3.1818 \(\int \frac{(A+B x) (d+e x)^{5/2}}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx\)

Optimal. Leaf size=250 \[ -\frac{5 e^2 (-7 a B e+A b e+6 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{8 b^{9/2} \sqrt{b d-a e}}+\frac{5 e^2 \sqrt{d+e x} (-7 a B e+A b e+6 b B d)}{8 b^4 (b d-a e)}-\frac{5 e (d+e x)^{3/2} (-7 a B e+A b e+6 b B d)}{24 b^3 (a+b x) (b d-a e)}-\frac{(d+e x)^{5/2} (-7 a B e+A b e+6 b B d)}{12 b^2 (a+b x)^2 (b d-a e)}-\frac{(d+e x)^{7/2} (A b-a B)}{3 b (a+b x)^3 (b d-a e)} \]

[Out]

(5*e^2*(6*b*B*d + A*b*e - 7*a*B*e)*Sqrt[d + e*x])/(8*b^4*(b*d - a*e)) - (5*e*(6*
b*B*d + A*b*e - 7*a*B*e)*(d + e*x)^(3/2))/(24*b^3*(b*d - a*e)*(a + b*x)) - ((6*b
*B*d + A*b*e - 7*a*B*e)*(d + e*x)^(5/2))/(12*b^2*(b*d - a*e)*(a + b*x)^2) - ((A*
b - a*B)*(d + e*x)^(7/2))/(3*b*(b*d - a*e)*(a + b*x)^3) - (5*e^2*(6*b*B*d + A*b*
e - 7*a*B*e)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(8*b^(9/2)*Sqrt[b
*d - a*e])

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Rubi [A]  time = 0.466601, antiderivative size = 250, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ -\frac{5 e^2 (-7 a B e+A b e+6 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{8 b^{9/2} \sqrt{b d-a e}}+\frac{5 e^2 \sqrt{d+e x} (-7 a B e+A b e+6 b B d)}{8 b^4 (b d-a e)}-\frac{5 e (d+e x)^{3/2} (-7 a B e+A b e+6 b B d)}{24 b^3 (a+b x) (b d-a e)}-\frac{(d+e x)^{5/2} (-7 a B e+A b e+6 b B d)}{12 b^2 (a+b x)^2 (b d-a e)}-\frac{(d+e x)^{7/2} (A b-a B)}{3 b (a+b x)^3 (b d-a e)} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(d + e*x)^(5/2))/(a^2 + 2*a*b*x + b^2*x^2)^2,x]

[Out]

(5*e^2*(6*b*B*d + A*b*e - 7*a*B*e)*Sqrt[d + e*x])/(8*b^4*(b*d - a*e)) - (5*e*(6*
b*B*d + A*b*e - 7*a*B*e)*(d + e*x)^(3/2))/(24*b^3*(b*d - a*e)*(a + b*x)) - ((6*b
*B*d + A*b*e - 7*a*B*e)*(d + e*x)^(5/2))/(12*b^2*(b*d - a*e)*(a + b*x)^2) - ((A*
b - a*B)*(d + e*x)^(7/2))/(3*b*(b*d - a*e)*(a + b*x)^3) - (5*e^2*(6*b*B*d + A*b*
e - 7*a*B*e)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(8*b^(9/2)*Sqrt[b
*d - a*e])

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Rubi in Sympy [A]  time = 94.4335, size = 235, normalized size = 0.94 \[ \frac{\left (d + e x\right )^{\frac{7}{2}} \left (A b - B a\right )}{3 b \left (a + b x\right )^{3} \left (a e - b d\right )} + \frac{\left (d + e x\right )^{\frac{5}{2}} \left (A b e - 7 B a e + 6 B b d\right )}{12 b^{2} \left (a + b x\right )^{2} \left (a e - b d\right )} + \frac{5 e \left (d + e x\right )^{\frac{3}{2}} \left (A b e - 7 B a e + 6 B b d\right )}{24 b^{3} \left (a + b x\right ) \left (a e - b d\right )} - \frac{5 e^{2} \sqrt{d + e x} \left (A b e - 7 B a e + 6 B b d\right )}{8 b^{4} \left (a e - b d\right )} + \frac{5 e^{2} \left (A b e - 7 B a e + 6 B b d\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{d + e x}}{\sqrt{a e - b d}} \right )}}{8 b^{\frac{9}{2}} \sqrt{a e - b d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(e*x+d)**(5/2)/(b**2*x**2+2*a*b*x+a**2)**2,x)

[Out]

(d + e*x)**(7/2)*(A*b - B*a)/(3*b*(a + b*x)**3*(a*e - b*d)) + (d + e*x)**(5/2)*(
A*b*e - 7*B*a*e + 6*B*b*d)/(12*b**2*(a + b*x)**2*(a*e - b*d)) + 5*e*(d + e*x)**(
3/2)*(A*b*e - 7*B*a*e + 6*B*b*d)/(24*b**3*(a + b*x)*(a*e - b*d)) - 5*e**2*sqrt(d
+ e*x)*(A*b*e - 7*B*a*e + 6*B*b*d)/(8*b**4*(a*e - b*d)) + 5*e**2*(A*b*e - 7*B*a
*e + 6*B*b*d)*atan(sqrt(b)*sqrt(d + e*x)/sqrt(a*e - b*d))/(8*b**(9/2)*sqrt(a*e -
b*d))

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Mathematica [A]  time = 0.600348, size = 177, normalized size = 0.71 \[ -\frac{5 e^2 (-7 a B e+A b e+6 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{8 b^{9/2} \sqrt{b d-a e}}-\frac{\sqrt{d+e x} \left (\frac{3 e (-29 a B e+11 A b e+18 b B d)}{a+b x}+\frac{2 (b d-a e) (-19 a B e+13 A b e+6 b B d)}{(a+b x)^2}+\frac{8 (A b-a B) (b d-a e)^2}{(a+b x)^3}-48 B e^2\right )}{24 b^4} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(d + e*x)^(5/2))/(a^2 + 2*a*b*x + b^2*x^2)^2,x]

[Out]

-(Sqrt[d + e*x]*(-48*B*e^2 + (8*(A*b - a*B)*(b*d - a*e)^2)/(a + b*x)^3 + (2*(b*d
- a*e)*(6*b*B*d + 13*A*b*e - 19*a*B*e))/(a + b*x)^2 + (3*e*(18*b*B*d + 11*A*b*e
- 29*a*B*e))/(a + b*x)))/(24*b^4) - (5*e^2*(6*b*B*d + A*b*e - 7*a*B*e)*ArcTanh[
(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(8*b^(9/2)*Sqrt[b*d - a*e])

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Maple [B]  time = 0.033, size = 573, normalized size = 2.3 \[ 2\,{\frac{{e}^{2}B\sqrt{ex+d}}{{b}^{4}}}-{\frac{11\,{e}^{3}A}{8\,b \left ( bex+ae \right ) ^{3}} \left ( ex+d \right ) ^{{\frac{5}{2}}}}+{\frac{29\,{e}^{3}Ba}{8\,{b}^{2} \left ( bex+ae \right ) ^{3}} \left ( ex+d \right ) ^{{\frac{5}{2}}}}-{\frac{9\,{e}^{2}Bd}{4\,b \left ( bex+ae \right ) ^{3}} \left ( ex+d \right ) ^{{\frac{5}{2}}}}-{\frac{5\,{e}^{4}Aa}{3\,{b}^{2} \left ( bex+ae \right ) ^{3}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}+{\frac{5\,{e}^{3}Ad}{3\,b \left ( bex+ae \right ) ^{3}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}+{\frac{17\,B{e}^{4}{a}^{2}}{3\,{b}^{3} \left ( bex+ae \right ) ^{3}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}-{\frac{29\,{e}^{3}Bad}{3\,{b}^{2} \left ( bex+ae \right ) ^{3}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}+4\,{\frac{{e}^{2}B \left ( ex+d \right ) ^{3/2}{d}^{2}}{b \left ( bex+ae \right ) ^{3}}}-{\frac{5\,A{a}^{2}{e}^{5}}{8\,{b}^{3} \left ( bex+ae \right ) ^{3}}\sqrt{ex+d}}+{\frac{5\,{e}^{4}Aad}{4\,{b}^{2} \left ( bex+ae \right ) ^{3}}\sqrt{ex+d}}-{\frac{5\,A{d}^{2}{e}^{3}}{8\,b \left ( bex+ae \right ) ^{3}}\sqrt{ex+d}}+{\frac{19\,{e}^{5}B{a}^{3}}{8\,{b}^{4} \left ( bex+ae \right ) ^{3}}\sqrt{ex+d}}-{\frac{13\,Bd{a}^{2}{e}^{4}}{2\,{b}^{3} \left ( bex+ae \right ) ^{3}}\sqrt{ex+d}}+{\frac{47\,{e}^{3}Ba{d}^{2}}{8\,{b}^{2} \left ( bex+ae \right ) ^{3}}\sqrt{ex+d}}-{\frac{7\,B{d}^{3}{e}^{2}}{4\,b \left ( bex+ae \right ) ^{3}}\sqrt{ex+d}}+{\frac{5\,{e}^{3}A}{8\,{b}^{3}}\arctan \left ({b\sqrt{ex+d}{\frac{1}{\sqrt{b \left ( ae-bd \right ) }}}} \right ){\frac{1}{\sqrt{b \left ( ae-bd \right ) }}}}-{\frac{35\,{e}^{3}Ba}{8\,{b}^{4}}\arctan \left ({b\sqrt{ex+d}{\frac{1}{\sqrt{b \left ( ae-bd \right ) }}}} \right ){\frac{1}{\sqrt{b \left ( ae-bd \right ) }}}}+{\frac{15\,{e}^{2}Bd}{4\,{b}^{3}}\arctan \left ({b\sqrt{ex+d}{\frac{1}{\sqrt{b \left ( ae-bd \right ) }}}} \right ){\frac{1}{\sqrt{b \left ( ae-bd \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(e*x+d)^(5/2)/(b^2*x^2+2*a*b*x+a^2)^2,x)

[Out]

2*e^2*B/b^4*(e*x+d)^(1/2)-11/8*e^3/b/(b*e*x+a*e)^3*(e*x+d)^(5/2)*A+29/8*e^3/b^2/
(b*e*x+a*e)^3*(e*x+d)^(5/2)*B*a-9/4*e^2/b/(b*e*x+a*e)^3*(e*x+d)^(5/2)*B*d-5/3*e^
4/b^2/(b*e*x+a*e)^3*A*(e*x+d)^(3/2)*a+5/3*e^3/b/(b*e*x+a*e)^3*A*(e*x+d)^(3/2)*d+
17/3*e^4/b^3/(b*e*x+a*e)^3*B*(e*x+d)^(3/2)*a^2-29/3*e^3/b^2/(b*e*x+a*e)^3*B*(e*x
+d)^(3/2)*a*d+4*e^2/b/(b*e*x+a*e)^3*B*(e*x+d)^(3/2)*d^2-5/8*e^5/b^3/(b*e*x+a*e)^
3*(e*x+d)^(1/2)*A*a^2+5/4*e^4/b^2/(b*e*x+a*e)^3*(e*x+d)^(1/2)*A*a*d-5/8*e^3/b/(b
*e*x+a*e)^3*(e*x+d)^(1/2)*A*d^2+19/8*e^5/b^4/(b*e*x+a*e)^3*(e*x+d)^(1/2)*B*a^3-1
3/2*e^4/b^3/(b*e*x+a*e)^3*(e*x+d)^(1/2)*B*a^2*d+47/8*e^3/b^2/(b*e*x+a*e)^3*(e*x+
d)^(1/2)*B*a*d^2-7/4*e^2/b/(b*e*x+a*e)^3*(e*x+d)^(1/2)*B*d^3+5/8*e^3/b^3/(b*(a*e
-b*d))^(1/2)*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*A-35/8*e^3/b^4/(b*(a*e-
b*d))^(1/2)*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*a*B+15/4*e^2/b^3/(b*(a*e
-b*d))^(1/2)*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(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)^(5/2)/(b^2*x^2 + 2*a*b*x + a^2)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.300619, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(e*x + d)^(5/2)/(b^2*x^2 + 2*a*b*x + a^2)^2,x, algorithm="fricas")

[Out]

[1/48*(2*(48*B*b^3*e^2*x^3 - 4*(B*a*b^2 + 2*A*b^3)*d^2 - 10*(2*B*a^2*b + A*a*b^2
)*d*e + 15*(7*B*a^3 - A*a^2*b)*e^2 - 3*(18*B*b^3*d*e - 11*(7*B*a*b^2 - A*b^3)*e^
2)*x^2 - 2*(6*B*b^3*d^2 + (29*B*a*b^2 + 13*A*b^3)*d*e - 20*(7*B*a^2*b - A*a*b^2)
*e^2)*x)*sqrt(b^2*d - a*b*e)*sqrt(e*x + d) + 15*(6*B*a^3*b*d*e^2 - (7*B*a^4 - A*
a^3*b)*e^3 + (6*B*b^4*d*e^2 - (7*B*a*b^3 - A*b^4)*e^3)*x^3 + 3*(6*B*a*b^3*d*e^2
- (7*B*a^2*b^2 - A*a*b^3)*e^3)*x^2 + 3*(6*B*a^2*b^2*d*e^2 - (7*B*a^3*b - A*a^2*b
^2)*e^3)*x)*log((sqrt(b^2*d - a*b*e)*(b*e*x + 2*b*d - a*e) - 2*(b^2*d - a*b*e)*s
qrt(e*x + d))/(b*x + a)))/((b^7*x^3 + 3*a*b^6*x^2 + 3*a^2*b^5*x + a^3*b^4)*sqrt(
b^2*d - a*b*e)), 1/24*((48*B*b^3*e^2*x^3 - 4*(B*a*b^2 + 2*A*b^3)*d^2 - 10*(2*B*a
^2*b + A*a*b^2)*d*e + 15*(7*B*a^3 - A*a^2*b)*e^2 - 3*(18*B*b^3*d*e - 11*(7*B*a*b
^2 - A*b^3)*e^2)*x^2 - 2*(6*B*b^3*d^2 + (29*B*a*b^2 + 13*A*b^3)*d*e - 20*(7*B*a^
2*b - A*a*b^2)*e^2)*x)*sqrt(-b^2*d + a*b*e)*sqrt(e*x + d) - 15*(6*B*a^3*b*d*e^2
- (7*B*a^4 - A*a^3*b)*e^3 + (6*B*b^4*d*e^2 - (7*B*a*b^3 - A*b^4)*e^3)*x^3 + 3*(6
*B*a*b^3*d*e^2 - (7*B*a^2*b^2 - A*a*b^3)*e^3)*x^2 + 3*(6*B*a^2*b^2*d*e^2 - (7*B*
a^3*b - A*a^2*b^2)*e^3)*x)*arctan(-(b*d - a*e)/(sqrt(-b^2*d + a*b*e)*sqrt(e*x +
d))))/((b^7*x^3 + 3*a*b^6*x^2 + 3*a^2*b^5*x + a^3*b^4)*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)**(5/2)/(b**2*x**2+2*a*b*x+a**2)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.304839, size = 500, normalized size = 2. \[ \frac{2 \, \sqrt{x e + d} B e^{2}}{b^{4}} + \frac{5 \,{\left (6 \, B b d e^{2} - 7 \, B a e^{3} + A b e^{3}\right )} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right )}{8 \, \sqrt{-b^{2} d + a b e} b^{4}} - \frac{54 \,{\left (x e + d\right )}^{\frac{5}{2}} B b^{3} d e^{2} - 96 \,{\left (x e + d\right )}^{\frac{3}{2}} B b^{3} d^{2} e^{2} + 42 \, \sqrt{x e + d} B b^{3} d^{3} e^{2} - 87 \,{\left (x e + d\right )}^{\frac{5}{2}} B a b^{2} e^{3} + 33 \,{\left (x e + d\right )}^{\frac{5}{2}} A b^{3} e^{3} + 232 \,{\left (x e + d\right )}^{\frac{3}{2}} B a b^{2} d e^{3} - 40 \,{\left (x e + d\right )}^{\frac{3}{2}} A b^{3} d e^{3} - 141 \, \sqrt{x e + d} B a b^{2} d^{2} e^{3} + 15 \, \sqrt{x e + d} A b^{3} d^{2} e^{3} - 136 \,{\left (x e + d\right )}^{\frac{3}{2}} B a^{2} b e^{4} + 40 \,{\left (x e + d\right )}^{\frac{3}{2}} A a b^{2} e^{4} + 156 \, \sqrt{x e + d} B a^{2} b d e^{4} - 30 \, \sqrt{x e + d} A a b^{2} d e^{4} - 57 \, \sqrt{x e + d} B a^{3} e^{5} + 15 \, \sqrt{x e + d} A a^{2} b e^{5}}{24 \,{\left ({\left (x e + d\right )} b - b d + a e\right )}^{3} b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(e*x + d)^(5/2)/(b^2*x^2 + 2*a*b*x + a^2)^2,x, algorithm="giac")

[Out]

2*sqrt(x*e + d)*B*e^2/b^4 + 5/8*(6*B*b*d*e^2 - 7*B*a*e^3 + A*b*e^3)*arctan(sqrt(
x*e + d)*b/sqrt(-b^2*d + a*b*e))/(sqrt(-b^2*d + a*b*e)*b^4) - 1/24*(54*(x*e + d)
^(5/2)*B*b^3*d*e^2 - 96*(x*e + d)^(3/2)*B*b^3*d^2*e^2 + 42*sqrt(x*e + d)*B*b^3*d
^3*e^2 - 87*(x*e + d)^(5/2)*B*a*b^2*e^3 + 33*(x*e + d)^(5/2)*A*b^3*e^3 + 232*(x*
e + d)^(3/2)*B*a*b^2*d*e^3 - 40*(x*e + d)^(3/2)*A*b^3*d*e^3 - 141*sqrt(x*e + d)*
B*a*b^2*d^2*e^3 + 15*sqrt(x*e + d)*A*b^3*d^2*e^3 - 136*(x*e + d)^(3/2)*B*a^2*b*e
^4 + 40*(x*e + d)^(3/2)*A*a*b^2*e^4 + 156*sqrt(x*e + d)*B*a^2*b*d*e^4 - 30*sqrt(
x*e + d)*A*a*b^2*d*e^4 - 57*sqrt(x*e + d)*B*a^3*e^5 + 15*sqrt(x*e + d)*A*a^2*b*e
^5)/(((x*e + d)*b - b*d + a*e)^3*b^4)