3.1241 \(\int \frac{(A+B x) (d+e x)^{3/2}}{\left (b x+c x^2\right )^2} \, dx\)

Optimal. Leaf size=184 \[ -\frac{\sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) (3 A b e-4 A c d+2 b B d)}{b^3}+\frac{\sqrt{d+e x} (b B-2 A c) (c d-b e)}{b^2 c (b+c x)}-\frac{\sqrt{c d-b e} \left (-b c (A e+2 B d)+4 A c^2 d+b^2 (-B) e\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b^3 c^{3/2}}-\frac{A (d+e x)^{3/2}}{b x (b+c x)} \]

[Out]

((b*B - 2*A*c)*(c*d - b*e)*Sqrt[d + e*x])/(b^2*c*(b + c*x)) - (A*(d + e*x)^(3/2)
)/(b*x*(b + c*x)) - (Sqrt[d]*(2*b*B*d - 4*A*c*d + 3*A*b*e)*ArcTanh[Sqrt[d + e*x]
/Sqrt[d]])/b^3 - (Sqrt[c*d - b*e]*(4*A*c^2*d - b^2*B*e - b*c*(2*B*d + A*e))*ArcT
anh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(b^3*c^(3/2))

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Rubi [A]  time = 0.715651, antiderivative size = 184, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -\frac{\sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) (3 A b e-4 A c d+2 b B d)}{b^3}+\frac{\sqrt{d+e x} (b B-2 A c) (c d-b e)}{b^2 c (b+c x)}-\frac{\sqrt{c d-b e} \left (-b c (A e+2 B d)+4 A c^2 d+b^2 (-B) e\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b^3 c^{3/2}}-\frac{A (d+e x)^{3/2}}{b x (b+c x)} \]

Antiderivative was successfully verified.

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

[Out]

((b*B - 2*A*c)*(c*d - b*e)*Sqrt[d + e*x])/(b^2*c*(b + c*x)) - (A*(d + e*x)^(3/2)
)/(b*x*(b + c*x)) - (Sqrt[d]*(2*b*B*d - 4*A*c*d + 3*A*b*e)*ArcTanh[Sqrt[d + e*x]
/Sqrt[d]])/b^3 - (Sqrt[c*d - b*e]*(4*A*c^2*d - b^2*B*e - b*c*(2*B*d + A*e))*ArcT
anh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(b^3*c^(3/2))

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Rubi in Sympy [A]  time = 81.9025, size = 168, normalized size = 0.91 \[ \frac{\left (d + e x\right )^{\frac{3}{2}} \left (A c - B b\right )}{b c x \left (b + c x\right )} - \frac{d \sqrt{d + e x} \left (2 A c - B b\right )}{b^{2} c x} - \frac{\sqrt{d} \left (3 A b e - 4 A c d + 2 B b d\right ) \operatorname{atanh}{\left (\frac{\sqrt{d + e x}}{\sqrt{d}} \right )}}{b^{3}} - \frac{\sqrt{b e - c d} \left (- A b c e + 4 A c^{2} d - B b^{2} e - 2 B b c d\right ) \operatorname{atan}{\left (\frac{\sqrt{c} \sqrt{d + e x}}{\sqrt{b e - c d}} \right )}}{b^{3} c^{\frac{3}{2}}} \]

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)**2,x)

[Out]

(d + e*x)**(3/2)*(A*c - B*b)/(b*c*x*(b + c*x)) - d*sqrt(d + e*x)*(2*A*c - B*b)/(
b**2*c*x) - sqrt(d)*(3*A*b*e - 4*A*c*d + 2*B*b*d)*atanh(sqrt(d + e*x)/sqrt(d))/b
**3 - sqrt(b*e - c*d)*(-A*b*c*e + 4*A*c**2*d - B*b**2*e - 2*B*b*c*d)*atan(sqrt(c
)*sqrt(d + e*x)/sqrt(b*e - c*d))/(b**3*c**(3/2))

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Mathematica [A]  time = 0.606684, size = 162, normalized size = 0.88 \[ -\frac{\frac{\sqrt{c d-b e} \left (-b c (A e+2 B d)+4 A c^2 d+b^2 (-B) e\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{c^{3/2}}+b \sqrt{d+e x} \left (\frac{(A c-b B) (c d-b e)}{c (b+c x)}+\frac{A d}{x}\right )+\sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) (3 A b e-4 A c d+2 b B d)}{b^3} \]

Antiderivative was successfully verified.

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

[Out]

-((b*Sqrt[d + e*x]*((A*d)/x + ((-(b*B) + A*c)*(c*d - b*e))/(c*(b + c*x))) + Sqrt
[d]*(2*b*B*d - 4*A*c*d + 3*A*b*e)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]] + (Sqrt[c*d - b
*e]*(4*A*c^2*d - b^2*B*e - b*c*(2*B*d + A*e))*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sq
rt[c*d - b*e]])/c^(3/2))/b^3)

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Maple [B]  time = 0.029, size = 443, normalized size = 2.4 \[{\frac{{e}^{2}A}{b \left ( cex+be \right ) }\sqrt{ex+d}}-{\frac{Acde}{{b}^{2} \left ( cex+be \right ) }\sqrt{ex+d}}-{\frac{{e}^{2}B}{c \left ( cex+be \right ) }\sqrt{ex+d}}+{\frac{Bed}{b \left ( cex+be \right ) }\sqrt{ex+d}}+{\frac{{e}^{2}A}{b}\arctan \left ({c\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( be-cd \right ) c}}}} \right ){\frac{1}{\sqrt{ \left ( be-cd \right ) c}}}}-5\,{\frac{Acde}{{b}^{2}\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{c\sqrt{ex+d}}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }+4\,{\frac{A{c}^{2}{d}^{2}}{{b}^{3}\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{c\sqrt{ex+d}}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }+{\frac{{e}^{2}B}{c}\arctan \left ({c\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( be-cd \right ) c}}}} \right ){\frac{1}{\sqrt{ \left ( be-cd \right ) c}}}}+{\frac{Bed}{b}\arctan \left ({c\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( be-cd \right ) c}}}} \right ){\frac{1}{\sqrt{ \left ( be-cd \right ) c}}}}-2\,{\frac{Bc{d}^{2}}{{b}^{2}\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{c\sqrt{ex+d}}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }-{\frac{dA}{{b}^{2}x}\sqrt{ex+d}}-3\,{\frac{e\sqrt{d}A}{{b}^{2}}{\it Artanh} \left ({\frac{\sqrt{ex+d}}{\sqrt{d}}} \right ) }+4\,{\frac{{d}^{3/2}Ac}{{b}^{3}}{\it Artanh} \left ({\frac{\sqrt{ex+d}}{\sqrt{d}}} \right ) }-2\,{\frac{{d}^{3/2}B}{{b}^{2}}{\it Artanh} \left ({\frac{\sqrt{ex+d}}{\sqrt{d}}} \right ) } \]

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)^2,x)

[Out]

e^2/b*(e*x+d)^(1/2)/(c*e*x+b*e)*A-e/b^2*(e*x+d)^(1/2)/(c*e*x+b*e)*A*c*d-e^2/c*(e
*x+d)^(1/2)/(c*e*x+b*e)*B+e/b*(e*x+d)^(1/2)/(c*e*x+b*e)*B*d+e^2/b/((b*e-c*d)*c)^
(1/2)*arctan(c*(e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2))*A-5*e/b^2/((b*e-c*d)*c)^(1/2)*
arctan(c*(e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2))*A*c*d+4/b^3*c^2/((b*e-c*d)*c)^(1/2)*
arctan(c*(e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2))*A*d^2+e^2/c/((b*e-c*d)*c)^(1/2)*arct
an(c*(e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2))*B+e/b/((b*e-c*d)*c)^(1/2)*arctan(c*(e*x+
d)^(1/2)/((b*e-c*d)*c)^(1/2))*B*d-2/b^2/((b*e-c*d)*c)^(1/2)*arctan(c*(e*x+d)^(1/
2)/((b*e-c*d)*c)^(1/2))*B*d^2*c-d/b^2*A*(e*x+d)^(1/2)/x-3*e*d^(1/2)/b^2*arctanh(
(e*x+d)^(1/2)/d^(1/2))*A+4*d^(3/2)/b^3*arctanh((e*x+d)^(1/2)/d^(1/2))*A*c-2*d^(3
/2)/b^2*arctanh((e*x+d)^(1/2)/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)*(e*x + d)^(3/2)/(c*x^2 + b*x)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.01742, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

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)^2,x, algorithm="fricas")

[Out]

[1/2*(((2*(B*b*c^2 - 2*A*c^3)*d + (B*b^2*c + A*b*c^2)*e)*x^2 + (2*(B*b^2*c - 2*A
*b*c^2)*d + (B*b^3 + A*b^2*c)*e)*x)*sqrt((c*d - b*e)/c)*log((c*e*x + 2*c*d - b*e
+ 2*sqrt(e*x + d)*c*sqrt((c*d - b*e)/c))/(c*x + b)) + ((3*A*b*c^2*e + 2*(B*b*c^
2 - 2*A*c^3)*d)*x^2 + (3*A*b^2*c*e + 2*(B*b^2*c - 2*A*b*c^2)*d)*x)*sqrt(d)*log((
e*x - 2*sqrt(e*x + d)*sqrt(d) + 2*d)/x) - 2*(A*b^2*c*d - ((B*b^2*c - 2*A*b*c^2)*
d - (B*b^3 - A*b^2*c)*e)*x)*sqrt(e*x + d))/(b^3*c^2*x^2 + b^4*c*x), 1/2*(2*((2*(
B*b*c^2 - 2*A*c^3)*d + (B*b^2*c + A*b*c^2)*e)*x^2 + (2*(B*b^2*c - 2*A*b*c^2)*d +
(B*b^3 + A*b^2*c)*e)*x)*sqrt(-(c*d - b*e)/c)*arctan(sqrt(e*x + d)/sqrt(-(c*d -
b*e)/c)) + ((3*A*b*c^2*e + 2*(B*b*c^2 - 2*A*c^3)*d)*x^2 + (3*A*b^2*c*e + 2*(B*b^
2*c - 2*A*b*c^2)*d)*x)*sqrt(d)*log((e*x - 2*sqrt(e*x + d)*sqrt(d) + 2*d)/x) - 2*
(A*b^2*c*d - ((B*b^2*c - 2*A*b*c^2)*d - (B*b^3 - A*b^2*c)*e)*x)*sqrt(e*x + d))/(
b^3*c^2*x^2 + b^4*c*x), -1/2*(2*((3*A*b*c^2*e + 2*(B*b*c^2 - 2*A*c^3)*d)*x^2 + (
3*A*b^2*c*e + 2*(B*b^2*c - 2*A*b*c^2)*d)*x)*sqrt(-d)*arctan(sqrt(e*x + d)/sqrt(-
d)) - ((2*(B*b*c^2 - 2*A*c^3)*d + (B*b^2*c + A*b*c^2)*e)*x^2 + (2*(B*b^2*c - 2*A
*b*c^2)*d + (B*b^3 + A*b^2*c)*e)*x)*sqrt((c*d - b*e)/c)*log((c*e*x + 2*c*d - b*e
+ 2*sqrt(e*x + d)*c*sqrt((c*d - b*e)/c))/(c*x + b)) + 2*(A*b^2*c*d - ((B*b^2*c
- 2*A*b*c^2)*d - (B*b^3 - A*b^2*c)*e)*x)*sqrt(e*x + d))/(b^3*c^2*x^2 + b^4*c*x),
-(((3*A*b*c^2*e + 2*(B*b*c^2 - 2*A*c^3)*d)*x^2 + (3*A*b^2*c*e + 2*(B*b^2*c - 2*
A*b*c^2)*d)*x)*sqrt(-d)*arctan(sqrt(e*x + d)/sqrt(-d)) - ((2*(B*b*c^2 - 2*A*c^3)
*d + (B*b^2*c + A*b*c^2)*e)*x^2 + (2*(B*b^2*c - 2*A*b*c^2)*d + (B*b^3 + A*b^2*c)
*e)*x)*sqrt(-(c*d - b*e)/c)*arctan(sqrt(e*x + d)/sqrt(-(c*d - b*e)/c)) + (A*b^2*
c*d - ((B*b^2*c - 2*A*b*c^2)*d - (B*b^3 - A*b^2*c)*e)*x)*sqrt(e*x + d))/(b^3*c^2
*x^2 + b^4*c*x)]

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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)/(c*x**2+b*x)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.291661, size = 451, normalized size = 2.45 \[ \frac{{\left (2 \, B b d^{2} - 4 \, A c d^{2} + 3 \, A b d e\right )} \arctan \left (\frac{\sqrt{x e + d}}{\sqrt{-d}}\right )}{b^{3} \sqrt{-d}} - \frac{{\left (2 \, B b c^{2} d^{2} - 4 \, A c^{3} d^{2} - B b^{2} c d e + 5 \, A b c^{2} d e - B b^{3} e^{2} - A b^{2} c e^{2}\right )} \arctan \left (\frac{\sqrt{x e + d} c}{\sqrt{-c^{2} d + b c e}}\right )}{\sqrt{-c^{2} d + b c e} b^{3} c} + \frac{{\left (x e + d\right )}^{\frac{3}{2}} B b c d e - 2 \,{\left (x e + d\right )}^{\frac{3}{2}} A c^{2} d e - \sqrt{x e + d} B b c d^{2} e + 2 \, \sqrt{x e + d} A c^{2} d^{2} e -{\left (x e + d\right )}^{\frac{3}{2}} B b^{2} e^{2} +{\left (x e + d\right )}^{\frac{3}{2}} A b c e^{2} + \sqrt{x e + d} B b^{2} d e^{2} - 2 \, \sqrt{x e + d} A b c d e^{2}}{{\left ({\left (x e + d\right )}^{2} c - 2 \,{\left (x e + d\right )} c d + c d^{2} +{\left (x e + d\right )} b e - b d e\right )} b^{2} c} \]

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)^2,x, algorithm="giac")

[Out]

(2*B*b*d^2 - 4*A*c*d^2 + 3*A*b*d*e)*arctan(sqrt(x*e + d)/sqrt(-d))/(b^3*sqrt(-d)
) - (2*B*b*c^2*d^2 - 4*A*c^3*d^2 - B*b^2*c*d*e + 5*A*b*c^2*d*e - B*b^3*e^2 - A*b
^2*c*e^2)*arctan(sqrt(x*e + d)*c/sqrt(-c^2*d + b*c*e))/(sqrt(-c^2*d + b*c*e)*b^3
*c) + ((x*e + d)^(3/2)*B*b*c*d*e - 2*(x*e + d)^(3/2)*A*c^2*d*e - sqrt(x*e + d)*B
*b*c*d^2*e + 2*sqrt(x*e + d)*A*c^2*d^2*e - (x*e + d)^(3/2)*B*b^2*e^2 + (x*e + d)
^(3/2)*A*b*c*e^2 + sqrt(x*e + d)*B*b^2*d*e^2 - 2*sqrt(x*e + d)*A*b*c*d*e^2)/(((x
*e + d)^2*c - 2*(x*e + d)*c*d + c*d^2 + (x*e + d)*b*e - b*d*e)*b^2*c)