3} + 3 \,{\left (16 \, b^{3} d^{2} e^{2} - 30 \, a b^{2} d e^{3} + 15 \, a^{2} b e^{4}\right )} x^{2} + 3 \,{\left (64 \, b^{3} d^{3} e - 120 \, a b^{2} d^{2} e^{2} + 60 \, a^{2} b d e^{3} - 5 \, a^{3} e^{4}\right )} x\right )} B}{15 \,{\left (e^{6} x + d e^{5}\right )} \sqrt{e x + d}} \]

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 = 0.29193, size = 369, normalized size = 1.21 \[ \frac{2 \,{\left (3 \, B b^{3} e^{4} x^{4} + 128 \, B b^{3} d^{4} - 5 \, A a^{3} e^{4} - 80 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e + 120 \,{\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} - 10 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3} -{\left (8 \, B b^{3} d e^{3} - 5 \,{\left (3 \, B a b^{2} + A b^{3}\right )} e^{4}\right )} x^{3} + 3 \,{\left (16 \, B b^{3} d^{2} e^{2} - 10 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d e^{3} + 15 \,{\left (B a^{2} b + A a b^{2}\right )} e^{4}\right )} x^{2} + 3 \,{\left (64 \, B b^{3} d^{3} e - 40 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{2} + 60 \,{\left (B a^{2} b + A a b^{2}\right )} d e^{3} - 5 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} e^{4}\right )} x\right )}}{15 \,{\left (e^{6} x + d e^{5}\right )} \sqrt{e x + d}} \]

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*b^3*e^4*x^4 + 128*B*b^3*d^4 - 5*A*a^3*e^4 - 80*(3*B*a*b^2 + A*b^3)*d^3
*e + 120*(B*a^2*b + A*a*b^2)*d^2*e^2 - 10*(B*a^3 + 3*A*a^2*b)*d*e^3 - (8*B*b^3*d
*e^3 - 5*(3*B*a*b^2 + A*b^3)*e^4)*x^3 + 3*(16*B*b^3*d^2*e^2 - 10*(3*B*a*b^2 + A*
b^3)*d*e^3 + 15*(B*a^2*b + A*a*b^2)*e^4)*x^2 + 3*(64*B*b^3*d^3*e - 40*(3*B*a*b^2
+ A*b^3)*d^2*e^2 + 60*(B*a^2*b + A*a*b^2)*d*e^3 - 5*(B*a^3 + 3*A*a^2*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 = 0.301585, size = 687, normalized size = 2.26 \[ \frac{2}{15} \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} B b^{3} e^{20}{\rm sign}\left (b x + a\right ) - 20 \,{\left (x e + d\right )}^{\frac{3}{2}} B b^{3} d e^{20}{\rm sign}\left (b x + a\right ) + 90 \, \sqrt{x e + d} B b^{3} d^{2} e^{20}{\rm sign}\left (b x + a\right ) + 15 \,{\left (x e + d\right )}^{\frac{3}{2}} B a b^{2} e^{21}{\rm sign}\left (b x + a\right ) + 5 \,{\left (x e + d\right )}^{\frac{3}{2}} A b^{3} e^{21}{\rm sign}\left (b x + a\right ) - 135 \, \sqrt{x e + d} B a b^{2} d e^{21}{\rm sign}\left (b x + a\right ) - 45 \, \sqrt{x e + d} A b^{3} d e^{21}{\rm sign}\left (b x + a\right ) + 45 \, \sqrt{x e + d} B a^{2} b e^{22}{\rm sign}\left (b x + a\right ) + 45 \, \sqrt{x e + d} A a b^{2} e^{22}{\rm sign}\left (b x + a\right )\right )} e^{\left (-25\right )} + \frac{2 \,{\left (12 \,{\left (x e + d\right )} B b^{3} d^{3}{\rm sign}\left (b x + a\right ) - B b^{3} d^{4}{\rm sign}\left (b x + a\right ) - 27 \,{\left (x e + d\right )} B a b^{2} d^{2} e{\rm sign}\left (b x + a\right ) - 9 \,{\left (x e + d\right )} A b^{3} d^{2} e{\rm sign}\left (b x + a\right ) + 3 \, B a b^{2} d^{3} e{\rm sign}\left (b x + a\right ) + A b^{3} d^{3} e{\rm sign}\left (b x + a\right ) + 18 \,{\left (x e + d\right )} B a^{2} b d e^{2}{\rm sign}\left (b x + a\right ) + 18 \,{\left (x e + d\right )} A a b^{2} d e^{2}{\rm sign}\left (b x + a\right ) - 3 \, B a^{2} b d^{2} e^{2}{\rm sign}\left (b x + a\right ) - 3 \, A a b^{2} d^{2} e^{2}{\rm sign}\left (b x + a\right ) - 3 \,{\left (x e + d\right )} B a^{3} e^{3}{\rm sign}\left (b x + a\right ) - 9 \,{\left (x e + d\right )} A a^{2} b e^{3}{\rm sign}\left (b x + a\right ) + B a^{3} d e^{3}{\rm sign}\left (b x + a\right ) + 3 \, A a^{2} b d e^{3}{\rm sign}\left (b x + a\right ) - A a^{3} e^{4}{\rm sign}\left (b x + a\right )\right )} e^{\left (-5\right )}}{3 \,{\left (x e + d\right )}^{\frac{3}{2}}} \]

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*b^3*e^20*sign(b*x + a) - 20*(x*e + d)^(3/2)*B*b^3*d*e^
20*sign(b*x + a) + 90*sqrt(x*e + d)*B*b^3*d^2*e^20*sign(b*x + a) + 15*(x*e + d)^
(3/2)*B*a*b^2*e^21*sign(b*x + a) + 5*(x*e + d)^(3/2)*A*b^3*e^21*sign(b*x + a) -
135*sqrt(x*e + d)*B*a*b^2*d*e^21*sign(b*x + a) - 45*sqrt(x*e + d)*A*b^3*d*e^21*s
ign(b*x + a) + 45*sqrt(x*e + d)*B*a^2*b*e^22*sign(b*x + a) + 45*sqrt(x*e + d)*A*
a*b^2*e^22*sign(b*x + a))*e^(-25) + 2/3*(12*(x*e + d)*B*b^3*d^3*sign(b*x + a) -
B*b^3*d^4*sign(b*x + a) - 27*(x*e + d)*B*a*b^2*d^2*e*sign(b*x + a) - 9*(x*e + d)
*A*b^3*d^2*e*sign(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) + 18*(x*e + d)*B*a^2*b*d*e^2*sign(b*x + a) + 18*(x*e + d)*A*a*b^2*d*e^2*s
ign(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)
- 3*(x*e + d)*B*a^3*e^3*sign(b*x + a) - 9*(x*e + d)*A*a^2*b*e^3*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)/(x*e + d)^(3/2)