3.1203 \(\int \frac{A+B x}{(d+e x) \left (b x+c x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=141 \[ -\frac{2 (c x (2 A c d-b (A e+B d))+A b (c d-b e))}{b^2 d \sqrt{b x+c x^2} (c d-b e)}-\frac{e (B d-A e) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{d^{3/2} (c d-b e)^{3/2}} \]

[Out]

(-2*(A*b*(c*d - b*e) + c*(2*A*c*d - b*(B*d + A*e))*x))/(b^2*d*(c*d - b*e)*Sqrt[b
*x + c*x^2]) - (e*(B*d - A*e)*ArcTanh[(b*d + (2*c*d - b*e)*x)/(2*Sqrt[d]*Sqrt[c*
d - b*e]*Sqrt[b*x + c*x^2])])/(d^(3/2)*(c*d - b*e)^(3/2))

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Rubi [A]  time = 0.336542, antiderivative size = 141, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ -\frac{2 (c x (2 A c d-b (A e+B d))+A b (c d-b e))}{b^2 d \sqrt{b x+c x^2} (c d-b e)}-\frac{e (B d-A e) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{d^{3/2} (c d-b e)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(A*b*(c*d - b*e) + c*(2*A*c*d - b*(B*d + A*e))*x))/(b^2*d*(c*d - b*e)*Sqrt[b
*x + c*x^2]) - (e*(B*d - A*e)*ArcTanh[(b*d + (2*c*d - b*e)*x)/(2*Sqrt[d]*Sqrt[c*
d - b*e]*Sqrt[b*x + c*x^2])])/(d^(3/2)*(c*d - b*e)^(3/2))

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Rubi in Sympy [A]  time = 42.647, size = 126, normalized size = 0.89 \[ - \frac{e \left (A e - B d\right ) \operatorname{atan}{\left (\frac{- b d + x \left (b e - 2 c d\right )}{2 \sqrt{d} \sqrt{b e - c d} \sqrt{b x + c x^{2}}} \right )}}{d^{\frac{3}{2}} \left (b e - c d\right )^{\frac{3}{2}}} - \frac{2 \left (A b \left (b e - c d\right ) + c x \left (A b e - 2 A c d + B b d\right )\right )}{b^{2} d \left (b e - c d\right ) \sqrt{b x + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-e*(A*e - B*d)*atan((-b*d + x*(b*e - 2*c*d))/(2*sqrt(d)*sqrt(b*e - c*d)*sqrt(b*x
+ c*x**2)))/(d**(3/2)*(b*e - c*d)**(3/2)) - 2*(A*b*(b*e - c*d) + c*x*(A*b*e - 2
*A*c*d + B*b*d))/(b**2*d*(b*e - c*d)*sqrt(b*x + c*x**2))

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Mathematica [A]  time = 0.395481, size = 144, normalized size = 1.02 \[ -\frac{2 \left (\sqrt{d} \sqrt{b e-c d} \left (A \left (b^2 e-b c d+b c e x-2 c^2 d x\right )+b B c d x\right )+b^2 e \sqrt{x} \sqrt{b+c x} (A e-B d) \tan ^{-1}\left (\frac{\sqrt{x} \sqrt{b e-c d}}{\sqrt{d} \sqrt{b+c x}}\right )\right )}{b^2 d^{3/2} \sqrt{x (b+c x)} (b e-c d)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(Sqrt[d]*Sqrt[-(c*d) + b*e]*(b*B*c*d*x + A*(-(b*c*d) + b^2*e - 2*c^2*d*x + b
*c*e*x)) + b^2*e*(-(B*d) + A*e)*Sqrt[x]*Sqrt[b + c*x]*ArcTan[(Sqrt[-(c*d) + b*e]
*Sqrt[x])/(Sqrt[d]*Sqrt[b + c*x])]))/(b^2*d^(3/2)*(-(c*d) + b*e)^(3/2)*Sqrt[x*(b
+ c*x)])

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Maple [B]  time = 0.013, size = 837, normalized size = 5.9 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*B/e*(2*c*x+b)/b^2/(c*x^2+b*x)^(1/2)-2*e/d/(b*e-c*d)/(c*(d/e+x)^2+(b*e-2*c*d)/
e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*A+2/(b*e-c*d)/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x
)-d*(b*e-c*d)/e^2)^(1/2)*B-2*e/d/(b*e-c*d)/b/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-
d*(b*e-c*d)/e^2)^(1/2)*x*c*A+2/(b*e-c*d)/b/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*
(b*e-c*d)/e^2)^(1/2)*x*c*B+4/(b*e-c*d)/b^2/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*
(b*e-c*d)/e^2)^(1/2)*x*c^2*A-4/e/(b*e-c*d)/b^2/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x
)-d*(b*e-c*d)/e^2)^(1/2)*x*c^2*B*d+2/(b*e-c*d)/b/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e
+x)-d*(b*e-c*d)/e^2)^(1/2)*c*A-2/e/(b*e-c*d)/b/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x
)-d*(b*e-c*d)/e^2)^(1/2)*c*B*d+e/d/(b*e-c*d)/(-d*(b*e-c*d)/e^2)^(1/2)*ln((-2*d*(
b*e-c*d)/e^2+(b*e-2*c*d)/e*(d/e+x)+2*(-d*(b*e-c*d)/e^2)^(1/2)*(c*(d/e+x)^2+(b*e-
2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2))/(d/e+x))*A-1/(b*e-c*d)/(-d*(b*e-c*d)/e^
2)^(1/2)*ln((-2*d*(b*e-c*d)/e^2+(b*e-2*c*d)/e*(d/e+x)+2*(-d*(b*e-c*d)/e^2)^(1/2)
*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2))/(d/e+x))*B

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.294003, size = 1, normalized size = 0.01 \[ \left [\frac{{\left (B b^{2} d e - A b^{2} e^{2}\right )} \sqrt{c x^{2} + b x} \log \left (-\frac{2 \,{\left (c d^{2} - b d e\right )} \sqrt{c x^{2} + b x} - \sqrt{c d^{2} - b d e}{\left (b d +{\left (2 \, c d - b e\right )} x\right )}}{e x + d}\right ) - 2 \,{\left (A b c d - A b^{2} e -{\left (A b c e +{\left (B b c - 2 \, A c^{2}\right )} d\right )} x\right )} \sqrt{c d^{2} - b d e}}{{\left (b^{2} c d^{2} - b^{3} d e\right )} \sqrt{c d^{2} - b d e} \sqrt{c x^{2} + b x}}, \frac{2 \,{\left ({\left (B b^{2} d e - A b^{2} e^{2}\right )} \sqrt{c x^{2} + b x} \arctan \left (-\frac{\sqrt{-c d^{2} + b d e} \sqrt{c x^{2} + b x}}{{\left (c d - b e\right )} x}\right ) -{\left (A b c d - A b^{2} e -{\left (A b c e +{\left (B b c - 2 \, A c^{2}\right )} d\right )} x\right )} \sqrt{-c d^{2} + b d e}\right )}}{{\left (b^{2} c d^{2} - b^{3} d e\right )} \sqrt{-c d^{2} + b d e} \sqrt{c x^{2} + b x}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[((B*b^2*d*e - A*b^2*e^2)*sqrt(c*x^2 + b*x)*log(-(2*(c*d^2 - b*d*e)*sqrt(c*x^2 +
b*x) - sqrt(c*d^2 - b*d*e)*(b*d + (2*c*d - b*e)*x))/(e*x + d)) - 2*(A*b*c*d - A
*b^2*e - (A*b*c*e + (B*b*c - 2*A*c^2)*d)*x)*sqrt(c*d^2 - b*d*e))/((b^2*c*d^2 - b
^3*d*e)*sqrt(c*d^2 - b*d*e)*sqrt(c*x^2 + b*x)), 2*((B*b^2*d*e - A*b^2*e^2)*sqrt(
c*x^2 + b*x)*arctan(-sqrt(-c*d^2 + b*d*e)*sqrt(c*x^2 + b*x)/((c*d - b*e)*x)) - (
A*b*c*d - A*b^2*e - (A*b*c*e + (B*b*c - 2*A*c^2)*d)*x)*sqrt(-c*d^2 + b*d*e))/((b
^2*c*d^2 - b^3*d*e)*sqrt(-c*d^2 + b*d*e)*sqrt(c*x^2 + b*x))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{A + B x}{\left (x \left (b + c x\right )\right )^{\frac{3}{2}} \left (d + e x\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((A + B*x)/((x*(b + c*x))**(3/2)*(d + e*x)), x)

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GIAC/XCAS [A]  time = 0.302235, size = 252, normalized size = 1.79 \[ \frac{2 \,{\left (\frac{{\left (B b c d^{2} - 2 \, A c^{2} d^{2} + A b c d e\right )} x}{b^{2} c d^{3} - b^{3} d^{2} e} - \frac{A b c d^{2} - A b^{2} d e}{b^{2} c d^{3} - b^{3} d^{2} e}\right )}}{\sqrt{c x^{2} + b x}} + \frac{2 \,{\left (B d e - A e^{2}\right )} \arctan \left (\frac{{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} e + \sqrt{c} d}{\sqrt{-c d^{2} + b d e}}\right )}{{\left (c d^{2} - b d e\right )} \sqrt{-c d^{2} + b d e}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*((B*b*c*d^2 - 2*A*c^2*d^2 + A*b*c*d*e)*x/(b^2*c*d^3 - b^3*d^2*e) - (A*b*c*d^2
- A*b^2*d*e)/(b^2*c*d^3 - b^3*d^2*e))/sqrt(c*x^2 + b*x) + 2*(B*d*e - A*e^2)*arct
an(((sqrt(c)*x - sqrt(c*x^2 + b*x))*e + sqrt(c)*d)/sqrt(-c*d^2 + b*d*e))/((c*d^2
- b*d*e)*sqrt(-c*d^2 + b*d*e))