3.1745 \(\int \frac{A+B x}{(a+b x)^3 \sqrt{d+e x}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.336153, 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 \[ \frac{e (-a B e-3 A b e+4 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{4 b^{3/2} (b d-a e)^{5/2}}-\frac{\sqrt{d+e x} (A b-a B)}{2 b (a+b x)^2 (b d-a e)}-\frac{\sqrt{d+e x} (-a B e-3 A b e+4 b B d)}{4 b (a+b x) (b d-a e)^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 29.6263, size = 138, normalized size = 0.88 \[ \frac{\sqrt{d + e x} \left (3 A b e + B a e - 4 B b d\right )}{4 b \left (a + b x\right ) \left (a e - b d\right )^{2}} + \frac{\sqrt{d + e x} \left (A b - B a\right )}{2 b \left (a + b x\right )^{2} \left (a e - b d\right )} + \frac{e \left (3 A b e + B a e - 4 B b d\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{d + e x}}{\sqrt{a e - b d}} \right )}}{4 b^{\frac{3}{2}} \left (a e - b d\right )^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.308606, size = 143, normalized size = 0.91 \[ -\frac{\sqrt{d+e x} \left (B \left (a^2 e+a b (2 d-e x)+4 b^2 d x\right )+A b (-5 a e+2 b d-3 b e x)\right )}{4 b (a+b x)^2 (b d-a e)^2}-\frac{e (a B e+3 A b e-4 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{4 b^{3/2} (b d-a e)^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.025, size = 436, normalized size = 2.8 \[{\frac{3\,Ab{e}^{2}}{4\, \left ( bxe+ae \right ) ^{2} \left ({a}^{2}{e}^{2}-2\,abde+{b}^{2}{d}^{2} \right ) } \left ( ex+d \right ) ^{{\frac{3}{2}}}}+{\frac{Ba{e}^{2}}{4\, \left ( bxe+ae \right ) ^{2} \left ({a}^{2}{e}^{2}-2\,abde+{b}^{2}{d}^{2} \right ) } \left ( ex+d \right ) ^{{\frac{3}{2}}}}-{\frac{bBde}{ \left ( bxe+ae \right ) ^{2} \left ({a}^{2}{e}^{2}-2\,abde+{b}^{2}{d}^{2} \right ) } \left ( ex+d \right ) ^{{\frac{3}{2}}}}+{\frac{5\,A{e}^{2}}{4\, \left ( bxe+ae \right ) ^{2} \left ( ae-bd \right ) }\sqrt{ex+d}}-{\frac{Ba{e}^{2}}{4\, \left ( bxe+ae \right ) ^{2} \left ( ae-bd \right ) b}\sqrt{ex+d}}-{\frac{eBd}{ \left ( bxe+ae \right ) ^{2} \left ( ae-bd \right ) }\sqrt{ex+d}}+{\frac{3\,A{e}^{2}}{4\,{a}^{2}{e}^{2}-8\,abde+4\,{b}^{2}{d}^{2}}\arctan \left ({b\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( ae-bd \right ) b}}}} \right ){\frac{1}{\sqrt{ \left ( ae-bd \right ) b}}}}+{\frac{Ba{e}^{2}}{ \left ( 4\,{a}^{2}{e}^{2}-8\,abde+4\,{b}^{2}{d}^{2} \right ) b}\arctan \left ({b\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( ae-bd \right ) b}}}} \right ){\frac{1}{\sqrt{ \left ( ae-bd \right ) b}}}}-{\frac{eBd}{{a}^{2}{e}^{2}-2\,abde+{b}^{2}{d}^{2}}\arctan \left ({b\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( ae-bd \right ) b}}}} \right ){\frac{1}{\sqrt{ \left ( ae-bd \right ) b}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.226826, size = 1, normalized size = 0.01 \[ \left [-\frac{2 \, \sqrt{b^{2} d - a b e}{\left (2 \,{\left (B a b + A b^{2}\right )} d +{\left (B a^{2} - 5 \, A a b\right )} e +{\left (4 \, B b^{2} d -{\left (B a b + 3 \, A b^{2}\right )} e\right )} x\right )} \sqrt{e x + d} +{\left (4 \, B a^{2} b d e -{\left (B a^{3} + 3 \, A a^{2} b\right )} e^{2} +{\left (4 \, B b^{3} d e -{\left (B a b^{2} + 3 \, A b^{3}\right )} e^{2}\right )} x^{2} + 2 \,{\left (4 \, B a b^{2} d e -{\left (B a^{2} b + 3 \, A a b^{2}\right )} e^{2}\right )} x\right )} \log \left (\frac{\sqrt{b^{2} d - a b e}{\left (b e x + 2 \, b d - a e\right )} - 2 \,{\left (b^{2} d - a b e\right )} \sqrt{e x + d}}{b x + a}\right )}{8 \,{\left (a^{2} b^{3} d^{2} - 2 \, a^{3} b^{2} d e + a^{4} b e^{2} +{\left (b^{5} d^{2} - 2 \, a b^{4} d e + a^{2} b^{3} e^{2}\right )} x^{2} + 2 \,{\left (a b^{4} d^{2} - 2 \, a^{2} b^{3} d e + a^{3} b^{2} e^{2}\right )} x\right )} \sqrt{b^{2} d - a b e}}, -\frac{\sqrt{-b^{2} d + a b e}{\left (2 \,{\left (B a b + A b^{2}\right )} d +{\left (B a^{2} - 5 \, A a b\right )} e +{\left (4 \, B b^{2} d -{\left (B a b + 3 \, A b^{2}\right )} e\right )} x\right )} \sqrt{e x + d} -{\left (4 \, B a^{2} b d e -{\left (B a^{3} + 3 \, A a^{2} b\right )} e^{2} +{\left (4 \, B b^{3} d e -{\left (B a b^{2} + 3 \, A b^{3}\right )} e^{2}\right )} x^{2} + 2 \,{\left (4 \, B a b^{2} d e -{\left (B a^{2} b + 3 \, A a b^{2}\right )} e^{2}\right )} x\right )} \arctan \left (-\frac{b d - a e}{\sqrt{-b^{2} d + a b e} \sqrt{e x + d}}\right )}{4 \,{\left (a^{2} b^{3} d^{2} - 2 \, a^{3} b^{2} d e + a^{4} b e^{2} +{\left (b^{5} d^{2} - 2 \, a b^{4} d e + a^{2} b^{3} e^{2}\right )} x^{2} + 2 \,{\left (a b^{4} d^{2} - 2 \, a^{2} b^{3} d e + a^{3} b^{2} e^{2}\right )} x\right )} \sqrt{-b^{2} d + a b e}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.219294, size = 359, normalized size = 2.29 \[ -\frac{{\left (4 \, B b d e - B a e^{2} - 3 \, A b e^{2}\right )} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right )}{4 \,{\left (b^{3} d^{2} - 2 \, a b^{2} d e + a^{2} b e^{2}\right )} \sqrt{-b^{2} d + a b e}} - \frac{4 \,{\left (x e + d\right )}^{\frac{3}{2}} B b^{2} d e - 4 \, \sqrt{x e + d} B b^{2} d^{2} e -{\left (x e + d\right )}^{\frac{3}{2}} B a b e^{2} - 3 \,{\left (x e + d\right )}^{\frac{3}{2}} A b^{2} e^{2} + 3 \, \sqrt{x e + d} B a b d e^{2} + 5 \, \sqrt{x e + d} A b^{2} d e^{2} + \sqrt{x e + d} B a^{2} e^{3} - 5 \, \sqrt{x e + d} A a b e^{3}}{4 \,{\left (b^{3} d^{2} - 2 \, a b^{2} d e + a^{2} b e^{2}\right )}{\left ({\left (x e + d\right )} b - b d + a e\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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