3.1822 \(\int \frac{A+B x}{(d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^2} \, dx\)

Optimal. Leaf size=250 \[ \frac{5 e^2 (a B e-7 A b e+6 b B d)}{8 b \sqrt{d+e x} (b d-a e)^4}-\frac{5 e^2 (a B e-7 A b e+6 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{8 \sqrt{b} (b d-a e)^{9/2}}+\frac{5 e (a B e-7 A b e+6 b B d)}{24 b (a+b x) \sqrt{d+e x} (b d-a e)^3}-\frac{a B e-7 A b e+6 b B d}{12 b (a+b x)^2 \sqrt{d+e x} (b d-a e)^2}-\frac{A b-a B}{3 b (a+b x)^3 \sqrt{d+e x} (b d-a e)} \]

[Out]

(5*e^2*(6*b*B*d - 7*A*b*e + a*B*e))/(8*b*(b*d - a*e)^4*Sqrt[d + e*x]) - (A*b - a
*B)/(3*b*(b*d - a*e)*(a + b*x)^3*Sqrt[d + e*x]) - (6*b*B*d - 7*A*b*e + a*B*e)/(1
2*b*(b*d - a*e)^2*(a + b*x)^2*Sqrt[d + e*x]) + (5*e*(6*b*B*d - 7*A*b*e + a*B*e))
/(24*b*(b*d - a*e)^3*(a + b*x)*Sqrt[d + e*x]) - (5*e^2*(6*b*B*d - 7*A*b*e + a*B*
e)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(8*Sqrt[b]*(b*d - a*e)^(9/2
))

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Rubi [A]  time = 0.562196, antiderivative size = 250, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{5 e^2 (a B e-7 A b e+6 b B d)}{8 b \sqrt{d+e x} (b d-a e)^4}-\frac{5 e^2 (a B e-7 A b e+6 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{8 \sqrt{b} (b d-a e)^{9/2}}+\frac{5 e (a B e-7 A b e+6 b B d)}{24 b (a+b x) \sqrt{d+e x} (b d-a e)^3}-\frac{a B e-7 A b e+6 b B d}{12 b (a+b x)^2 \sqrt{d+e x} (b d-a e)^2}-\frac{A b-a B}{3 b (a+b x)^3 \sqrt{d+e x} (b d-a e)} \]

Antiderivative was successfully verified.

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

[Out]

(5*e^2*(6*b*B*d - 7*A*b*e + a*B*e))/(8*b*(b*d - a*e)^4*Sqrt[d + e*x]) - (A*b - a
*B)/(3*b*(b*d - a*e)*(a + b*x)^3*Sqrt[d + e*x]) - (6*b*B*d - 7*A*b*e + a*B*e)/(1
2*b*(b*d - a*e)^2*(a + b*x)^2*Sqrt[d + e*x]) + (5*e*(6*b*B*d - 7*A*b*e + a*B*e))
/(24*b*(b*d - a*e)^3*(a + b*x)*Sqrt[d + e*x]) - (5*e^2*(6*b*B*d - 7*A*b*e + a*B*
e)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(8*Sqrt[b]*(b*d - a*e)^(9/2
))

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Rubi in Sympy [A]  time = 109.408, size = 236, normalized size = 0.94 \[ - \frac{5 e \sqrt{d + e x} \left (7 A b e - B a e - 6 B b d\right )}{8 \left (a + b x\right ) \left (a e - b d\right )^{4}} - \frac{5 e \left (7 A b e - B a e - 6 B b d\right )}{12 b \left (a + b x\right ) \sqrt{d + e x} \left (a e - b d\right )^{3}} + \frac{7 A b e - B a e - 6 B b d}{12 b \left (a + b x\right )^{2} \sqrt{d + e x} \left (a e - b d\right )^{2}} + \frac{A b - B a}{3 b \left (a + b x\right )^{3} \sqrt{d + e x} \left (a e - b d\right )} - \frac{5 e^{2} \left (7 A b e - B a e - 6 B b d\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{d + e x}}{\sqrt{a e - b d}} \right )}}{8 \sqrt{b} \left (a e - b d\right )^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5*e*sqrt(d + e*x)*(7*A*b*e - B*a*e - 6*B*b*d)/(8*(a + b*x)*(a*e - b*d)**4) - 5*
e*(7*A*b*e - B*a*e - 6*B*b*d)/(12*b*(a + b*x)*sqrt(d + e*x)*(a*e - b*d)**3) + (7
*A*b*e - B*a*e - 6*B*b*d)/(12*b*(a + b*x)**2*sqrt(d + e*x)*(a*e - b*d)**2) + (A*
b - B*a)/(3*b*(a + b*x)**3*sqrt(d + e*x)*(a*e - b*d)) - 5*e**2*(7*A*b*e - B*a*e
- 6*B*b*d)*atan(sqrt(b)*sqrt(d + e*x)/sqrt(a*e - b*d))/(8*sqrt(b)*(a*e - b*d)**(
9/2))

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Mathematica [A]  time = 0.974258, size = 198, normalized size = 0.79 \[ -\frac{\sqrt{d+e x} \left (\frac{3 e (-5 a B e+19 A b e-14 b B d)}{a+b x}+\frac{2 (b d-a e) (5 a B e-11 A b e+6 b B d)}{(a+b x)^2}+\frac{8 (A b-a B) (b d-a e)^2}{(a+b x)^3}+\frac{48 e^2 (A e-B d)}{d+e x}\right )}{24 (b d-a e)^4}-\frac{5 e^2 (a B e-7 A b e+6 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{8 \sqrt{b} (b d-a e)^{9/2}} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[d + e*x]*((8*(A*b - a*B)*(b*d - a*e)^2)/(a + b*x)^3 + (2*(b*d - a*e)*(6*b
*B*d - 11*A*b*e + 5*a*B*e))/(a + b*x)^2 + (3*e*(-14*b*B*d + 19*A*b*e - 5*a*B*e))
/(a + b*x) + (48*e^2*(-(B*d) + A*e))/(d + e*x)))/(24*(b*d - a*e)^4) - (5*e^2*(6*
b*B*d - 7*A*b*e + a*B*e)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(8*Sq
rt[b]*(b*d - a*e)^(9/2))

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Maple [B]  time = 0.041, size = 768, normalized size = 3.1 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*e^3/(a*e-b*d)^4/(e*x+d)^(1/2)*A+2*e^2/(a*e-b*d)^4/(e*x+d)^(1/2)*B*d-19/8*e^3/
(a*e-b*d)^4/(b*e*x+a*e)^3*(e*x+d)^(5/2)*A*b^3+7/4*e^2/(a*e-b*d)^4/(b*e*x+a*e)^3*
(e*x+d)^(5/2)*B*d*b^3+5/8*e^3/(a*e-b*d)^4/(b*e*x+a*e)^3*(e*x+d)^(5/2)*B*a*b^2-17
/3*e^4/(a*e-b*d)^4/(b*e*x+a*e)^3*A*(e*x+d)^(3/2)*a*b^2+17/3*e^3/(a*e-b*d)^4/(b*e
*x+a*e)^3*A*(e*x+d)^(3/2)*b^3*d+5/3*e^4/(a*e-b*d)^4/(b*e*x+a*e)^3*B*(e*x+d)^(3/2
)*a^2*b+7/3*e^3/(a*e-b*d)^4/(b*e*x+a*e)^3*B*(e*x+d)^(3/2)*a*b^2*d-4*e^2/(a*e-b*d
)^4/(b*e*x+a*e)^3*B*(e*x+d)^(3/2)*b^3*d^2-29/8*e^5/(a*e-b*d)^4/(b*e*x+a*e)^3*(e*
x+d)^(1/2)*A*a^2*b+29/4*e^4/(a*e-b*d)^4/(b*e*x+a*e)^3*(e*x+d)^(1/2)*A*a*b^2*d-29
/8*e^3/(a*e-b*d)^4/(b*e*x+a*e)^3*(e*x+d)^(1/2)*A*b^3*d^2+11/8*e^5/(a*e-b*d)^4/(b
*e*x+a*e)^3*(e*x+d)^(1/2)*B*a^3-1/2*e^4/(a*e-b*d)^4/(b*e*x+a*e)^3*(e*x+d)^(1/2)*
B*a^2*b*d-25/8*e^3/(a*e-b*d)^4/(b*e*x+a*e)^3*(e*x+d)^(1/2)*B*a*b^2*d^2+9/4*e^2/(
a*e-b*d)^4/(b*e*x+a*e)^3*(e*x+d)^(1/2)*B*b^3*d^3-35/8*e^3/(a*e-b*d)^4/(b*(a*e-b*
d))^(1/2)*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*A*b+5/8*e^3/(a*e-b*d)^4/(b
*(a*e-b*d))^(1/2)*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*a*B+15/4*e^2/(a*e-
b*d)^4/(b*(a*e-b*d))^(1/2)*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*B*b*d

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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)/((b^2*x^2 + 2*a*b*x + a^2)^2*(e*x + d)^(3/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.331306, size = 1, normalized size = 0. \[ \text{result too large to display} \]

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)^2*(e*x + d)^(3/2)),x, algorithm="fricas")

[Out]

[-1/48*(15*(6*B*a^3*b*d*e^2 + (B*a^4 - 7*A*a^3*b)*e^3 + (6*B*b^4*d*e^2 + (B*a*b^
3 - 7*A*b^4)*e^3)*x^3 + 3*(6*B*a*b^3*d*e^2 + (B*a^2*b^2 - 7*A*a*b^3)*e^3)*x^2 +
3*(6*B*a^2*b^2*d*e^2 + (B*a^3*b - 7*A*a^2*b^2)*e^3)*x)*sqrt(e*x + d)*log((sqrt(b
^2*d - a*b*e)*(b*e*x + 2*b*d - a*e) + 2*(b^2*d - a*b*e)*sqrt(e*x + d))/(b*x + a)
) + 2*(48*A*a^3*e^3 + 4*(B*a*b^2 + 2*A*b^3)*d^3 - 2*(14*B*a^2*b + 19*A*a*b^2)*d^
2*e - 3*(27*B*a^3 - 29*A*a^2*b)*d*e^2 - 15*(6*B*b^3*d*e^2 + (B*a*b^2 - 7*A*b^3)*
e^3)*x^3 - 5*(6*B*b^3*d^2*e + 7*(7*B*a*b^2 - A*b^3)*d*e^2 + 8*(B*a^2*b - 7*A*a*b
^2)*e^3)*x^2 + (12*B*b^3*d^3 - 2*(41*B*a*b^2 + 7*A*b^3)*d^2*e - 2*(106*B*a^2*b -
49*A*a*b^2)*d*e^2 - 33*(B*a^3 - 7*A*a^2*b)*e^3)*x)*sqrt(b^2*d - a*b*e))/((a^3*b
^4*d^4 - 4*a^4*b^3*d^3*e + 6*a^5*b^2*d^2*e^2 - 4*a^6*b*d*e^3 + a^7*e^4 + (b^7*d^
4 - 4*a*b^6*d^3*e + 6*a^2*b^5*d^2*e^2 - 4*a^3*b^4*d*e^3 + a^4*b^3*e^4)*x^3 + 3*(
a*b^6*d^4 - 4*a^2*b^5*d^3*e + 6*a^3*b^4*d^2*e^2 - 4*a^4*b^3*d*e^3 + a^5*b^2*e^4)
*x^2 + 3*(a^2*b^5*d^4 - 4*a^3*b^4*d^3*e + 6*a^4*b^3*d^2*e^2 - 4*a^5*b^2*d*e^3 +
a^6*b*e^4)*x)*sqrt(b^2*d - a*b*e)*sqrt(e*x + d)), -1/24*(15*(6*B*a^3*b*d*e^2 + (
B*a^4 - 7*A*a^3*b)*e^3 + (6*B*b^4*d*e^2 + (B*a*b^3 - 7*A*b^4)*e^3)*x^3 + 3*(6*B*
a*b^3*d*e^2 + (B*a^2*b^2 - 7*A*a*b^3)*e^3)*x^2 + 3*(6*B*a^2*b^2*d*e^2 + (B*a^3*b
- 7*A*a^2*b^2)*e^3)*x)*sqrt(e*x + d)*arctan(-(b*d - a*e)/(sqrt(-b^2*d + a*b*e)*
sqrt(e*x + d))) + (48*A*a^3*e^3 + 4*(B*a*b^2 + 2*A*b^3)*d^3 - 2*(14*B*a^2*b + 19
*A*a*b^2)*d^2*e - 3*(27*B*a^3 - 29*A*a^2*b)*d*e^2 - 15*(6*B*b^3*d*e^2 + (B*a*b^2
- 7*A*b^3)*e^3)*x^3 - 5*(6*B*b^3*d^2*e + 7*(7*B*a*b^2 - A*b^3)*d*e^2 + 8*(B*a^2
*b - 7*A*a*b^2)*e^3)*x^2 + (12*B*b^3*d^3 - 2*(41*B*a*b^2 + 7*A*b^3)*d^2*e - 2*(1
06*B*a^2*b - 49*A*a*b^2)*d*e^2 - 33*(B*a^3 - 7*A*a^2*b)*e^3)*x)*sqrt(-b^2*d + a*
b*e))/((a^3*b^4*d^4 - 4*a^4*b^3*d^3*e + 6*a^5*b^2*d^2*e^2 - 4*a^6*b*d*e^3 + a^7*
e^4 + (b^7*d^4 - 4*a*b^6*d^3*e + 6*a^2*b^5*d^2*e^2 - 4*a^3*b^4*d*e^3 + a^4*b^3*e
^4)*x^3 + 3*(a*b^6*d^4 - 4*a^2*b^5*d^3*e + 6*a^3*b^4*d^2*e^2 - 4*a^4*b^3*d*e^3 +
a^5*b^2*e^4)*x^2 + 3*(a^2*b^5*d^4 - 4*a^3*b^4*d^3*e + 6*a^4*b^3*d^2*e^2 - 4*a^5
*b^2*d*e^3 + a^6*b*e^4)*x)*sqrt(-b^2*d + a*b*e)*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)/(e*x+d)**(3/2)/(b**2*x**2+2*a*b*x+a**2)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.31229, size = 697, normalized size = 2.79 \[ \frac{5 \,{\left (6 \, B b d e^{2} + B a e^{3} - 7 \, A b e^{3}\right )} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right )}{8 \,{\left (b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}\right )} \sqrt{-b^{2} d + a b e}} + \frac{2 \,{\left (B d e^{2} - A e^{3}\right )}}{{\left (b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}\right )} \sqrt{x e + d}} + \frac{42 \,{\left (x e + d\right )}^{\frac{5}{2}} B b^{3} d e^{2} - 96 \,{\left (x e + d\right )}^{\frac{3}{2}} B b^{3} d^{2} e^{2} + 54 \, \sqrt{x e + d} B b^{3} d^{3} e^{2} + 15 \,{\left (x e + d\right )}^{\frac{5}{2}} B a b^{2} e^{3} - 57 \,{\left (x e + d\right )}^{\frac{5}{2}} A b^{3} e^{3} + 56 \,{\left (x e + d\right )}^{\frac{3}{2}} B a b^{2} d e^{3} + 136 \,{\left (x e + d\right )}^{\frac{3}{2}} A b^{3} d e^{3} - 75 \, \sqrt{x e + d} B a b^{2} d^{2} e^{3} - 87 \, \sqrt{x e + d} A b^{3} d^{2} e^{3} + 40 \,{\left (x e + d\right )}^{\frac{3}{2}} B a^{2} b e^{4} - 136 \,{\left (x e + d\right )}^{\frac{3}{2}} A a b^{2} e^{4} - 12 \, \sqrt{x e + d} B a^{2} b d e^{4} + 174 \, \sqrt{x e + d} A a b^{2} d e^{4} + 33 \, \sqrt{x e + d} B a^{3} e^{5} - 87 \, \sqrt{x e + d} A a^{2} b e^{5}}{24 \,{\left (b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}\right )}{\left ({\left (x e + d\right )} b - b d + a e\right )}^{3}} \]

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)^2*(e*x + d)^(3/2)),x, algorithm="giac")

[Out]

5/8*(6*B*b*d*e^2 + B*a*e^3 - 7*A*b*e^3)*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b
*e))/((b^4*d^4 - 4*a*b^3*d^3*e + 6*a^2*b^2*d^2*e^2 - 4*a^3*b*d*e^3 + a^4*e^4)*sq
rt(-b^2*d + a*b*e)) + 2*(B*d*e^2 - A*e^3)/((b^4*d^4 - 4*a*b^3*d^3*e + 6*a^2*b^2*
d^2*e^2 - 4*a^3*b*d*e^3 + a^4*e^4)*sqrt(x*e + d)) + 1/24*(42*(x*e + d)^(5/2)*B*b
^3*d*e^2 - 96*(x*e + d)^(3/2)*B*b^3*d^2*e^2 + 54*sqrt(x*e + d)*B*b^3*d^3*e^2 + 1
5*(x*e + d)^(5/2)*B*a*b^2*e^3 - 57*(x*e + d)^(5/2)*A*b^3*e^3 + 56*(x*e + d)^(3/2
)*B*a*b^2*d*e^3 + 136*(x*e + d)^(3/2)*A*b^3*d*e^3 - 75*sqrt(x*e + d)*B*a*b^2*d^2
*e^3 - 87*sqrt(x*e + d)*A*b^3*d^2*e^3 + 40*(x*e + d)^(3/2)*B*a^2*b*e^4 - 136*(x*
e + d)^(3/2)*A*a*b^2*e^4 - 12*sqrt(x*e + d)*B*a^2*b*d*e^4 + 174*sqrt(x*e + d)*A*
a*b^2*d*e^4 + 33*sqrt(x*e + d)*B*a^3*e^5 - 87*sqrt(x*e + d)*A*a^2*b*e^5)/((b^4*d
^4 - 4*a*b^3*d^3*e + 6*a^2*b^2*d^2*e^2 - 4*a^3*b*d*e^3 + a^4*e^4)*((x*e + d)*b -
b*d + a*e)^3)