3.1742 \(\int \frac{(A+B x) (d+e x)^{5/2}}{(a+b x)^3} \, dx\)

Optimal. Leaf size=233 \[ -\frac{5 e \sqrt{b d-a e} (-7 a B e+3 A b e+4 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{4 b^{9/2}}+\frac{5 e \sqrt{d+e x} (-7 a B e+3 A b e+4 b B d)}{4 b^4}+\frac{5 e (d+e x)^{3/2} (-7 a B e+3 A b e+4 b B d)}{12 b^3 (b d-a e)}-\frac{(d+e x)^{5/2} (-7 a B e+3 A b e+4 b B d)}{4 b^2 (a+b x) (b d-a e)}-\frac{(d+e x)^{7/2} (A b-a B)}{2 b (a+b x)^2 (b d-a e)} \]

[Out]

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

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Rubi [A]  time = 0.449489, antiderivative size = 233, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ -\frac{5 e \sqrt{b d-a e} (-7 a B e+3 A b e+4 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{4 b^{9/2}}+\frac{5 e \sqrt{d+e x} (-7 a B e+3 A b e+4 b B d)}{4 b^4}+\frac{5 e (d+e x)^{3/2} (-7 a B e+3 A b e+4 b B d)}{12 b^3 (b d-a e)}-\frac{(d+e x)^{5/2} (-7 a B e+3 A b e+4 b B d)}{4 b^2 (a+b x) (b d-a e)}-\frac{(d+e x)^{7/2} (A b-a B)}{2 b (a+b x)^2 (b d-a e)} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(d + e*x)^(5/2))/(a + b*x)^3,x]

[Out]

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

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Rubi in Sympy [A]  time = 48.8593, size = 224, normalized size = 0.96 \[ \frac{\left (d + e x\right )^{\frac{7}{2}} \left (A b - B a\right )}{2 b \left (a + b x\right )^{2} \left (a e - b d\right )} + \frac{\left (d + e x\right )^{\frac{5}{2}} \left (3 A b e - 7 B a e + 4 B b d\right )}{4 b^{2} \left (a + b x\right ) \left (a e - b d\right )} - \frac{5 e \left (d + e x\right )^{\frac{3}{2}} \left (3 A b e - 7 B a e + 4 B b d\right )}{12 b^{3} \left (a e - b d\right )} + \frac{5 e \sqrt{d + e x} \left (3 A b e - 7 B a e + 4 B b d\right )}{4 b^{4}} - \frac{5 e \sqrt{a e - b d} \left (3 A b e - 7 B a e + 4 B b d\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{d + e x}}{\sqrt{a e - b d}} \right )}}{4 b^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(e*x+d)**(5/2)/(b*x+a)**3,x)

[Out]

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

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Mathematica [A]  time = 0.527278, size = 171, normalized size = 0.73 \[ \frac{\sqrt{d+e x} \left (-\frac{3 (b d-a e) (-13 a B e+9 A b e+4 b B d)}{a+b x}-\frac{6 (A b-a B) (b d-a e)^2}{(a+b x)^2}+8 e (-9 a B e+3 A b e+7 b B d)+8 b B e^2 x\right )}{12 b^4}-\frac{5 e \sqrt{b d-a e} (-7 a B e+3 A b e+4 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{4 b^{9/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(d + e*x)^(5/2))/(a + b*x)^3,x]

[Out]

(Sqrt[d + e*x]*(8*e*(7*b*B*d + 3*A*b*e - 9*a*B*e) + 8*b*B*e^2*x - (6*(A*b - a*B)
*(b*d - a*e)^2)/(a + b*x)^2 - (3*(b*d - a*e)*(4*b*B*d + 9*A*b*e - 13*a*B*e))/(a
+ b*x)))/(12*b^4) - (5*e*Sqrt[b*d - a*e]*(4*b*B*d + 3*A*b*e - 7*a*B*e)*ArcTanh[(
Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(4*b^(9/2))

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Maple [B]  time = 0.03, size = 626, normalized size = 2.7 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(e*x+d)^(5/2)/(b*x+a)^3,x)

[Out]

2/3*e/b^3*B*(e*x+d)^(3/2)+2/b^3*A*e^2*(e*x+d)^(1/2)-6/b^4*B*a*e^2*(e*x+d)^(1/2)+
4*e/b^3*B*d*(e*x+d)^(1/2)+9/4/b^2/(b*e*x+a*e)^2*(e*x+d)^(3/2)*A*a*e^3-9/4/b/(b*e
*x+a*e)^2*(e*x+d)^(3/2)*A*d*e^2-13/4/b^3/(b*e*x+a*e)^2*(e*x+d)^(3/2)*B*a^2*e^3+1
7/4/b^2/(b*e*x+a*e)^2*(e*x+d)^(3/2)*B*a*d*e^2-e/b/(b*e*x+a*e)^2*(e*x+d)^(3/2)*B*
d^2+7/4/b^3/(b*e*x+a*e)^2*(e*x+d)^(1/2)*A*a^2*e^4-7/2/b^2/(b*e*x+a*e)^2*(e*x+d)^
(1/2)*A*a*d*e^3+7/4/b/(b*e*x+a*e)^2*(e*x+d)^(1/2)*A*d^2*e^2-11/4/b^4/(b*e*x+a*e)
^2*(e*x+d)^(1/2)*B*a^3*e^4+13/2/b^3/(b*e*x+a*e)^2*(e*x+d)^(1/2)*B*a^2*d*e^3-19/4
/b^2/(b*e*x+a*e)^2*(e*x+d)^(1/2)*B*a*d^2*e^2+e/b/(b*e*x+a*e)^2*(e*x+d)^(1/2)*B*d
^3-15/4/b^3/((a*e-b*d)*b)^(1/2)*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*A*a*
e^3+15/4/b^2/((a*e-b*d)*b)^(1/2)*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*A*d
*e^2+35/4/b^4/((a*e-b*d)*b)^(1/2)*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*B*
a^2*e^3-55/4/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*d*e^2+5*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^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)^(5/2)/(b*x + a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.226386, size = 1, normalized size = 0. \[ \left [-\frac{15 \,{\left (4 \, B a^{2} b d e -{\left (7 \, B a^{3} - 3 \, A a^{2} b\right )} e^{2} +{\left (4 \, B b^{3} d e -{\left (7 \, B a b^{2} - 3 \, A b^{3}\right )} e^{2}\right )} x^{2} + 2 \,{\left (4 \, B a b^{2} d e -{\left (7 \, B a^{2} b - 3 \, A a b^{2}\right )} e^{2}\right )} x\right )} \sqrt{\frac{b d - a e}{b}} \log \left (\frac{b e x + 2 \, b d - a e + 2 \, \sqrt{e x + d} b \sqrt{\frac{b d - a e}{b}}}{b x + a}\right ) - 2 \,{\left (8 \, B b^{3} e^{2} x^{3} - 6 \,{\left (B a b^{2} + A b^{3}\right )} d^{2} + 5 \,{\left (19 \, B a^{2} b - 3 \, A a b^{2}\right )} d e - 15 \,{\left (7 \, B a^{3} - 3 \, A a^{2} b\right )} e^{2} + 8 \,{\left (7 \, B b^{3} d e -{\left (7 \, B a b^{2} - 3 \, A b^{3}\right )} e^{2}\right )} x^{2} -{\left (12 \, B b^{3} d^{2} -{\left (163 \, B a b^{2} - 27 \, A b^{3}\right )} d e + 25 \,{\left (7 \, B a^{2} b - 3 \, A a b^{2}\right )} e^{2}\right )} x\right )} \sqrt{e x + d}}{24 \,{\left (b^{6} x^{2} + 2 \, a b^{5} x + a^{2} b^{4}\right )}}, -\frac{15 \,{\left (4 \, B a^{2} b d e -{\left (7 \, B a^{3} - 3 \, A a^{2} b\right )} e^{2} +{\left (4 \, B b^{3} d e -{\left (7 \, B a b^{2} - 3 \, A b^{3}\right )} e^{2}\right )} x^{2} + 2 \,{\left (4 \, B a b^{2} d e -{\left (7 \, B a^{2} b - 3 \, A a b^{2}\right )} e^{2}\right )} x\right )} \sqrt{-\frac{b d - a e}{b}} \arctan \left (\frac{\sqrt{e x + d}}{\sqrt{-\frac{b d - a e}{b}}}\right ) -{\left (8 \, B b^{3} e^{2} x^{3} - 6 \,{\left (B a b^{2} + A b^{3}\right )} d^{2} + 5 \,{\left (19 \, B a^{2} b - 3 \, A a b^{2}\right )} d e - 15 \,{\left (7 \, B a^{3} - 3 \, A a^{2} b\right )} e^{2} + 8 \,{\left (7 \, B b^{3} d e -{\left (7 \, B a b^{2} - 3 \, A b^{3}\right )} e^{2}\right )} x^{2} -{\left (12 \, B b^{3} d^{2} -{\left (163 \, B a b^{2} - 27 \, A b^{3}\right )} d e + 25 \,{\left (7 \, B a^{2} b - 3 \, A a b^{2}\right )} e^{2}\right )} x\right )} \sqrt{e x + d}}{12 \,{\left (b^{6} x^{2} + 2 \, a b^{5} x + a^{2} b^{4}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(e*x + d)^(5/2)/(b*x + a)^3,x, algorithm="fricas")

[Out]

[-1/24*(15*(4*B*a^2*b*d*e - (7*B*a^3 - 3*A*a^2*b)*e^2 + (4*B*b^3*d*e - (7*B*a*b^
2 - 3*A*b^3)*e^2)*x^2 + 2*(4*B*a*b^2*d*e - (7*B*a^2*b - 3*A*a*b^2)*e^2)*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*(8*B*b^3*e^2*x^3 - 6*(B*a*b^2 + A*b^3)*d^2 + 5*(19*B*a^2*b - 3*A
*a*b^2)*d*e - 15*(7*B*a^3 - 3*A*a^2*b)*e^2 + 8*(7*B*b^3*d*e - (7*B*a*b^2 - 3*A*b
^3)*e^2)*x^2 - (12*B*b^3*d^2 - (163*B*a*b^2 - 27*A*b^3)*d*e + 25*(7*B*a^2*b - 3*
A*a*b^2)*e^2)*x)*sqrt(e*x + d))/(b^6*x^2 + 2*a*b^5*x + a^2*b^4), -1/12*(15*(4*B*
a^2*b*d*e - (7*B*a^3 - 3*A*a^2*b)*e^2 + (4*B*b^3*d*e - (7*B*a*b^2 - 3*A*b^3)*e^2
)*x^2 + 2*(4*B*a*b^2*d*e - (7*B*a^2*b - 3*A*a*b^2)*e^2)*x)*sqrt(-(b*d - a*e)/b)*
arctan(sqrt(e*x + d)/sqrt(-(b*d - a*e)/b)) - (8*B*b^3*e^2*x^3 - 6*(B*a*b^2 + A*b
^3)*d^2 + 5*(19*B*a^2*b - 3*A*a*b^2)*d*e - 15*(7*B*a^3 - 3*A*a^2*b)*e^2 + 8*(7*B
*b^3*d*e - (7*B*a*b^2 - 3*A*b^3)*e^2)*x^2 - (12*B*b^3*d^2 - (163*B*a*b^2 - 27*A*
b^3)*d*e + 25*(7*B*a^2*b - 3*A*a*b^2)*e^2)*x)*sqrt(e*x + d))/(b^6*x^2 + 2*a*b^5*
x + a^2*b^4)]

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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*x+a)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.227147, size = 540, normalized size = 2.32 \[ \frac{5 \,{\left (4 \, B b^{2} d^{2} e - 11 \, B a b d e^{2} + 3 \, A b^{2} d e^{2} + 7 \, B a^{2} e^{3} - 3 \, A a b e^{3}\right )} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right )}{4 \, \sqrt{-b^{2} d + a b e} b^{4}} - \frac{4 \,{\left (x e + d\right )}^{\frac{3}{2}} B b^{3} d^{2} e - 4 \, \sqrt{x e + d} B b^{3} d^{3} e - 17 \,{\left (x e + d\right )}^{\frac{3}{2}} B a b^{2} d e^{2} + 9 \,{\left (x e + d\right )}^{\frac{3}{2}} A b^{3} d e^{2} + 19 \, \sqrt{x e + d} B a b^{2} d^{2} e^{2} - 7 \, \sqrt{x e + d} A b^{3} d^{2} e^{2} + 13 \,{\left (x e + d\right )}^{\frac{3}{2}} B a^{2} b e^{3} - 9 \,{\left (x e + d\right )}^{\frac{3}{2}} A a b^{2} e^{3} - 26 \, \sqrt{x e + d} B a^{2} b d e^{3} + 14 \, \sqrt{x e + d} A a b^{2} d e^{3} + 11 \, \sqrt{x e + d} B a^{3} e^{4} - 7 \, \sqrt{x e + d} A a^{2} b e^{4}}{4 \,{\left ({\left (x e + d\right )} b - b d + a e\right )}^{2} b^{4}} + \frac{2 \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} B b^{6} e + 6 \, \sqrt{x e + d} B b^{6} d e - 9 \, \sqrt{x e + d} B a b^{5} e^{2} + 3 \, \sqrt{x e + d} A b^{6} e^{2}\right )}}{3 \, b^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(e*x + d)^(5/2)/(b*x + a)^3,x, algorithm="giac")

[Out]

5/4*(4*B*b^2*d^2*e - 11*B*a*b*d*e^2 + 3*A*b^2*d*e^2 + 7*B*a^2*e^3 - 3*A*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) - 1/4*(
4*(x*e + d)^(3/2)*B*b^3*d^2*e - 4*sqrt(x*e + d)*B*b^3*d^3*e - 17*(x*e + d)^(3/2)
*B*a*b^2*d*e^2 + 9*(x*e + d)^(3/2)*A*b^3*d*e^2 + 19*sqrt(x*e + d)*B*a*b^2*d^2*e^
2 - 7*sqrt(x*e + d)*A*b^3*d^2*e^2 + 13*(x*e + d)^(3/2)*B*a^2*b*e^3 - 9*(x*e + d)
^(3/2)*A*a*b^2*e^3 - 26*sqrt(x*e + d)*B*a^2*b*d*e^3 + 14*sqrt(x*e + d)*A*a*b^2*d
*e^3 + 11*sqrt(x*e + d)*B*a^3*e^4 - 7*sqrt(x*e + d)*A*a^2*b*e^4)/(((x*e + d)*b -
b*d + a*e)^2*b^4) + 2/3*((x*e + d)^(3/2)*B*b^6*e + 6*sqrt(x*e + d)*B*b^6*d*e -
9*sqrt(x*e + d)*B*a*b^5*e^2 + 3*sqrt(x*e + d)*A*b^6*e^2)/b^9