3.1234 \(\int \frac{A+B x}{(d+e x)^{3/2} \left (b x+c x^2\right )} \, dx\)

Optimal. Leaf size=118 \[ \frac{2 (B d-A e)}{d \sqrt{d+e x} (c d-b e)}-\frac{2 \sqrt{c} (b B-A c) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b (c d-b e)^{3/2}}-\frac{2 A \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{b d^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.436587, antiderivative size = 118, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{2 (B d-A e)}{d \sqrt{d+e x} (c d-b e)}-\frac{2 \sqrt{c} (b B-A c) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b (c d-b e)^{3/2}}-\frac{2 A \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{b d^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 43.2999, size = 102, normalized size = 0.86 \[ - \frac{2 A \operatorname{atanh}{\left (\frac{\sqrt{d + e x}}{\sqrt{d}} \right )}}{b d^{\frac{3}{2}}} + \frac{2 \left (A e - B d\right )}{d \sqrt{d + e x} \left (b e - c d\right )} + \frac{2 \sqrt{c} \left (A c - B b\right ) \operatorname{atan}{\left (\frac{\sqrt{c} \sqrt{d + e x}}{\sqrt{b e - c d}} \right )}}{b \left (b e - c d\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.613155, size = 118, normalized size = 1. \[ \frac{2 (B d-A e)}{d \sqrt{d+e x} (c d-b e)}+\frac{2 \sqrt{c} (A c-b B) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b (c d-b e)^{3/2}}-\frac{2 A \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{b d^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.022, size = 168, normalized size = 1.4 \[ 2\,{\frac{A{c}^{2}}{ \left ( be-cd \right ) b\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{c\sqrt{ex+d}}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }-2\,{\frac{Bc}{ \left ( be-cd \right ) \sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{c\sqrt{ex+d}}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }-2\,{\frac{A}{b{d}^{3/2}}{\it Artanh} \left ({\frac{\sqrt{ex+d}}{\sqrt{d}}} \right ) }+2\,{\frac{Ae}{d \left ( be-cd \right ) \sqrt{ex+d}}}-2\,{\frac{B}{ \left ( be-cd \right ) \sqrt{ex+d}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/(b*e-c*d)*c^2/b/((b*e-c*d)*c)^(1/2)*arctan(c*(e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2)
)*A-2/(b*e-c*d)*c/((b*e-c*d)*c)^(1/2)*arctan(c*(e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2)
)*B-2*A*arctanh((e*x+d)^(1/2)/d^(1/2))/b/d^(3/2)+2/d/(b*e-c*d)/(e*x+d)^(1/2)*A*e
-2/(b*e-c*d)/(e*x+d)^(1/2)*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)*(e*x + d)^(3/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.573063, size = 1, normalized size = 0.01 \[ \left [\frac{{\left (B b - A c\right )} \sqrt{e x + d} d^{\frac{3}{2}} \sqrt{\frac{c}{c d - b e}} \log \left (\frac{c e x + 2 \, c d - b e - 2 \,{\left (c d - b e\right )} \sqrt{e x + d} \sqrt{\frac{c}{c d - b e}}}{c x + b}\right ) +{\left (A c d - A b e\right )} \sqrt{e x + d} \log \left (\frac{{\left (e x + 2 \, d\right )} \sqrt{d} - 2 \, \sqrt{e x + d} d}{x}\right ) + 2 \,{\left (B b d - A b e\right )} \sqrt{d}}{{\left (b c d^{2} - b^{2} d e\right )} \sqrt{e x + d} \sqrt{d}}, -\frac{2 \,{\left (B b - A c\right )} \sqrt{e x + d} d^{\frac{3}{2}} \sqrt{-\frac{c}{c d - b e}} \arctan \left (-\frac{{\left (c d - b e\right )} \sqrt{-\frac{c}{c d - b e}}}{\sqrt{e x + d} c}\right ) -{\left (A c d - A b e\right )} \sqrt{e x + d} \log \left (\frac{{\left (e x + 2 \, d\right )} \sqrt{d} - 2 \, \sqrt{e x + d} d}{x}\right ) - 2 \,{\left (B b d - A b e\right )} \sqrt{d}}{{\left (b c d^{2} - b^{2} d e\right )} \sqrt{e x + d} \sqrt{d}}, \frac{{\left (B b - A c\right )} \sqrt{e x + d} \sqrt{-d} d \sqrt{\frac{c}{c d - b e}} \log \left (\frac{c e x + 2 \, c d - b e - 2 \,{\left (c d - b e\right )} \sqrt{e x + d} \sqrt{\frac{c}{c d - b e}}}{c x + b}\right ) + 2 \,{\left (A c d - A b e\right )} \sqrt{e x + d} \arctan \left (\frac{d}{\sqrt{e x + d} \sqrt{-d}}\right ) + 2 \,{\left (B b d - A b e\right )} \sqrt{-d}}{{\left (b c d^{2} - b^{2} d e\right )} \sqrt{e x + d} \sqrt{-d}}, -\frac{2 \,{\left ({\left (B b - A c\right )} \sqrt{e x + d} \sqrt{-d} d \sqrt{-\frac{c}{c d - b e}} \arctan \left (-\frac{{\left (c d - b e\right )} \sqrt{-\frac{c}{c d - b e}}}{\sqrt{e x + d} c}\right ) -{\left (A c d - A b e\right )} \sqrt{e x + d} \arctan \left (\frac{d}{\sqrt{e x + d} \sqrt{-d}}\right ) -{\left (B b d - A b e\right )} \sqrt{-d}\right )}}{{\left (b c d^{2} - b^{2} d e\right )} \sqrt{e x + d} \sqrt{-d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.282963, size = 174, normalized size = 1.47 \[ \frac{2 \,{\left (B b c - A c^{2}\right )} \arctan \left (\frac{\sqrt{x e + d} c}{\sqrt{-c^{2} d + b c e}}\right )}{{\left (b c d - b^{2} e\right )} \sqrt{-c^{2} d + b c e}} + \frac{2 \,{\left (B d - A e\right )}}{{\left (c d^{2} - b d e\right )} \sqrt{x e + d}} + \frac{2 \, A \arctan \left (\frac{\sqrt{x e + d}}{\sqrt{-d}}\right )}{b \sqrt{-d} d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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