3.1734 \(\int \frac{(A+B x) (d+e x)^{5/2}}{(a+b x)^2} \, dx\)

Optimal. Leaf size=214 \[ -\frac{(b d-a e)^{3/2} (-7 a B e+5 A b e+2 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{9/2}}+\frac{\sqrt{d+e x} (b d-a e) (-7 a B e+5 A b e+2 b B d)}{b^4}+\frac{(d+e x)^{3/2} (-7 a B e+5 A b e+2 b B d)}{3 b^3}+\frac{(d+e x)^{5/2} (-7 a B e+5 A b e+2 b B d)}{5 b^2 (b d-a e)}-\frac{(d+e x)^{7/2} (A b-a B)}{b (a+b x) (b d-a e)} \]

[Out]

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

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Rubi [A]  time = 0.480962, antiderivative size = 214, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ -\frac{(b d-a e)^{3/2} (-7 a B e+5 A b e+2 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{9/2}}+\frac{\sqrt{d+e x} (b d-a e) (-7 a B e+5 A b e+2 b B d)}{b^4}+\frac{(d+e x)^{3/2} (-7 a B e+5 A b e+2 b B d)}{3 b^3}+\frac{(d+e x)^{5/2} (-7 a B e+5 A b e+2 b B d)}{5 b^2 (b d-a e)}-\frac{(d+e x)^{7/2} (A b-a B)}{b (a+b x) (b d-a e)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 49.0773, size = 202, normalized size = 0.94 \[ \frac{\left (d + e x\right )^{\frac{7}{2}} \left (A b - B a\right )}{b \left (a + b x\right ) \left (a e - b d\right )} - \frac{\left (d + e x\right )^{\frac{5}{2}} \left (5 A b e - 7 B a e + 2 B b d\right )}{5 b^{2} \left (a e - b d\right )} + \frac{\left (d + e x\right )^{\frac{3}{2}} \left (5 A b e - 7 B a e + 2 B b d\right )}{3 b^{3}} - \frac{\sqrt{d + e x} \left (a e - b d\right ) \left (5 A b e - 7 B a e + 2 B b d\right )}{b^{4}} + \frac{\left (a e - b d\right )^{\frac{3}{2}} \left (5 A b e - 7 B a e + 2 B b d\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{d + e x}}{\sqrt{a e - b d}} \right )}}{b^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.652234, size = 179, normalized size = 0.84 \[ \frac{\sqrt{d+e x} \left (90 a^2 B e^2+2 b e x (-10 a B e+5 A b e+11 b B d)-\frac{15 (A b-a B) (b d-a e)^2}{a+b x}-20 a b e (3 A e+7 B d)+2 b^2 d (35 A e+23 B d)+6 b^2 B e^2 x^2\right )}{15 b^4}-\frac{(b d-a e)^{3/2} (-7 a B e+5 A b e+2 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{9/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[d + e*x]*(90*a^2*B*e^2 - 20*a*b*e*(7*B*d + 3*A*e) + 2*b^2*d*(23*B*d + 35*A
*e) + 2*b*e*(11*b*B*d + 5*A*b*e - 10*a*B*e)*x + 6*b^2*B*e^2*x^2 - (15*(A*b - a*B
)*(b*d - a*e)^2)/(a + b*x)))/(15*b^4) - ((b*d - a*e)^(3/2)*(2*b*B*d + 5*A*b*e -
7*a*B*e)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/b^(9/2)

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Maple [B]  time = 0.025, size = 626, normalized size = 2.9 \[{\frac{2\,B}{5\,{b}^{2}} \left ( ex+d \right ) ^{{\frac{5}{2}}}}+{\frac{2\,Ae}{3\,{b}^{2}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}-{\frac{4\,Bae}{3\,{b}^{3}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}+{\frac{2\,Bd}{3\,{b}^{2}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}-4\,{\frac{aA{e}^{2}\sqrt{ex+d}}{{b}^{3}}}+4\,{\frac{Ade\sqrt{ex+d}}{{b}^{2}}}+6\,{\frac{B{a}^{2}{e}^{2}\sqrt{ex+d}}{{b}^{4}}}-8\,{\frac{Bade\sqrt{ex+d}}{{b}^{3}}}+2\,{\frac{B{d}^{2}\sqrt{ex+d}}{{b}^{2}}}-{\frac{A{a}^{2}{e}^{3}}{{b}^{3} \left ( bxe+ae \right ) }\sqrt{ex+d}}+2\,{\frac{\sqrt{ex+d}Aad{e}^{2}}{{b}^{2} \left ( bxe+ae \right ) }}-{\frac{A{d}^{2}e}{b \left ( bxe+ae \right ) }\sqrt{ex+d}}+{\frac{B{a}^{3}{e}^{3}}{{b}^{4} \left ( bxe+ae \right ) }\sqrt{ex+d}}-2\,{\frac{\sqrt{ex+d}B{a}^{2}d{e}^{2}}{{b}^{3} \left ( bxe+ae \right ) }}+{\frac{Ba{d}^{2}e}{{b}^{2} \left ( bxe+ae \right ) }\sqrt{ex+d}}+5\,{\frac{A{a}^{2}{e}^{3}}{{b}^{3}\sqrt{ \left ( ae-bd \right ) b}}\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{ \left ( ae-bd \right ) b}}} \right ) }-10\,{\frac{aAd{e}^{2}}{{b}^{2}\sqrt{ \left ( ae-bd \right ) b}}\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{ \left ( ae-bd \right ) b}}} \right ) }+5\,{\frac{A{d}^{2}e}{b\sqrt{ \left ( ae-bd \right ) b}}\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{ \left ( ae-bd \right ) b}}} \right ) }-7\,{\frac{B{a}^{3}{e}^{3}}{{b}^{4}\sqrt{ \left ( ae-bd \right ) b}}\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{ \left ( ae-bd \right ) b}}} \right ) }+16\,{\frac{B{a}^{2}d{e}^{2}}{{b}^{3}\sqrt{ \left ( ae-bd \right ) b}}\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{ \left ( ae-bd \right ) b}}} \right ) }-11\,{\frac{Ba{d}^{2}e}{{b}^{2}\sqrt{ \left ( ae-bd \right ) b}}\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{ \left ( ae-bd \right ) b}}} \right ) }+2\,{\frac{B{d}^{3}}{b\sqrt{ \left ( ae-bd \right ) b}}\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{ \left ( ae-bd \right ) b}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/5/b^2*B*(e*x+d)^(5/2)+2/3/b^2*A*(e*x+d)^(3/2)*e-4/3/b^3*B*(e*x+d)^(3/2)*a*e+2/
3/b^2*B*(e*x+d)^(3/2)*d-4/b^3*A*a*e^2*(e*x+d)^(1/2)+4/b^2*A*d*e*(e*x+d)^(1/2)+6/
b^4*B*a^2*e^2*(e*x+d)^(1/2)-8/b^3*B*a*d*e*(e*x+d)^(1/2)+2/b^2*B*d^2*(e*x+d)^(1/2
)-1/b^3*(e*x+d)^(1/2)/(b*e*x+a*e)*A*a^2*e^3+2/b^2*(e*x+d)^(1/2)/(b*e*x+a*e)*A*a*
d*e^2-1/b*(e*x+d)^(1/2)/(b*e*x+a*e)*A*d^2*e+1/b^4*(e*x+d)^(1/2)/(b*e*x+a*e)*B*a^
3*e^3-2/b^3*(e*x+d)^(1/2)/(b*e*x+a*e)*B*a^2*d*e^2+1/b^2*(e*x+d)^(1/2)/(b*e*x+a*e
)*B*a*d^2*e+5/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^2*e^3-10/b^2/((a*e-b*d)*b)^(1/2)*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2
))*A*a*d*e^2+5/b/((a*e-b*d)*b)^(1/2)*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))
*A*d^2*e-7/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^3*e^3+16/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^2*d*e^2-11/b^2/((a*e-b*d)*b)^(1/2)*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2
))*B*a*d^2*e+2/b/((a*e-b*d)*b)^(1/2)*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))
*B*d^3

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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)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.228565, size = 1, normalized size = 0. \[ \left [-\frac{15 \,{\left (2 \, B a b^{2} d^{2} -{\left (9 \, B a^{2} b - 5 \, A a b^{2}\right )} d e +{\left (7 \, B a^{3} - 5 \, A a^{2} b\right )} e^{2} +{\left (2 \, B b^{3} d^{2} -{\left (9 \, B a b^{2} - 5 \, A b^{3}\right )} d e +{\left (7 \, B a^{2} b - 5 \, A a b^{2}\right )} e^{2}\right )} x\right )} \sqrt{\frac{b d - a e}{b}} \log \left (\frac{b e x + 2 \, b d - a e + 2 \, \sqrt{e x + d} b \sqrt{\frac{b d - a e}{b}}}{b x + a}\right ) - 2 \,{\left (6 \, B b^{3} e^{2} x^{3} +{\left (61 \, B a b^{2} - 15 \, A b^{3}\right )} d^{2} - 10 \,{\left (17 \, B a^{2} b - 10 \, A a b^{2}\right )} d e + 15 \,{\left (7 \, B a^{3} - 5 \, A a^{2} b\right )} e^{2} + 2 \,{\left (11 \, B b^{3} d e -{\left (7 \, B a b^{2} - 5 \, A b^{3}\right )} e^{2}\right )} x^{2} + 2 \,{\left (23 \, B b^{3} d^{2} -{\left (59 \, B a b^{2} - 35 \, A b^{3}\right )} d e + 5 \,{\left (7 \, B a^{2} b - 5 \, A a b^{2}\right )} e^{2}\right )} x\right )} \sqrt{e x + d}}{30 \,{\left (b^{5} x + a b^{4}\right )}}, -\frac{15 \,{\left (2 \, B a b^{2} d^{2} -{\left (9 \, B a^{2} b - 5 \, A a b^{2}\right )} d e +{\left (7 \, B a^{3} - 5 \, A a^{2} b\right )} e^{2} +{\left (2 \, B b^{3} d^{2} -{\left (9 \, B a b^{2} - 5 \, A b^{3}\right )} d e +{\left (7 \, B a^{2} b - 5 \, A a b^{2}\right )} e^{2}\right )} x\right )} \sqrt{-\frac{b d - a e}{b}} \arctan \left (\frac{\sqrt{e x + d}}{\sqrt{-\frac{b d - a e}{b}}}\right ) -{\left (6 \, B b^{3} e^{2} x^{3} +{\left (61 \, B a b^{2} - 15 \, A b^{3}\right )} d^{2} - 10 \,{\left (17 \, B a^{2} b - 10 \, A a b^{2}\right )} d e + 15 \,{\left (7 \, B a^{3} - 5 \, A a^{2} b\right )} e^{2} + 2 \,{\left (11 \, B b^{3} d e -{\left (7 \, B a b^{2} - 5 \, A b^{3}\right )} e^{2}\right )} x^{2} + 2 \,{\left (23 \, B b^{3} d^{2} -{\left (59 \, B a b^{2} - 35 \, A b^{3}\right )} d e + 5 \,{\left (7 \, B a^{2} b - 5 \, A a b^{2}\right )} e^{2}\right )} x\right )} \sqrt{e x + d}}{15 \,{\left (b^{5} x + a b^{4}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/30*(15*(2*B*a*b^2*d^2 - (9*B*a^2*b - 5*A*a*b^2)*d*e + (7*B*a^3 - 5*A*a^2*b)*
e^2 + (2*B*b^3*d^2 - (9*B*a*b^2 - 5*A*b^3)*d*e + (7*B*a^2*b - 5*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*(6*B*b^3*e^2*x^3 + (61*B*a*b^2 - 15*A*b^3)*d^2 - 10*(17*B*
a^2*b - 10*A*a*b^2)*d*e + 15*(7*B*a^3 - 5*A*a^2*b)*e^2 + 2*(11*B*b^3*d*e - (7*B*
a*b^2 - 5*A*b^3)*e^2)*x^2 + 2*(23*B*b^3*d^2 - (59*B*a*b^2 - 35*A*b^3)*d*e + 5*(7
*B*a^2*b - 5*A*a*b^2)*e^2)*x)*sqrt(e*x + d))/(b^5*x + a*b^4), -1/15*(15*(2*B*a*b
^2*d^2 - (9*B*a^2*b - 5*A*a*b^2)*d*e + (7*B*a^3 - 5*A*a^2*b)*e^2 + (2*B*b^3*d^2
- (9*B*a*b^2 - 5*A*b^3)*d*e + (7*B*a^2*b - 5*A*a*b^2)*e^2)*x)*sqrt(-(b*d - a*e)/
b)*arctan(sqrt(e*x + d)/sqrt(-(b*d - a*e)/b)) - (6*B*b^3*e^2*x^3 + (61*B*a*b^2 -
15*A*b^3)*d^2 - 10*(17*B*a^2*b - 10*A*a*b^2)*d*e + 15*(7*B*a^3 - 5*A*a^2*b)*e^2
+ 2*(11*B*b^3*d*e - (7*B*a*b^2 - 5*A*b^3)*e^2)*x^2 + 2*(23*B*b^3*d^2 - (59*B*a*
b^2 - 35*A*b^3)*d*e + 5*(7*B*a^2*b - 5*A*a*b^2)*e^2)*x)*sqrt(e*x + d))/(b^5*x +
a*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)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.222899, size = 540, normalized size = 2.52 \[ \frac{{\left (2 \, B b^{3} d^{3} - 11 \, B a b^{2} d^{2} e + 5 \, A b^{3} d^{2} e + 16 \, B a^{2} b d e^{2} - 10 \, A a b^{2} d e^{2} - 7 \, B a^{3} e^{3} + 5 \, A a^{2} b e^{3}\right )} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right )}{\sqrt{-b^{2} d + a b e} b^{4}} + \frac{\sqrt{x e + d} B a b^{2} d^{2} e - \sqrt{x e + d} A b^{3} d^{2} e - 2 \, \sqrt{x e + d} B a^{2} b d e^{2} + 2 \, \sqrt{x e + d} A a b^{2} d e^{2} + \sqrt{x e + d} B a^{3} e^{3} - \sqrt{x e + d} A a^{2} b e^{3}}{{\left ({\left (x e + d\right )} b - b d + a e\right )} b^{4}} + \frac{2 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} B b^{8} + 5 \,{\left (x e + d\right )}^{\frac{3}{2}} B b^{8} d + 15 \, \sqrt{x e + d} B b^{8} d^{2} - 10 \,{\left (x e + d\right )}^{\frac{3}{2}} B a b^{7} e + 5 \,{\left (x e + d\right )}^{\frac{3}{2}} A b^{8} e - 60 \, \sqrt{x e + d} B a b^{7} d e + 30 \, \sqrt{x e + d} A b^{8} d e + 45 \, \sqrt{x e + d} B a^{2} b^{6} e^{2} - 30 \, \sqrt{x e + d} A a b^{7} e^{2}\right )}}{15 \, b^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(2*B*b^3*d^3 - 11*B*a*b^2*d^2*e + 5*A*b^3*d^2*e + 16*B*a^2*b*d*e^2 - 10*A*a*b^2*
d*e^2 - 7*B*a^3*e^3 + 5*A*a^2*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) + (sqrt(x*e + d)*B*a*b^2*d^2*e - sqrt(x*e + d)*A*b^
3*d^2*e - 2*sqrt(x*e + d)*B*a^2*b*d*e^2 + 2*sqrt(x*e + d)*A*a*b^2*d*e^2 + sqrt(x
*e + d)*B*a^3*e^3 - sqrt(x*e + d)*A*a^2*b*e^3)/(((x*e + d)*b - b*d + a*e)*b^4) +
2/15*(3*(x*e + d)^(5/2)*B*b^8 + 5*(x*e + d)^(3/2)*B*b^8*d + 15*sqrt(x*e + d)*B*
b^8*d^2 - 10*(x*e + d)^(3/2)*B*a*b^7*e + 5*(x*e + d)^(3/2)*A*b^8*e - 60*sqrt(x*e
+ d)*B*a*b^7*d*e + 30*sqrt(x*e + d)*A*b^8*d*e + 45*sqrt(x*e + d)*B*a^2*b^6*e^2
- 30*sqrt(x*e + d)*A*a*b^7*e^2)/b^10