3.1867 \(\int \frac{(A+B x) (d+e x)^{5/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=341 \[ -\frac{(d+e x)^{7/2} (A b-a B)}{2 b (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac{(d+e x)^{5/2} (-7 a B e+3 A b e+4 b B d)}{4 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac{5 e (a+b x) \sqrt{b d-a e} (-7 a B e+3 A b e+4 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{4 b^{9/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{5 e (a+b x) \sqrt{d+e x} (-7 a B e+3 A b e+4 b B d)}{4 b^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{5 e (a+b x) (d+e x)^{3/2} (-7 a B e+3 A b e+4 b B d)}{12 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)} \]

[Out]

(5*e*(4*b*B*d + 3*A*b*e - 7*a*B*e)*(a + b*x)*Sqrt[d + e*x])/(4*b^4*Sqrt[a^2 + 2*
a*b*x + b^2*x^2]) + (5*e*(4*b*B*d + 3*A*b*e - 7*a*B*e)*(a + b*x)*(d + e*x)^(3/2)
)/(12*b^3*(b*d - a*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - ((4*b*B*d + 3*A*b*e - 7*a
*B*e)*(d + e*x)^(5/2))/(4*b^2*(b*d - a*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - ((A*b
- a*B)*(d + e*x)^(7/2))/(2*b*(b*d - a*e)*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2
]) - (5*e*Sqrt[b*d - a*e]*(4*b*B*d + 3*A*b*e - 7*a*B*e)*(a + b*x)*ArcTanh[(Sqrt[
b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(4*b^(9/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])

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Rubi [A]  time = 0.724562, antiderivative size = 341, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.171 \[ -\frac{(d+e x)^{7/2} (A b-a B)}{2 b (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac{(d+e x)^{5/2} (-7 a B e+3 A b e+4 b B d)}{4 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac{5 e (a+b x) \sqrt{b d-a e} (-7 a B e+3 A b e+4 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{4 b^{9/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{5 e (a+b x) \sqrt{d+e x} (-7 a B e+3 A b e+4 b B d)}{4 b^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{5 e (a+b x) (d+e x)^{3/2} (-7 a B e+3 A b e+4 b B d)}{12 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)} \]

Antiderivative was successfully verified.

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

[Out]

(5*e*(4*b*B*d + 3*A*b*e - 7*a*B*e)*(a + b*x)*Sqrt[d + e*x])/(4*b^4*Sqrt[a^2 + 2*
a*b*x + b^2*x^2]) + (5*e*(4*b*B*d + 3*A*b*e - 7*a*B*e)*(a + b*x)*(d + e*x)^(3/2)
)/(12*b^3*(b*d - a*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - ((4*b*B*d + 3*A*b*e - 7*a
*B*e)*(d + e*x)^(5/2))/(4*b^2*(b*d - a*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - ((A*b
- a*B)*(d + e*x)^(7/2))/(2*b*(b*d - a*e)*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2
]) - (5*e*Sqrt[b*d - a*e]*(4*b*B*d + 3*A*b*e - 7*a*B*e)*(a + b*x)*ArcTanh[(Sqrt[
b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(4*b^(9/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])

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Rubi in Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: RecursionError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: RecursionError

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Mathematica [A]  time = 0.789502, size = 191, normalized size = 0.56 \[ \frac{(a+b x)^3 \left (\frac{\sqrt{d+e x} \left (-\frac{3 (b d-a e) (-13 a B e+9 A b e+4 b B d)}{a+b x}-\frac{6 (A b-a B) (b d-a e)^2}{(a+b x)^2}+8 e (-9 a B e+3 A b e+7 b B d)+8 b B e^2 x\right )}{3 b^4}-\frac{5 e \sqrt{b d-a e} (-7 a B e+3 A b e+4 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{9/2}}\right )}{4 \left ((a+b x)^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

((a + b*x)^3*((Sqrt[d + e*x]*(8*e*(7*b*B*d + 3*A*b*e - 9*a*B*e) + 8*b*B*e^2*x -
(6*(A*b - a*B)*(b*d - a*e)^2)/(a + b*x)^2 - (3*(b*d - a*e)*(4*b*B*d + 9*A*b*e -
13*a*B*e))/(a + b*x)))/(3*b^4) - (5*e*Sqrt[b*d - a*e]*(4*b*B*d + 3*A*b*e - 7*a*B
*e)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/b^(9/2)))/(4*((a + b*x)^2)
^(3/2))

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Maple [B]  time = 0.036, size = 1150, normalized size = 3.4 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12*(105*B*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*a^4*e^4-105*B*(b*(a*e-b*
d))^(1/2)*(e*x+d)^(1/2)*a^3*e^3-45*A*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))
*x^2*a*b^3*e^4+45*A*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x^2*b^4*d*e^3+8*
B*(b*(a*e-b*d))^(1/2)*(e*x+d)^(3/2)*x^2*b^3*e^2+105*B*arctan((e*x+d)^(1/2)*b/(b*
(a*e-b*d))^(1/2))*x^2*a^2*b^2*e^4+60*B*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2
))*x^2*b^4*d^2*e^2+24*A*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*x^2*b^3*e^3-90*A*arcta
n((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x*a^2*b^2*e^4+210*B*arctan((e*x+d)^(1/2)*
b/(b*(a*e-b*d))^(1/2))*x*a^3*b*e^4+51*B*(b*(a*e-b*d))^(1/2)*(e*x+d)^(3/2)*a*b^2*
d*e-42*A*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*a*b^2*d*e^2-12*B*(b*(a*e-b*d))^(1/2)*
(e*x+d)^(3/2)*b^3*d^2+12*B*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*b^3*d^3-57*B*(b*(a*
e-b*d))^(1/2)*(e*x+d)^(1/2)*a*b^2*d^2*e+60*B*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d)
)^(1/2))*a^2*b^2*d^2*e^2+45*A*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*a^2*b*e^3+27*A*(
b*(a*e-b*d))^(1/2)*(e*x+d)^(3/2)*a*b^2*e^2-27*A*(b*(a*e-b*d))^(1/2)*(e*x+d)^(3/2
)*b^3*d*e+45*A*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*a^2*b^2*d*e^3-31*B*(b
*(a*e-b*d))^(1/2)*(e*x+d)^(3/2)*a^2*b*e^2-165*B*arctan((e*x+d)^(1/2)*b/(b*(a*e-b
*d))^(1/2))*a^3*b*d*e^3+120*B*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x*a*b^
3*d^2*e^2+48*A*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*x*a*b^2*e^3-144*B*(b*(a*e-b*d))
^(1/2)*(e*x+d)^(1/2)*x*a^2*b*e^3-165*B*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2
))*x^2*a*b^3*d*e^3+90*A*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x*a*b^3*d*e^
3+16*B*(b*(a*e-b*d))^(1/2)*(e*x+d)^(3/2)*x*a*b^2*e^2-72*B*(b*(a*e-b*d))^(1/2)*(e
*x+d)^(1/2)*x^2*a*b^2*e^3+96*B*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*x*a*b^2*d*e^2+4
8*B*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*x^2*b^3*d*e^2-45*A*arctan((e*x+d)^(1/2)*b/
(b*(a*e-b*d))^(1/2))*a^3*b*e^4-330*B*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))
*x*a^2*b^2*d*e^3+21*A*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*b^3*d^2*e+126*B*(b*(a*e-
b*d))^(1/2)*(e*x+d)^(1/2)*a^2*b*d*e^2)/e*(b*x+a)/(b*(a*e-b*d))^(1/2)/b^4/((b*x+a
)^2)^(3/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.29341, size = 1, normalized size = 0. \[ \left [-\frac{15 \,{\left (4 \, B a^{2} b d e -{\left (7 \, B a^{3} - 3 \, A a^{2} b\right )} e^{2} +{\left (4 \, B b^{3} d e -{\left (7 \, B a b^{2} - 3 \, A b^{3}\right )} e^{2}\right )} x^{2} + 2 \,{\left (4 \, B a b^{2} d e -{\left (7 \, B a^{2} b - 3 \, A a b^{2}\right )} e^{2}\right )} x\right )} \sqrt{\frac{b d - a e}{b}} \log \left (\frac{b e x + 2 \, b d - a e + 2 \, \sqrt{e x + d} b \sqrt{\frac{b d - a e}{b}}}{b x + a}\right ) - 2 \,{\left (8 \, B b^{3} e^{2} x^{3} - 6 \,{\left (B a b^{2} + A b^{3}\right )} d^{2} + 5 \,{\left (19 \, B a^{2} b - 3 \, A a b^{2}\right )} d e - 15 \,{\left (7 \, B a^{3} - 3 \, A a^{2} b\right )} e^{2} + 8 \,{\left (7 \, B b^{3} d e -{\left (7 \, B a b^{2} - 3 \, A b^{3}\right )} e^{2}\right )} x^{2} -{\left (12 \, B b^{3} d^{2} -{\left (163 \, B a b^{2} - 27 \, A b^{3}\right )} d e + 25 \,{\left (7 \, B a^{2} b - 3 \, A a b^{2}\right )} e^{2}\right )} x\right )} \sqrt{e x + d}}{24 \,{\left (b^{6} x^{2} + 2 \, a b^{5} x + a^{2} b^{4}\right )}}, -\frac{15 \,{\left (4 \, B a^{2} b d e -{\left (7 \, B a^{3} - 3 \, A a^{2} b\right )} e^{2} +{\left (4 \, B b^{3} d e -{\left (7 \, B a b^{2} - 3 \, A b^{3}\right )} e^{2}\right )} x^{2} + 2 \,{\left (4 \, B a b^{2} d e -{\left (7 \, B a^{2} b - 3 \, A a b^{2}\right )} e^{2}\right )} x\right )} \sqrt{-\frac{b d - a e}{b}} \arctan \left (\frac{\sqrt{e x + d}}{\sqrt{-\frac{b d - a e}{b}}}\right ) -{\left (8 \, B b^{3} e^{2} x^{3} - 6 \,{\left (B a b^{2} + A b^{3}\right )} d^{2} + 5 \,{\left (19 \, B a^{2} b - 3 \, A a b^{2}\right )} d e - 15 \,{\left (7 \, B a^{3} - 3 \, A a^{2} b\right )} e^{2} + 8 \,{\left (7 \, B b^{3} d e -{\left (7 \, B a b^{2} - 3 \, A b^{3}\right )} e^{2}\right )} x^{2} -{\left (12 \, B b^{3} d^{2} -{\left (163 \, B a b^{2} - 27 \, A b^{3}\right )} d e + 25 \,{\left (7 \, B a^{2} b - 3 \, A a b^{2}\right )} e^{2}\right )} x\right )} \sqrt{e x + d}}{12 \,{\left (b^{6} x^{2} + 2 \, a b^{5} x + a^{2} b^{4}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/24*(15*(4*B*a^2*b*d*e - (7*B*a^3 - 3*A*a^2*b)*e^2 + (4*B*b^3*d*e - (7*B*a*b^
2 - 3*A*b^3)*e^2)*x^2 + 2*(4*B*a*b^2*d*e - (7*B*a^2*b - 3*A*a*b^2)*e^2)*x)*sqrt(
(b*d - a*e)/b)*log((b*e*x + 2*b*d - a*e + 2*sqrt(e*x + d)*b*sqrt((b*d - a*e)/b))
/(b*x + a)) - 2*(8*B*b^3*e^2*x^3 - 6*(B*a*b^2 + A*b^3)*d^2 + 5*(19*B*a^2*b - 3*A
*a*b^2)*d*e - 15*(7*B*a^3 - 3*A*a^2*b)*e^2 + 8*(7*B*b^3*d*e - (7*B*a*b^2 - 3*A*b
^3)*e^2)*x^2 - (12*B*b^3*d^2 - (163*B*a*b^2 - 27*A*b^3)*d*e + 25*(7*B*a^2*b - 3*
A*a*b^2)*e^2)*x)*sqrt(e*x + d))/(b^6*x^2 + 2*a*b^5*x + a^2*b^4), -1/12*(15*(4*B*
a^2*b*d*e - (7*B*a^3 - 3*A*a^2*b)*e^2 + (4*B*b^3*d*e - (7*B*a*b^2 - 3*A*b^3)*e^2
)*x^2 + 2*(4*B*a*b^2*d*e - (7*B*a^2*b - 3*A*a*b^2)*e^2)*x)*sqrt(-(b*d - a*e)/b)*
arctan(sqrt(e*x + d)/sqrt(-(b*d - a*e)/b)) - (8*B*b^3*e^2*x^3 - 6*(B*a*b^2 + A*b
^3)*d^2 + 5*(19*B*a^2*b - 3*A*a*b^2)*d*e - 15*(7*B*a^3 - 3*A*a^2*b)*e^2 + 8*(7*B
*b^3*d*e - (7*B*a*b^2 - 3*A*b^3)*e^2)*x^2 - (12*B*b^3*d^2 - (163*B*a*b^2 - 27*A*
b^3)*d*e + 25*(7*B*a^2*b - 3*A*a*b^2)*e^2)*x)*sqrt(e*x + d))/(b^6*x^2 + 2*a*b^5*
x + a^2*b^4)]

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.322811, size = 649, normalized size = 1.9 \[ -\frac{5 \,{\left (4 \, B b^{2} d^{2} e^{2} - 11 \, B a b d e^{3} + 3 \, A b^{2} d e^{3} + 7 \, B a^{2} e^{4} - 3 \, A a b e^{4}\right )} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right ) e^{\left (-1\right )}}{4 \, \sqrt{-b^{2} d + a b e} b^{4}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right )} + \frac{{\left (4 \,{\left (x e + d\right )}^{\frac{3}{2}} B b^{3} d^{2} e^{2} - 4 \, \sqrt{x e + d} B b^{3} d^{3} e^{2} - 17 \,{\left (x e + d\right )}^{\frac{3}{2}} B a b^{2} d e^{3} + 9 \,{\left (x e + d\right )}^{\frac{3}{2}} A b^{3} d e^{3} + 19 \, \sqrt{x e + d} B a b^{2} d^{2} e^{3} - 7 \, \sqrt{x e + d} A b^{3} d^{2} e^{3} + 13 \,{\left (x e + d\right )}^{\frac{3}{2}} B a^{2} b e^{4} - 9 \,{\left (x e + d\right )}^{\frac{3}{2}} A a b^{2} e^{4} - 26 \, \sqrt{x e + d} B a^{2} b d e^{4} + 14 \, \sqrt{x e + d} A a b^{2} d e^{4} + 11 \, \sqrt{x e + d} B a^{3} e^{5} - 7 \, \sqrt{x e + d} A a^{2} b e^{5}\right )} e^{\left (-1\right )}}{4 \,{\left ({\left (x e + d\right )} b - b d + a e\right )}^{2} b^{4}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right )} - \frac{2 \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} B b^{6} e^{4} + 6 \, \sqrt{x e + d} B b^{6} d e^{4} - 9 \, \sqrt{x e + d} B a b^{5} e^{5} + 3 \, \sqrt{x e + d} A b^{6} e^{5}\right )} e^{\left (-3\right )}}{3 \, b^{9}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5/4*(4*B*b^2*d^2*e^2 - 11*B*a*b*d*e^3 + 3*A*b^2*d*e^3 + 7*B*a^2*e^4 - 3*A*a*b*e
^4)*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))*e^(-1)/(sqrt(-b^2*d + a*b*e)*b^
4*sign(-(x*e + d)*b*e + b*d*e - a*e^2)) + 1/4*(4*(x*e + d)^(3/2)*B*b^3*d^2*e^2 -
4*sqrt(x*e + d)*B*b^3*d^3*e^2 - 17*(x*e + d)^(3/2)*B*a*b^2*d*e^3 + 9*(x*e + d)^
(3/2)*A*b^3*d*e^3 + 19*sqrt(x*e + d)*B*a*b^2*d^2*e^3 - 7*sqrt(x*e + d)*A*b^3*d^2
*e^3 + 13*(x*e + d)^(3/2)*B*a^2*b*e^4 - 9*(x*e + d)^(3/2)*A*a*b^2*e^4 - 26*sqrt(
x*e + d)*B*a^2*b*d*e^4 + 14*sqrt(x*e + d)*A*a*b^2*d*e^4 + 11*sqrt(x*e + d)*B*a^3
*e^5 - 7*sqrt(x*e + d)*A*a^2*b*e^5)*e^(-1)/(((x*e + d)*b - b*d + a*e)^2*b^4*sign
(-(x*e + d)*b*e + b*d*e - a*e^2)) - 2/3*((x*e + d)^(3/2)*B*b^6*e^4 + 6*sqrt(x*e
+ d)*B*b^6*d*e^4 - 9*sqrt(x*e + d)*B*a*b^5*e^5 + 3*sqrt(x*e + d)*A*b^6*e^5)*e^(-
3)/(b^9*sign(-(x*e + d)*b*e + b*d*e - a*e^2))