3.1817 \(\int \frac{(A+B x) (d+e x)^{7/2}}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx\)

Optimal. Leaf size=284 \[ -\frac{35 e^2 \sqrt{b d-a e} (-3 a B e+A b e+2 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{8 b^{11/2}}+\frac{35 e^2 \sqrt{d+e x} (-3 a B e+A b e+2 b B d)}{8 b^5}+\frac{35 e^2 (d+e x)^{3/2} (-3 a B e+A b e+2 b B d)}{24 b^4 (b d-a e)}-\frac{7 e (d+e x)^{5/2} (-3 a B e+A b e+2 b B d)}{8 b^3 (a+b x) (b d-a e)}-\frac{(d+e x)^{7/2} (-3 a B e+A b e+2 b B d)}{4 b^2 (a+b x)^2 (b d-a e)}-\frac{(d+e x)^{9/2} (A b-a B)}{3 b (a+b x)^3 (b d-a e)} \]

[Out]

(35*e^2*(2*b*B*d + A*b*e - 3*a*B*e)*Sqrt[d + e*x])/(8*b^5) + (35*e^2*(2*b*B*d +
A*b*e - 3*a*B*e)*(d + e*x)^(3/2))/(24*b^4*(b*d - a*e)) - (7*e*(2*b*B*d + A*b*e -
3*a*B*e)*(d + e*x)^(5/2))/(8*b^3*(b*d - a*e)*(a + b*x)) - ((2*b*B*d + A*b*e - 3
*a*B*e)*(d + e*x)^(7/2))/(4*b^2*(b*d - a*e)*(a + b*x)^2) - ((A*b - a*B)*(d + e*x
)^(9/2))/(3*b*(b*d - a*e)*(a + b*x)^3) - (35*e^2*Sqrt[b*d - a*e]*(2*b*B*d + A*b*
e - 3*a*B*e)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(8*b^(11/2))

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Rubi [A]  time = 0.558294, antiderivative size = 284, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ -\frac{35 e^2 \sqrt{b d-a e} (-3 a B e+A b e+2 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{8 b^{11/2}}+\frac{35 e^2 \sqrt{d+e x} (-3 a B e+A b e+2 b B d)}{8 b^5}+\frac{35 e^2 (d+e x)^{3/2} (-3 a B e+A b e+2 b B d)}{24 b^4 (b d-a e)}-\frac{7 e (d+e x)^{5/2} (-3 a B e+A b e+2 b B d)}{8 b^3 (a+b x) (b d-a e)}-\frac{(d+e x)^{7/2} (-3 a B e+A b e+2 b B d)}{4 b^2 (a+b x)^2 (b d-a e)}-\frac{(d+e x)^{9/2} (A b-a B)}{3 b (a+b x)^3 (b d-a e)} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(d + e*x)^(7/2))/(a^2 + 2*a*b*x + b^2*x^2)^2,x]

[Out]

(35*e^2*(2*b*B*d + A*b*e - 3*a*B*e)*Sqrt[d + e*x])/(8*b^5) + (35*e^2*(2*b*B*d +
A*b*e - 3*a*B*e)*(d + e*x)^(3/2))/(24*b^4*(b*d - a*e)) - (7*e*(2*b*B*d + A*b*e -
3*a*B*e)*(d + e*x)^(5/2))/(8*b^3*(b*d - a*e)*(a + b*x)) - ((2*b*B*d + A*b*e - 3
*a*B*e)*(d + e*x)^(7/2))/(4*b^2*(b*d - a*e)*(a + b*x)^2) - ((A*b - a*B)*(d + e*x
)^(9/2))/(3*b*(b*d - a*e)*(a + b*x)^3) - (35*e^2*Sqrt[b*d - a*e]*(2*b*B*d + A*b*
e - 3*a*B*e)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(8*b^(11/2))

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Rubi in Sympy [A]  time = 107.982, size = 272, normalized size = 0.96 \[ \frac{\left (d + e x\right )^{\frac{9}{2}} \left (A b - B a\right )}{3 b \left (a + b x\right )^{3} \left (a e - b d\right )} + \frac{\left (d + e x\right )^{\frac{7}{2}} \left (A b e - 3 B a e + 2 B b d\right )}{4 b^{2} \left (a + b x\right )^{2} \left (a e - b d\right )} + \frac{7 e \left (d + e x\right )^{\frac{5}{2}} \left (A b e - 3 B a e + 2 B b d\right )}{8 b^{3} \left (a + b x\right ) \left (a e - b d\right )} - \frac{35 e^{2} \left (d + e x\right )^{\frac{3}{2}} \left (A b e - 3 B a e + 2 B b d\right )}{24 b^{4} \left (a e - b d\right )} + \frac{35 e^{2} \sqrt{d + e x} \left (A b e - 3 B a e + 2 B b d\right )}{8 b^{5}} - \frac{35 e^{2} \sqrt{a e - b d} \left (A b e - 3 B a e + 2 B b d\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{d + e x}}{\sqrt{a e - b d}} \right )}}{8 b^{\frac{11}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(e*x+d)**(7/2)/(b**2*x**2+2*a*b*x+a**2)**2,x)

[Out]

(d + e*x)**(9/2)*(A*b - B*a)/(3*b*(a + b*x)**3*(a*e - b*d)) + (d + e*x)**(7/2)*(
A*b*e - 3*B*a*e + 2*B*b*d)/(4*b**2*(a + b*x)**2*(a*e - b*d)) + 7*e*(d + e*x)**(5
/2)*(A*b*e - 3*B*a*e + 2*B*b*d)/(8*b**3*(a + b*x)*(a*e - b*d)) - 35*e**2*(d + e*
x)**(3/2)*(A*b*e - 3*B*a*e + 2*B*b*d)/(24*b**4*(a*e - b*d)) + 35*e**2*sqrt(d + e
*x)*(A*b*e - 3*B*a*e + 2*B*b*d)/(8*b**5) - 35*e**2*sqrt(a*e - b*d)*(A*b*e - 3*B*
a*e + 2*B*b*d)*atan(sqrt(b)*sqrt(d + e*x)/sqrt(a*e - b*d))/(8*b**(11/2))

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Mathematica [A]  time = 0.501149, size = 222, normalized size = 0.78 \[ -\frac{35 e^2 \sqrt{b d-a e} (-3 a B e+A b e+2 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{8 b^{11/2}}-\frac{\sqrt{d+e x} \left (-16 e^2 (a+b x)^3 (-12 a B e+3 A b e+10 b B d)+2 (a+b x) (b d-a e)^2 (-25 a B e+19 A b e+6 b B d)+3 e (a+b x)^2 (b d-a e) (-55 a B e+29 A b e+26 b B d)+8 (A b-a B) (b d-a e)^3-16 b B e^3 x (a+b x)^3\right )}{24 b^5 (a+b x)^3} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(d + e*x)^(7/2))/(a^2 + 2*a*b*x + b^2*x^2)^2,x]

[Out]

-(Sqrt[d + e*x]*(8*(A*b - a*B)*(b*d - a*e)^3 + 2*(b*d - a*e)^2*(6*b*B*d + 19*A*b
*e - 25*a*B*e)*(a + b*x) + 3*e*(b*d - a*e)*(26*b*B*d + 29*A*b*e - 55*a*B*e)*(a +
b*x)^2 - 16*e^2*(10*b*B*d + 3*A*b*e - 12*a*B*e)*(a + b*x)^3 - 16*b*B*e^3*x*(a +
b*x)^3))/(24*b^5*(a + b*x)^3) - (35*e^2*Sqrt[b*d - a*e]*(2*b*B*d + A*b*e - 3*a*
B*e)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(8*b^(11/2))

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Maple [B]  time = 0.043, size = 905, normalized size = 3.2 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(e*x+d)^(7/2)/(b^2*x^2+2*a*b*x+a^2)^2,x)

[Out]

6*e^2/b^4*B*d*(e*x+d)^(1/2)-8*e^3/b^5*a*B*(e*x+d)^(1/2)+19/8*e^6/b^4/(b*e*x+a*e)
^3*(e*x+d)^(1/2)*A*a^3-19/8*e^3/b/(b*e*x+a*e)^3*(e*x+d)^(1/2)*A*d^3+35/8*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*d+6*e^2/b/(b*
e*x+a*e)^3*B*(e*x+d)^(3/2)*d^3-11/4*e^2/b/(b*e*x+a*e)^3*(e*x+d)^(1/2)*B*d^4-35/8
*e^4/b^4/(b*(a*e-b*d))^(1/2)*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*A*a+105
/8*e^4/b^5/(b*(a*e-b*d))^(1/2)*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*a^2*B
-41/8*e^6/b^5/(b*e*x+a*e)^3*(e*x+d)^(1/2)*B*a^4+29/8*e^4/b^2/(b*e*x+a*e)^3*(e*x+
d)^(5/2)*A*a-29/8*e^3/b/(b*e*x+a*e)^3*(e*x+d)^(5/2)*A*d+35/4*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^2-13/4*e^2/b/(b*e*x+a*e
)^3*(e*x+d)^(5/2)*B*d^2-55/8*e^4/b^3/(b*e*x+a*e)^3*(e*x+d)^(5/2)*B*a^2+17/3*e^5/
b^3/(b*e*x+a*e)^3*A*(e*x+d)^(3/2)*a^2+17/3*e^3/b/(b*e*x+a*e)^3*A*(e*x+d)^(3/2)*d
^2-35/3*e^5/b^4/(b*e*x+a*e)^3*B*(e*x+d)^(3/2)*a^3-189/8*e^4/b^3/(b*e*x+a*e)^3*(e
*x+d)^(1/2)*B*d^2*a^2-175/8*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))*B*d*a+81/8*e^3/b^2/(b*e*x+a*e)^3*(e*x+d)^(5/2)*B*a*d-34/3*e^
4/b^2/(b*e*x+a*e)^3*A*(e*x+d)^(3/2)*a*d+88/3*e^4/b^3/(b*e*x+a*e)^3*B*(e*x+d)^(3/
2)*a^2*d-71/3*e^3/b^2/(b*e*x+a*e)^3*B*(e*x+d)^(3/2)*a*d^2-57/8*e^5/b^3/(b*e*x+a*
e)^3*(e*x+d)^(1/2)*A*d*a^2+57/8*e^4/b^2/(b*e*x+a*e)^3*(e*x+d)^(1/2)*A*a*d^2+145/
8*e^5/b^4/(b*e*x+a*e)^3*(e*x+d)^(1/2)*B*a^3*d+107/8*e^3/b^2/(b*e*x+a*e)^3*(e*x+d
)^(1/2)*B*a*d^3+2/3*e^2/b^4*B*(e*x+d)^(3/2)+2*e^3/b^4*A*(e*x+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)^(7/2)/(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 = 0.303218, 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)^(7/2)/(b^2*x^2 + 2*a*b*x + a^2)^2,x, algorithm="fricas")

[Out]

[-1/48*(105*(2*B*a^3*b*d*e^2 - (3*B*a^4 - A*a^3*b)*e^3 + (2*B*b^4*d*e^2 - (3*B*a
*b^3 - A*b^4)*e^3)*x^3 + 3*(2*B*a*b^3*d*e^2 - (3*B*a^2*b^2 - A*a*b^3)*e^3)*x^2 +
3*(2*B*a^2*b^2*d*e^2 - (3*B*a^3*b - A*a^2*b^2)*e^3)*x)*sqrt((b*d - a*e)/b)*log(
(b*e*x + 2*b*d - a*e + 2*sqrt(e*x + d)*b*sqrt((b*d - a*e)/b))/(b*x + a)) - 2*(16
*B*b^4*e^3*x^4 - 4*(B*a*b^3 + 2*A*b^4)*d^3 - 14*(2*B*a^2*b^2 + A*a*b^3)*d^2*e +
35*(9*B*a^3*b - A*a^2*b^2)*d*e^2 - 105*(3*B*a^4 - A*a^3*b)*e^3 + 16*(10*B*b^4*d*
e^2 - 3*(3*B*a*b^3 - A*b^4)*e^3)*x^3 - 3*(26*B*b^4*d^2*e - (241*B*a*b^3 - 29*A*b
^4)*d*e^2 + 77*(3*B*a^2*b^2 - A*a*b^3)*e^3)*x^2 - 2*(6*B*b^4*d^3 + (41*B*a*b^3 +
19*A*b^4)*d^2*e - 7*(61*B*a^2*b^2 - 7*A*a*b^3)*d*e^2 + 140*(3*B*a^3*b - A*a^2*b
^2)*e^3)*x)*sqrt(e*x + d))/(b^8*x^3 + 3*a*b^7*x^2 + 3*a^2*b^6*x + a^3*b^5), -1/2
4*(105*(2*B*a^3*b*d*e^2 - (3*B*a^4 - A*a^3*b)*e^3 + (2*B*b^4*d*e^2 - (3*B*a*b^3
- A*b^4)*e^3)*x^3 + 3*(2*B*a*b^3*d*e^2 - (3*B*a^2*b^2 - A*a*b^3)*e^3)*x^2 + 3*(2
*B*a^2*b^2*d*e^2 - (3*B*a^3*b - A*a^2*b^2)*e^3)*x)*sqrt(-(b*d - a*e)/b)*arctan(s
qrt(e*x + d)/sqrt(-(b*d - a*e)/b)) - (16*B*b^4*e^3*x^4 - 4*(B*a*b^3 + 2*A*b^4)*d
^3 - 14*(2*B*a^2*b^2 + A*a*b^3)*d^2*e + 35*(9*B*a^3*b - A*a^2*b^2)*d*e^2 - 105*(
3*B*a^4 - A*a^3*b)*e^3 + 16*(10*B*b^4*d*e^2 - 3*(3*B*a*b^3 - A*b^4)*e^3)*x^3 - 3
*(26*B*b^4*d^2*e - (241*B*a*b^3 - 29*A*b^4)*d*e^2 + 77*(3*B*a^2*b^2 - A*a*b^3)*e
^3)*x^2 - 2*(6*B*b^4*d^3 + (41*B*a*b^3 + 19*A*b^4)*d^2*e - 7*(61*B*a^2*b^2 - 7*A
*a*b^3)*d*e^2 + 140*(3*B*a^3*b - A*a^2*b^2)*e^3)*x)*sqrt(e*x + d))/(b^8*x^3 + 3*
a*b^7*x^2 + 3*a^2*b^6*x + a^3*b^5)]

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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)**(7/2)/(b**2*x**2+2*a*b*x+a**2)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.316795, size = 779, normalized size = 2.74 \[ \frac{35 \,{\left (2 \, B b^{2} d^{2} e^{2} - 5 \, B a b d e^{3} + A b^{2} d e^{3} + 3 \, B a^{2} e^{4} - A a b e^{4}\right )} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right )}{8 \, \sqrt{-b^{2} d + a b e} b^{5}} - \frac{78 \,{\left (x e + d\right )}^{\frac{5}{2}} B b^{4} d^{2} e^{2} - 144 \,{\left (x e + d\right )}^{\frac{3}{2}} B b^{4} d^{3} e^{2} + 66 \, \sqrt{x e + d} B b^{4} d^{4} e^{2} - 243 \,{\left (x e + d\right )}^{\frac{5}{2}} B a b^{3} d e^{3} + 87 \,{\left (x e + d\right )}^{\frac{5}{2}} A b^{4} d e^{3} + 568 \,{\left (x e + d\right )}^{\frac{3}{2}} B a b^{3} d^{2} e^{3} - 136 \,{\left (x e + d\right )}^{\frac{3}{2}} A b^{4} d^{2} e^{3} - 321 \, \sqrt{x e + d} B a b^{3} d^{3} e^{3} + 57 \, \sqrt{x e + d} A b^{4} d^{3} e^{3} + 165 \,{\left (x e + d\right )}^{\frac{5}{2}} B a^{2} b^{2} e^{4} - 87 \,{\left (x e + d\right )}^{\frac{5}{2}} A a b^{3} e^{4} - 704 \,{\left (x e + d\right )}^{\frac{3}{2}} B a^{2} b^{2} d e^{4} + 272 \,{\left (x e + d\right )}^{\frac{3}{2}} A a b^{3} d e^{4} + 567 \, \sqrt{x e + d} B a^{2} b^{2} d^{2} e^{4} - 171 \, \sqrt{x e + d} A a b^{3} d^{2} e^{4} + 280 \,{\left (x e + d\right )}^{\frac{3}{2}} B a^{3} b e^{5} - 136 \,{\left (x e + d\right )}^{\frac{3}{2}} A a^{2} b^{2} e^{5} - 435 \, \sqrt{x e + d} B a^{3} b d e^{5} + 171 \, \sqrt{x e + d} A a^{2} b^{2} d e^{5} + 123 \, \sqrt{x e + d} B a^{4} e^{6} - 57 \, \sqrt{x e + d} A a^{3} b e^{6}}{24 \,{\left ({\left (x e + d\right )} b - b d + a e\right )}^{3} b^{5}} + \frac{2 \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} B b^{8} e^{2} + 9 \, \sqrt{x e + d} B b^{8} d e^{2} - 12 \, \sqrt{x e + d} B a b^{7} e^{3} + 3 \, \sqrt{x e + d} A b^{8} e^{3}\right )}}{3 \, b^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(e*x + d)^(7/2)/(b^2*x^2 + 2*a*b*x + a^2)^2,x, algorithm="giac")

[Out]

35/8*(2*B*b^2*d^2*e^2 - 5*B*a*b*d*e^3 + A*b^2*d*e^3 + 3*B*a^2*e^4 - A*a*b*e^4)*a
rctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))/(sqrt(-b^2*d + a*b*e)*b^5) - 1/24*(7
8*(x*e + d)^(5/2)*B*b^4*d^2*e^2 - 144*(x*e + d)^(3/2)*B*b^4*d^3*e^2 + 66*sqrt(x*
e + d)*B*b^4*d^4*e^2 - 243*(x*e + d)^(5/2)*B*a*b^3*d*e^3 + 87*(x*e + d)^(5/2)*A*
b^4*d*e^3 + 568*(x*e + d)^(3/2)*B*a*b^3*d^2*e^3 - 136*(x*e + d)^(3/2)*A*b^4*d^2*
e^3 - 321*sqrt(x*e + d)*B*a*b^3*d^3*e^3 + 57*sqrt(x*e + d)*A*b^4*d^3*e^3 + 165*(
x*e + d)^(5/2)*B*a^2*b^2*e^4 - 87*(x*e + d)^(5/2)*A*a*b^3*e^4 - 704*(x*e + d)^(3
/2)*B*a^2*b^2*d*e^4 + 272*(x*e + d)^(3/2)*A*a*b^3*d*e^4 + 567*sqrt(x*e + d)*B*a^
2*b^2*d^2*e^4 - 171*sqrt(x*e + d)*A*a*b^3*d^2*e^4 + 280*(x*e + d)^(3/2)*B*a^3*b*
e^5 - 136*(x*e + d)^(3/2)*A*a^2*b^2*e^5 - 435*sqrt(x*e + d)*B*a^3*b*d*e^5 + 171*
sqrt(x*e + d)*A*a^2*b^2*d*e^5 + 123*sqrt(x*e + d)*B*a^4*e^6 - 57*sqrt(x*e + d)*A
*a^3*b*e^6)/(((x*e + d)*b - b*d + a*e)^3*b^5) + 2/3*((x*e + d)^(3/2)*B*b^8*e^2 +
9*sqrt(x*e + d)*B*b^8*d*e^2 - 12*sqrt(x*e + d)*B*a*b^7*e^3 + 3*sqrt(x*e + d)*A*
b^8*e^3)/b^12