3.2087 \(\int \frac{(a+b x) (d+e x)^{3/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx\)

Optimal. Leaf size=172 \[ -\frac{3 e^4 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{64 b^{5/2} (b d-a e)^{5/2}}+\frac{3 e^3 \sqrt{d+e x}}{64 b^2 (a+b x) (b d-a e)^2}-\frac{e^2 \sqrt{d+e x}}{32 b^2 (a+b x)^2 (b d-a e)}-\frac{e \sqrt{d+e x}}{8 b^2 (a+b x)^3}-\frac{(d+e x)^{3/2}}{4 b (a+b x)^4} \]

[Out]

-(e*Sqrt[d + e*x])/(8*b^2*(a + b*x)^3) - (e^2*Sqrt[d + e*x])/(32*b^2*(b*d - a*e)
*(a + b*x)^2) + (3*e^3*Sqrt[d + e*x])/(64*b^2*(b*d - a*e)^2*(a + b*x)) - (d + e*
x)^(3/2)/(4*b*(a + b*x)^4) - (3*e^4*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a
*e]])/(64*b^(5/2)*(b*d - a*e)^(5/2))

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Rubi [A]  time = 0.319127, antiderivative size = 172, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.152 \[ -\frac{3 e^4 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{64 b^{5/2} (b d-a e)^{5/2}}+\frac{3 e^3 \sqrt{d+e x}}{64 b^2 (a+b x) (b d-a e)^2}-\frac{e^2 \sqrt{d+e x}}{32 b^2 (a+b x)^2 (b d-a e)}-\frac{e \sqrt{d+e x}}{8 b^2 (a+b x)^3}-\frac{(d+e x)^{3/2}}{4 b (a+b x)^4} \]

Antiderivative was successfully verified.

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

[Out]

-(e*Sqrt[d + e*x])/(8*b^2*(a + b*x)^3) - (e^2*Sqrt[d + e*x])/(32*b^2*(b*d - a*e)
*(a + b*x)^2) + (3*e^3*Sqrt[d + e*x])/(64*b^2*(b*d - a*e)^2*(a + b*x)) - (d + e*
x)^(3/2)/(4*b*(a + b*x)^4) - (3*e^4*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a
*e]])/(64*b^(5/2)*(b*d - a*e)^(5/2))

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Rubi in Sympy [A]  time = 72.1614, size = 150, normalized size = 0.87 \[ - \frac{\left (d + e x\right )^{\frac{3}{2}}}{4 b \left (a + b x\right )^{4}} + \frac{3 e^{3} \sqrt{d + e x}}{64 b^{2} \left (a + b x\right ) \left (a e - b d\right )^{2}} + \frac{e^{2} \sqrt{d + e x}}{32 b^{2} \left (a + b x\right )^{2} \left (a e - b d\right )} - \frac{e \sqrt{d + e x}}{8 b^{2} \left (a + b x\right )^{3}} + \frac{3 e^{4} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{d + e x}}{\sqrt{a e - b d}} \right )}}{64 b^{\frac{5}{2}} \left (a e - b d\right )^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(d + e*x)**(3/2)/(4*b*(a + b*x)**4) + 3*e**3*sqrt(d + e*x)/(64*b**2*(a + b*x)*(
a*e - b*d)**2) + e**2*sqrt(d + e*x)/(32*b**2*(a + b*x)**2*(a*e - b*d)) - e*sqrt(
d + e*x)/(8*b**2*(a + b*x)**3) + 3*e**4*atan(sqrt(b)*sqrt(d + e*x)/sqrt(a*e - b*
d))/(64*b**(5/2)*(a*e - b*d)**(5/2))

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Mathematica [A]  time = 0.258451, size = 149, normalized size = 0.87 \[ -\frac{3 e^4 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{64 b^{5/2} (b d-a e)^{5/2}}-\frac{\sqrt{d+e x} \left (2 e^2 (a+b x)^2 (b d-a e)+24 e (a+b x) (b d-a e)^2+16 (b d-a e)^3-3 e^3 (a+b x)^3\right )}{64 b^2 (a+b x)^4 (b d-a e)^2} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[d + e*x]*(16*(b*d - a*e)^3 + 24*e*(b*d - a*e)^2*(a + b*x) + 2*e^2*(b*d -
a*e)*(a + b*x)^2 - 3*e^3*(a + b*x)^3))/(64*b^2*(b*d - a*e)^2*(a + b*x)^4) - (3*e
^4*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(64*b^(5/2)*(b*d - a*e)^(5/
2))

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Maple [A]  time = 0.022, size = 222, normalized size = 1.3 \[{\frac{3\,{e}^{4}b}{64\, \left ( bex+ae \right ) ^{4} \left ({a}^{2}{e}^{2}-2\,abde+{b}^{2}{d}^{2} \right ) } \left ( ex+d \right ) ^{{\frac{7}{2}}}}+{\frac{11\,{e}^{4}}{64\, \left ( bex+ae \right ) ^{4} \left ( ae-bd \right ) } \left ( ex+d \right ) ^{{\frac{5}{2}}}}-{\frac{11\,{e}^{4}}{64\, \left ( bex+ae \right ) ^{4}b} \left ( ex+d \right ) ^{{\frac{3}{2}}}}-{\frac{3\,a{e}^{5}}{64\, \left ( bex+ae \right ) ^{4}{b}^{2}}\sqrt{ex+d}}+{\frac{3\,{e}^{4}d}{64\, \left ( bex+ae \right ) ^{4}b}\sqrt{ex+d}}+{\frac{3\,{e}^{4}}{64\,{b}^{2} \left ({a}^{2}{e}^{2}-2\,abde+{b}^{2}{d}^{2} \right ) }\arctan \left ({b\sqrt{ex+d}{\frac{1}{\sqrt{b \left ( ae-bd \right ) }}}} \right ){\frac{1}{\sqrt{b \left ( ae-bd \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/64*e^4/(b*e*x+a*e)^4*b/(a^2*e^2-2*a*b*d*e+b^2*d^2)*(e*x+d)^(7/2)+11/64*e^4/(b*
e*x+a*e)^4/(a*e-b*d)*(e*x+d)^(5/2)-11/64*e^4/(b*e*x+a*e)^4/b*(e*x+d)^(3/2)-3/64*
e^5/(b*e*x+a*e)^4/b^2*(e*x+d)^(1/2)*a+3/64*e^4/(b*e*x+a*e)^4/b*(e*x+d)^(1/2)*d+3
/64*e^4/b^2/(a^2*e^2-2*a*b*d*e+b^2*d^2)/(b*(a*e-b*d))^(1/2)*arctan((e*x+d)^(1/2)
*b/(b*(a*e-b*d))^(1/2))

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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^2*x^2 + 2*a*b*x + a^2)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.310864, 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)/(b^2*x^2 + 2*a*b*x + a^2)^3,x, algorithm="fricas")

[Out]

[1/128*(2*(3*b^3*e^3*x^3 - 16*b^3*d^3 + 24*a*b^2*d^2*e - 2*a^2*b*d*e^2 - 3*a^3*e
^3 - (2*b^3*d*e^2 - 11*a*b^2*e^3)*x^2 - (24*b^3*d^2*e - 44*a*b^2*d*e^2 + 11*a^2*
b*e^3)*x)*sqrt(b^2*d - a*b*e)*sqrt(e*x + d) + 3*(b^4*e^4*x^4 + 4*a*b^3*e^4*x^3 +
6*a^2*b^2*e^4*x^2 + 4*a^3*b*e^4*x + a^4*e^4)*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^4*b^4*d^2 - 2*a^
5*b^3*d*e + a^6*b^2*e^2 + (b^8*d^2 - 2*a*b^7*d*e + a^2*b^6*e^2)*x^4 + 4*(a*b^7*d
^2 - 2*a^2*b^6*d*e + a^3*b^5*e^2)*x^3 + 6*(a^2*b^6*d^2 - 2*a^3*b^5*d*e + a^4*b^4
*e^2)*x^2 + 4*(a^3*b^5*d^2 - 2*a^4*b^4*d*e + a^5*b^3*e^2)*x)*sqrt(b^2*d - a*b*e)
), 1/64*((3*b^3*e^3*x^3 - 16*b^3*d^3 + 24*a*b^2*d^2*e - 2*a^2*b*d*e^2 - 3*a^3*e^
3 - (2*b^3*d*e^2 - 11*a*b^2*e^3)*x^2 - (24*b^3*d^2*e - 44*a*b^2*d*e^2 + 11*a^2*b
*e^3)*x)*sqrt(-b^2*d + a*b*e)*sqrt(e*x + d) - 3*(b^4*e^4*x^4 + 4*a*b^3*e^4*x^3 +
6*a^2*b^2*e^4*x^2 + 4*a^3*b*e^4*x + a^4*e^4)*arctan(-(b*d - a*e)/(sqrt(-b^2*d +
a*b*e)*sqrt(e*x + d))))/((a^4*b^4*d^2 - 2*a^5*b^3*d*e + a^6*b^2*e^2 + (b^8*d^2
- 2*a*b^7*d*e + a^2*b^6*e^2)*x^4 + 4*(a*b^7*d^2 - 2*a^2*b^6*d*e + a^3*b^5*e^2)*x
^3 + 6*(a^2*b^6*d^2 - 2*a^3*b^5*d*e + a^4*b^4*e^2)*x^2 + 4*(a^3*b^5*d^2 - 2*a^4*
b^4*d*e + a^5*b^3*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)*(e*x+d)**(3/2)/(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.293372, size = 390, normalized size = 2.27 \[ \frac{3 \, \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right ) e^{4}}{64 \,{\left (b^{4} d^{2} - 2 \, a b^{3} d e + a^{2} b^{2} e^{2}\right )} \sqrt{-b^{2} d + a b e}} + \frac{3 \,{\left (x e + d\right )}^{\frac{7}{2}} b^{3} e^{4} - 11 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{3} d e^{4} - 11 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{3} d^{2} e^{4} + 3 \, \sqrt{x e + d} b^{3} d^{3} e^{4} + 11 \,{\left (x e + d\right )}^{\frac{5}{2}} a b^{2} e^{5} + 22 \,{\left (x e + d\right )}^{\frac{3}{2}} a b^{2} d e^{5} - 9 \, \sqrt{x e + d} a b^{2} d^{2} e^{5} - 11 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{2} b e^{6} + 9 \, \sqrt{x e + d} a^{2} b d e^{6} - 3 \, \sqrt{x e + d} a^{3} e^{7}}{64 \,{\left (b^{4} d^{2} - 2 \, a b^{3} d e + a^{2} b^{2} e^{2}\right )}{\left ({\left (x e + d\right )} b - b d + a e\right )}^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/64*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))*e^4/((b^4*d^2 - 2*a*b^3*d*e +
a^2*b^2*e^2)*sqrt(-b^2*d + a*b*e)) + 1/64*(3*(x*e + d)^(7/2)*b^3*e^4 - 11*(x*e +
d)^(5/2)*b^3*d*e^4 - 11*(x*e + d)^(3/2)*b^3*d^2*e^4 + 3*sqrt(x*e + d)*b^3*d^3*e
^4 + 11*(x*e + d)^(5/2)*a*b^2*e^5 + 22*(x*e + d)^(3/2)*a*b^2*d*e^5 - 9*sqrt(x*e
+ d)*a*b^2*d^2*e^5 - 11*(x*e + d)^(3/2)*a^2*b*e^6 + 9*sqrt(x*e + d)*a^2*b*d*e^6
- 3*sqrt(x*e + d)*a^3*e^7)/((b^4*d^2 - 2*a*b^3*d*e + a^2*b^2*e^2)*((x*e + d)*b -
b*d + a*e)^4)