3.1743 \(\int \frac{(A+B x) (d+e x)^{3/2}}{(a+b x)^3} \, dx\)

Optimal. Leaf size=197 \[ -\frac{3 e (-5 a B e+A b e+4 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{4 b^{7/2} \sqrt{b d-a e}}+\frac{3 e \sqrt{d+e x} (-5 a B e+A b e+4 b B d)}{4 b^3 (b d-a e)}-\frac{(d+e x)^{3/2} (-5 a B e+A b e+4 b B d)}{4 b^2 (a+b x) (b d-a e)}-\frac{(d+e x)^{5/2} (A b-a B)}{2 b (a+b x)^2 (b d-a e)} \]

[Out]

(3*e*(4*b*B*d + A*b*e - 5*a*B*e)*Sqrt[d + e*x])/(4*b^3*(b*d - a*e)) - ((4*b*B*d
+ A*b*e - 5*a*B*e)*(d + e*x)^(3/2))/(4*b^2*(b*d - a*e)*(a + b*x)) - ((A*b - a*B)
*(d + e*x)^(5/2))/(2*b*(b*d - a*e)*(a + b*x)^2) - (3*e*(4*b*B*d + A*b*e - 5*a*B*
e)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(4*b^(7/2)*Sqrt[b*d - a*e])

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Rubi [A]  time = 0.359684, antiderivative size = 197, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ -\frac{3 e (-5 a B e+A b e+4 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{4 b^{7/2} \sqrt{b d-a e}}+\frac{3 e \sqrt{d+e x} (-5 a B e+A b e+4 b B d)}{4 b^3 (b d-a e)}-\frac{(d+e x)^{3/2} (-5 a B e+A b e+4 b B d)}{4 b^2 (a+b x) (b d-a e)}-\frac{(d+e x)^{5/2} (A b-a B)}{2 b (a+b x)^2 (b d-a e)} \]

Antiderivative was successfully verified.

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

[Out]

(3*e*(4*b*B*d + A*b*e - 5*a*B*e)*Sqrt[d + e*x])/(4*b^3*(b*d - a*e)) - ((4*b*B*d
+ A*b*e - 5*a*B*e)*(d + e*x)^(3/2))/(4*b^2*(b*d - a*e)*(a + b*x)) - ((A*b - a*B)
*(d + e*x)^(5/2))/(2*b*(b*d - a*e)*(a + b*x)^2) - (3*e*(4*b*B*d + A*b*e - 5*a*B*
e)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(4*b^(7/2)*Sqrt[b*d - a*e])

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Rubi in Sympy [A]  time = 39.3463, size = 182, normalized size = 0.92 \[ \frac{\left (d + e x\right )^{\frac{5}{2}} \left (A b - B a\right )}{2 b \left (a + b x\right )^{2} \left (a e - b d\right )} + \frac{\left (d + e x\right )^{\frac{3}{2}} \left (A b e - 5 B a e + 4 B b d\right )}{4 b^{2} \left (a + b x\right ) \left (a e - b d\right )} - \frac{3 e \sqrt{d + e x} \left (A b e - 5 B a e + 4 B b d\right )}{4 b^{3} \left (a e - b d\right )} + \frac{3 e \left (A b e - 5 B a e + 4 B b d\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{d + e x}}{\sqrt{a e - b d}} \right )}}{4 b^{\frac{7}{2}} \sqrt{a e - b d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.261223, size = 139, normalized size = 0.71 \[ -\frac{\sqrt{d+e x} \left (B \left (-15 a^2 e+a b (2 d-25 e x)+4 b^2 x (d-2 e x)\right )+A b (3 a e+2 b d+5 b e x)\right )}{4 b^3 (a+b x)^2}-\frac{3 e (-5 a B e+A b e+4 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{4 b^{7/2} \sqrt{b d-a e}} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[d + e*x]*(A*b*(2*b*d + 3*a*e + 5*b*e*x) + B*(-15*a^2*e + a*b*(2*d - 25*e*
x) + 4*b^2*x*(d - 2*e*x))))/(4*b^3*(a + b*x)^2) - (3*e*(4*b*B*d + A*b*e - 5*a*B*
e)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(4*b^(7/2)*Sqrt[b*d - a*e])

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Maple [B]  time = 0.026, size = 360, normalized size = 1.8 \[ 2\,{\frac{eB\sqrt{ex+d}}{{b}^{3}}}-{\frac{5\,A{e}^{2}}{4\,b \left ( bxe+ae \right ) ^{2}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}+{\frac{9\,Ba{e}^{2}}{4\,{b}^{2} \left ( bxe+ae \right ) ^{2}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}-{\frac{eBd}{b \left ( bxe+ae \right ) ^{2}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}-{\frac{3\,aA{e}^{3}}{4\,{b}^{2} \left ( bxe+ae \right ) ^{2}}\sqrt{ex+d}}+{\frac{3\,Ad{e}^{2}}{4\,b \left ( bxe+ae \right ) ^{2}}\sqrt{ex+d}}+{\frac{7\,B{a}^{2}{e}^{3}}{4\,{b}^{3} \left ( bxe+ae \right ) ^{2}}\sqrt{ex+d}}-{\frac{11\,Bad{e}^{2}}{4\,{b}^{2} \left ( bxe+ae \right ) ^{2}}\sqrt{ex+d}}+{\frac{eB{d}^{2}}{b \left ( bxe+ae \right ) ^{2}}\sqrt{ex+d}}+{\frac{3\,A{e}^{2}}{4\,{b}^{2}}\arctan \left ({b\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( ae-bd \right ) b}}}} \right ){\frac{1}{\sqrt{ \left ( ae-bd \right ) b}}}}-{\frac{15\,Ba{e}^{2}}{4\,{b}^{3}}\arctan \left ({b\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( ae-bd \right ) b}}}} \right ){\frac{1}{\sqrt{ \left ( ae-bd \right ) b}}}}+3\,{\frac{eBd}{{b}^{2}\sqrt{ \left ( ae-bd \right ) b}}\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{ \left ( ae-bd \right ) b}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*e*B/b^3*(e*x+d)^(1/2)-5/4/b/(b*e*x+a*e)^2*(e*x+d)^(3/2)*A*e^2+9/4/b^2/(b*e*x+a
*e)^2*(e*x+d)^(3/2)*B*a*e^2-e/b/(b*e*x+a*e)^2*(e*x+d)^(3/2)*B*d-3/4/b^2/(b*e*x+a
*e)^2*(e*x+d)^(1/2)*A*a*e^3+3/4/b/(b*e*x+a*e)^2*(e*x+d)^(1/2)*A*d*e^2+7/4/b^3/(b
*e*x+a*e)^2*(e*x+d)^(1/2)*B*a^2*e^3-11/4/b^2/(b*e*x+a*e)^2*(e*x+d)^(1/2)*B*a*d*e
^2+e/b/(b*e*x+a*e)^2*(e*x+d)^(1/2)*B*d^2+3/4/b^2/((a*e-b*d)*b)^(1/2)*arctan((e*x
+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*A*e^2-15/4/b^3/((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+3*e/b^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)*(e*x + d)^(3/2)/(b*x + a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.225263, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (8 \, B b^{2} e x^{2} - 2 \,{\left (B a b + A b^{2}\right )} d + 3 \,{\left (5 \, B a^{2} - A a b\right )} e -{\left (4 \, B b^{2} d - 5 \,{\left (5 \, B a b - A b^{2}\right )} e\right )} x\right )} \sqrt{b^{2} d - a b e} \sqrt{e x + d} + 3 \,{\left (4 \, B a^{2} b d e -{\left (5 \, B a^{3} - A a^{2} b\right )} e^{2} +{\left (4 \, B b^{3} d e -{\left (5 \, B a b^{2} - A b^{3}\right )} e^{2}\right )} x^{2} + 2 \,{\left (4 \, B a b^{2} d e -{\left (5 \, B a^{2} b - A a b^{2}\right )} e^{2}\right )} x\right )} \log \left (\frac{\sqrt{b^{2} d - a b e}{\left (b e x + 2 \, b d - a e\right )} - 2 \,{\left (b^{2} d - a b e\right )} \sqrt{e x + d}}{b x + a}\right )}{8 \,{\left (b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{3}\right )} \sqrt{b^{2} d - a b e}}, \frac{{\left (8 \, B b^{2} e x^{2} - 2 \,{\left (B a b + A b^{2}\right )} d + 3 \,{\left (5 \, B a^{2} - A a b\right )} e -{\left (4 \, B b^{2} d - 5 \,{\left (5 \, B a b - A b^{2}\right )} e\right )} x\right )} \sqrt{-b^{2} d + a b e} \sqrt{e x + d} - 3 \,{\left (4 \, B a^{2} b d e -{\left (5 \, B a^{3} - A a^{2} b\right )} e^{2} +{\left (4 \, B b^{3} d e -{\left (5 \, B a b^{2} - A b^{3}\right )} e^{2}\right )} x^{2} + 2 \,{\left (4 \, B a b^{2} d e -{\left (5 \, B a^{2} b - A a b^{2}\right )} e^{2}\right )} x\right )} \arctan \left (-\frac{b d - a e}{\sqrt{-b^{2} d + a b e} \sqrt{e x + d}}\right )}{4 \,{\left (b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{3}\right )} \sqrt{-b^{2} d + a b e}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.220284, size = 319, normalized size = 1.62 \[ \frac{2 \, \sqrt{x e + d} B e}{b^{3}} + \frac{3 \,{\left (4 \, B b d e - 5 \, B a e^{2} + A b e^{2}\right )} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right )}{4 \, \sqrt{-b^{2} d + a b e} b^{3}} - \frac{4 \,{\left (x e + d\right )}^{\frac{3}{2}} B b^{2} d e - 4 \, \sqrt{x e + d} B b^{2} d^{2} e - 9 \,{\left (x e + d\right )}^{\frac{3}{2}} B a b e^{2} + 5 \,{\left (x e + d\right )}^{\frac{3}{2}} A b^{2} e^{2} + 11 \, \sqrt{x e + d} B a b d e^{2} - 3 \, \sqrt{x e + d} A b^{2} d e^{2} - 7 \, \sqrt{x e + d} B a^{2} e^{3} + 3 \, \sqrt{x e + d} A a b e^{3}}{4 \,{\left ({\left (x e + d\right )} b - b d + a e\right )}^{2} b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(x*e + d)*B*e/b^3 + 3/4*(4*B*b*d*e - 5*B*a*e^2 + A*b*e^2)*arctan(sqrt(x*e
+ d)*b/sqrt(-b^2*d + a*b*e))/(sqrt(-b^2*d + a*b*e)*b^3) - 1/4*(4*(x*e + d)^(3/2)
*B*b^2*d*e - 4*sqrt(x*e + d)*B*b^2*d^2*e - 9*(x*e + d)^(3/2)*B*a*b*e^2 + 5*(x*e
+ d)^(3/2)*A*b^2*e^2 + 11*sqrt(x*e + d)*B*a*b*d*e^2 - 3*sqrt(x*e + d)*A*b^2*d*e^
2 - 7*sqrt(x*e + d)*B*a^2*e^3 + 3*sqrt(x*e + d)*A*a*b*e^3)/(((x*e + d)*b - b*d +
a*e)^2*b^3)