3.1744 \(\int \frac{(A+B x) \sqrt{d+e x}}{(a+b x)^3} \, dx\)

Optimal. Leaf size=157 \[ -\frac{e (-3 a B e-A b e+4 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{4 b^{5/2} (b d-a e)^{3/2}}-\frac{\sqrt{d+e x} (-3 a B e-A b e+4 b B d)}{4 b^2 (a+b x) (b d-a e)}-\frac{(d+e x)^{3/2} (A b-a B)}{2 b (a+b x)^2 (b d-a e)} \]

[Out]

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

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Rubi [A]  time = 0.285947, antiderivative size = 157, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ -\frac{e (-3 a B e-A b e+4 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{4 b^{5/2} (b d-a e)^{3/2}}-\frac{\sqrt{d+e x} (-3 a B e-A b e+4 b B d)}{4 b^2 (a+b x) (b d-a e)}-\frac{(d+e x)^{3/2} (A b-a B)}{2 b (a+b x)^2 (b d-a e)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 28.6493, size = 138, normalized size = 0.88 \[ \frac{\left (d + e x\right )^{\frac{3}{2}} \left (A b - B a\right )}{2 b \left (a + b x\right )^{2} \left (a e - b d\right )} - \frac{\sqrt{d + e x} \left (A b e + 3 B a e - 4 B b d\right )}{4 b^{2} \left (a + b x\right ) \left (a e - b d\right )} + \frac{e \left (A b e + 3 B a e - 4 B b d\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{d + e x}}{\sqrt{a e - b d}} \right )}}{4 b^{\frac{5}{2}} \left (a e - b d\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(e*x+d)**(1/2)/(b*x+a)**3,x)

[Out]

(d + e*x)**(3/2)*(A*b - B*a)/(2*b*(a + b*x)**2*(a*e - b*d)) - sqrt(d + e*x)*(A*b
*e + 3*B*a*e - 4*B*b*d)/(4*b**2*(a + b*x)*(a*e - b*d)) + e*(A*b*e + 3*B*a*e - 4*
B*b*d)*atan(sqrt(b)*sqrt(d + e*x)/sqrt(a*e - b*d))/(4*b**(5/2)*(a*e - b*d)**(3/2
))

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Mathematica [A]  time = 0.318498, size = 129, normalized size = 0.82 \[ \frac{e (3 a B e+A b e-4 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{4 b^{5/2} (b d-a e)^{3/2}}-\frac{\sqrt{d+e x} \left (\frac{(a+b x) (-5 a B e+A b e+4 b B d)}{b d-a e}-2 a B+2 A b\right )}{4 b^2 (a+b x)^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.021, size = 339, normalized size = 2.2 \[{\frac{A{e}^{2}}{4\, \left ( bxe+ae \right ) ^{2} \left ( ae-bd \right ) } \left ( ex+d \right ) ^{{\frac{3}{2}}}}-{\frac{5\,Ba{e}^{2}}{4\, \left ( bxe+ae \right ) ^{2} \left ( ae-bd \right ) b} \left ( ex+d \right ) ^{{\frac{3}{2}}}}+{\frac{eBd}{ \left ( bxe+ae \right ) ^{2} \left ( ae-bd \right ) } \left ( ex+d \right ) ^{{\frac{3}{2}}}}-{\frac{A{e}^{2}}{4\, \left ( bxe+ae \right ) ^{2}b}\sqrt{ex+d}}-{\frac{3\,Ba{e}^{2}}{4\, \left ( bxe+ae \right ) ^{2}{b}^{2}}\sqrt{ex+d}}+{\frac{eBd}{ \left ( bxe+ae \right ) ^{2}b}\sqrt{ex+d}}+{\frac{A{e}^{2}}{ \left ( 4\,ae-4\,bd \right ) b}\arctan \left ({b\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( ae-bd \right ) b}}}} \right ){\frac{1}{\sqrt{ \left ( ae-bd \right ) b}}}}+{\frac{3\,Ba{e}^{2}}{ \left ( 4\,ae-4\,bd \right ){b}^{2}}\arctan \left ({b\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( ae-bd \right ) b}}}} \right ){\frac{1}{\sqrt{ \left ( ae-bd \right ) b}}}}-{\frac{eBd}{ \left ( ae-bd \right ) b}\arctan \left ({b\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( ae-bd \right ) b}}}} \right ){\frac{1}{\sqrt{ \left ( ae-bd \right ) b}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(e*x+d)^(1/2)/(b*x+a)^3,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.226476, size = 1, normalized size = 0.01 \[ \left [-\frac{2 \, \sqrt{b^{2} d - a b e}{\left (2 \,{\left (B a b + A b^{2}\right )} d -{\left (3 \, B a^{2} + A a b\right )} e +{\left (4 \, B b^{2} d -{\left (5 \, B a b - A b^{2}\right )} e\right )} x\right )} \sqrt{e x + d} -{\left (4 \, B a^{2} b d e -{\left (3 \, B a^{3} + A a^{2} b\right )} e^{2} +{\left (4 \, B b^{3} d e -{\left (3 \, B a b^{2} + A b^{3}\right )} e^{2}\right )} x^{2} + 2 \,{\left (4 \, B a b^{2} d e -{\left (3 \, B a^{2} b + A a b^{2}\right )} e^{2}\right )} x\right )} \log \left (\frac{\sqrt{b^{2} d - a b e}{\left (b e x + 2 \, b d - a e\right )} - 2 \,{\left (b^{2} d - a b e\right )} \sqrt{e x + d}}{b x + a}\right )}{8 \,{\left (a^{2} b^{3} d - a^{3} b^{2} e +{\left (b^{5} d - a b^{4} e\right )} x^{2} + 2 \,{\left (a b^{4} d - a^{2} b^{3} e\right )} x\right )} \sqrt{b^{2} d - a b e}}, -\frac{\sqrt{-b^{2} d + a b e}{\left (2 \,{\left (B a b + A b^{2}\right )} d -{\left (3 \, B a^{2} + A a b\right )} e +{\left (4 \, B b^{2} d -{\left (5 \, B a b - A b^{2}\right )} e\right )} x\right )} \sqrt{e x + d} +{\left (4 \, B a^{2} b d e -{\left (3 \, B a^{3} + A a^{2} b\right )} e^{2} +{\left (4 \, B b^{3} d e -{\left (3 \, B a b^{2} + A b^{3}\right )} e^{2}\right )} x^{2} + 2 \,{\left (4 \, B a b^{2} d e -{\left (3 \, B a^{2} b + A a b^{2}\right )} e^{2}\right )} x\right )} \arctan \left (-\frac{b d - a e}{\sqrt{-b^{2} d + a b e} \sqrt{e x + d}}\right )}{4 \,{\left (a^{2} b^{3} d - a^{3} b^{2} e +{\left (b^{5} d - a b^{4} e\right )} x^{2} + 2 \,{\left (a b^{4} d - a^{2} b^{3} e\right )} x\right )} \sqrt{-b^{2} d + a b e}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/8*(2*sqrt(b^2*d - a*b*e)*(2*(B*a*b + A*b^2)*d - (3*B*a^2 + A*a*b)*e + (4*B*b
^2*d - (5*B*a*b - A*b^2)*e)*x)*sqrt(e*x + d) - (4*B*a^2*b*d*e - (3*B*a^3 + A*a^2
*b)*e^2 + (4*B*b^3*d*e - (3*B*a*b^2 + A*b^3)*e^2)*x^2 + 2*(4*B*a*b^2*d*e - (3*B*
a^2*b + A*a*b^2)*e^2)*x)*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)))/((a^2*b^3*d - a^3*b^2*e + (b^5*d - a*b^4*
e)*x^2 + 2*(a*b^4*d - a^2*b^3*e)*x)*sqrt(b^2*d - a*b*e)), -1/4*(sqrt(-b^2*d + a*
b*e)*(2*(B*a*b + A*b^2)*d - (3*B*a^2 + A*a*b)*e + (4*B*b^2*d - (5*B*a*b - A*b^2)
*e)*x)*sqrt(e*x + d) + (4*B*a^2*b*d*e - (3*B*a^3 + A*a^2*b)*e^2 + (4*B*b^3*d*e -
(3*B*a*b^2 + A*b^3)*e^2)*x^2 + 2*(4*B*a*b^2*d*e - (3*B*a^2*b + A*a*b^2)*e^2)*x)
*arctan(-(b*d - a*e)/(sqrt(-b^2*d + a*b*e)*sqrt(e*x + d))))/((a^2*b^3*d - a^3*b^
2*e + (b^5*d - a*b^4*e)*x^2 + 2*(a*b^4*d - a^2*b^3*e)*x)*sqrt(-b^2*d + a*b*e))]

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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)**(1/2)/(b*x+a)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.220532, size = 331, normalized size = 2.11 \[ \frac{{\left (4 \, B b d e - 3 \, B a e^{2} - A b e^{2}\right )} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right )}{4 \,{\left (b^{3} d - a b^{2} e\right )} \sqrt{-b^{2} d + a b e}} - \frac{4 \,{\left (x e + d\right )}^{\frac{3}{2}} B b^{2} d e - 4 \, \sqrt{x e + d} B b^{2} d^{2} e - 5 \,{\left (x e + d\right )}^{\frac{3}{2}} B a b e^{2} +{\left (x e + d\right )}^{\frac{3}{2}} A b^{2} e^{2} + 7 \, \sqrt{x e + d} B a b d e^{2} + \sqrt{x e + d} A b^{2} d e^{2} - 3 \, \sqrt{x e + d} B a^{2} e^{3} - \sqrt{x e + d} A a b e^{3}}{4 \,{\left (b^{3} d - a b^{2} e\right )}{\left ({\left (x e + d\right )} b - b d + a e\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*(4*B*b*d*e - 3*B*a*e^2 - A*b*e^2)*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e
))/((b^3*d - a*b^2*e)*sqrt(-b^2*d + a*b*e)) - 1/4*(4*(x*e + d)^(3/2)*B*b^2*d*e -
4*sqrt(x*e + d)*B*b^2*d^2*e - 5*(x*e + d)^(3/2)*B*a*b*e^2 + (x*e + d)^(3/2)*A*b
^2*e^2 + 7*sqrt(x*e + d)*B*a*b*d*e^2 + sqrt(x*e + d)*A*b^2*d*e^2 - 3*sqrt(x*e +
d)*B*a^2*e^3 - sqrt(x*e + d)*A*a*b*e^3)/((b^3*d - a*b^2*e)*((x*e + d)*b - b*d +
a*e)^2)