3.1727 \(\int \frac{(A+B x) (d+e x)^{3/2}}{a+b x} \, dx\)

Optimal. Leaf size=130 \[ -\frac{2 (A b-a B) (b d-a e)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{7/2}}+\frac{2 \sqrt{d+e x} (A b-a B) (b d-a e)}{b^3}+\frac{2 (d+e x)^{3/2} (A b-a B)}{3 b^2}+\frac{2 B (d+e x)^{5/2}}{5 b e} \]

[Out]

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

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Rubi [A]  time = 0.221491, antiderivative size = 130, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ -\frac{2 (A b-a B) (b d-a e)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{7/2}}+\frac{2 \sqrt{d+e x} (A b-a B) (b d-a e)}{b^3}+\frac{2 (d+e x)^{3/2} (A b-a B)}{3 b^2}+\frac{2 B (d+e x)^{5/2}}{5 b e} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 24.1211, size = 114, normalized size = 0.88 \[ \frac{2 B \left (d + e x\right )^{\frac{5}{2}}}{5 b e} + \frac{2 \left (d + e x\right )^{\frac{3}{2}} \left (A b - B a\right )}{3 b^{2}} - \frac{2 \sqrt{d + e x} \left (A b - B a\right ) \left (a e - b d\right )}{b^{3}} + \frac{2 \left (A b - B a\right ) \left (a e - b d\right )^{\frac{3}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{d + e x}}{\sqrt{a e - b d}} \right )}}{b^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.205054, size = 129, normalized size = 0.99 \[ \frac{2 \sqrt{d+e x} \left (15 a^2 B e^2-5 a b e (3 A e+4 B d+B e x)+b^2 \left (5 A e (4 d+e x)+3 B (d+e x)^2\right )\right )}{15 b^3 e}-\frac{2 (A b-a B) (b d-a e)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[d + e*x]*(15*a^2*B*e^2 - 5*a*b*e*(4*B*d + 3*A*e + B*e*x) + b^2*(3*B*(d +
e*x)^2 + 5*A*e*(4*d + e*x))))/(15*b^3*e) - (2*(A*b - a*B)*(b*d - a*e)^(3/2)*Arc
Tanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/b^(7/2)

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Maple [B]  time = 0.014, size = 370, normalized size = 2.9 \[{\frac{2\,B}{5\,be} \left ( ex+d \right ) ^{{\frac{5}{2}}}}+{\frac{2\,A}{3\,b} \left ( ex+d \right ) ^{{\frac{3}{2}}}}-{\frac{2\,Ba}{3\,{b}^{2}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}-2\,{\frac{Aae\sqrt{ex+d}}{{b}^{2}}}+2\,{\frac{Ad\sqrt{ex+d}}{b}}+2\,{\frac{eB{a}^{2}\sqrt{ex+d}}{{b}^{3}}}-2\,{\frac{Bad\sqrt{ex+d}}{{b}^{2}}}+2\,{\frac{{a}^{2}A{e}^{2}}{{b}^{2}\sqrt{ \left ( ae-bd \right ) b}}\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{ \left ( ae-bd \right ) b}}} \right ) }-4\,{\frac{aAde}{b\sqrt{ \left ( ae-bd \right ) b}}\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{ \left ( ae-bd \right ) b}}} \right ) }+2\,{\frac{A{d}^{2}}{\sqrt{ \left ( ae-bd \right ) b}}\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{ \left ( ae-bd \right ) b}}} \right ) }-2\,{\frac{B{a}^{3}{e}^{2}}{{b}^{3}\sqrt{ \left ( ae-bd \right ) b}}\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{ \left ( ae-bd \right ) b}}} \right ) }+4\,{\frac{eB{a}^{2}d}{{b}^{2}\sqrt{ \left ( ae-bd \right ) b}}\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{ \left ( ae-bd \right ) b}}} \right ) }-2\,{\frac{Ba{d}^{2}}{b\sqrt{ \left ( ae-bd \right ) b}}\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{ \left ( ae-bd \right ) b}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/5*B*(e*x+d)^(5/2)/b/e+2/3/b*A*(e*x+d)^(3/2)-2/3/b^2*B*(e*x+d)^(3/2)*a-2*e/b^2*
A*a*(e*x+d)^(1/2)+2/b*A*d*(e*x+d)^(1/2)+2*e/b^3*B*a^2*(e*x+d)^(1/2)-2/b^2*B*a*d*
(e*x+d)^(1/2)+2*e^2/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^2-4*e/b/((a*e-b*d)*b)^(1/2)*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/
2))*A*a*d+2/((a*e-b*d)*b)^(1/2)*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*A*d^
2-2*e^2/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^
3+4*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*a^2*
d-2/b/((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

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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*x + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.222078, size = 1, normalized size = 0.01 \[ \left [-\frac{15 \,{\left ({\left (B a b - A b^{2}\right )} d e -{\left (B a^{2} - A a b\right )} e^{2}\right )} \sqrt{\frac{b d - a e}{b}} \log \left (\frac{b e x + 2 \, b d - a e - 2 \, \sqrt{e x + d} b \sqrt{\frac{b d - a e}{b}}}{b x + a}\right ) - 2 \,{\left (3 \, B b^{2} e^{2} x^{2} + 3 \, B b^{2} d^{2} - 20 \,{\left (B a b - A b^{2}\right )} d e + 15 \,{\left (B a^{2} - A a b\right )} e^{2} +{\left (6 \, B b^{2} d e - 5 \,{\left (B a b - A b^{2}\right )} e^{2}\right )} x\right )} \sqrt{e x + d}}{15 \, b^{3} e}, \frac{2 \,{\left (15 \,{\left ({\left (B a b - A b^{2}\right )} d e -{\left (B a^{2} - A a b\right )} e^{2}\right )} \sqrt{-\frac{b d - a e}{b}} \arctan \left (\frac{\sqrt{e x + d}}{\sqrt{-\frac{b d - a e}{b}}}\right ) +{\left (3 \, B b^{2} e^{2} x^{2} + 3 \, B b^{2} d^{2} - 20 \,{\left (B a b - A b^{2}\right )} d e + 15 \,{\left (B a^{2} - A a b\right )} e^{2} +{\left (6 \, B b^{2} d e - 5 \,{\left (B a b - A b^{2}\right )} e^{2}\right )} x\right )} \sqrt{e x + d}\right )}}{15 \, b^{3} e}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/15*(15*((B*a*b - A*b^2)*d*e - (B*a^2 - A*a*b)*e^2)*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*(3*B
*b^2*e^2*x^2 + 3*B*b^2*d^2 - 20*(B*a*b - A*b^2)*d*e + 15*(B*a^2 - A*a*b)*e^2 + (
6*B*b^2*d*e - 5*(B*a*b - A*b^2)*e^2)*x)*sqrt(e*x + d))/(b^3*e), 2/15*(15*((B*a*b
- A*b^2)*d*e - (B*a^2 - A*a*b)*e^2)*sqrt(-(b*d - a*e)/b)*arctan(sqrt(e*x + d)/s
qrt(-(b*d - a*e)/b)) + (3*B*b^2*e^2*x^2 + 3*B*b^2*d^2 - 20*(B*a*b - A*b^2)*d*e +
15*(B*a^2 - A*a*b)*e^2 + (6*B*b^2*d*e - 5*(B*a*b - A*b^2)*e^2)*x)*sqrt(e*x + d)
)/(b^3*e)]

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Sympy [A]  time = 46.673, size = 258, normalized size = 1.98 \[ \frac{2 B \left (d + e x\right )^{\frac{5}{2}}}{5 b e} + \frac{\left (d + e x\right )^{\frac{3}{2}} \left (2 A b - 2 B a\right )}{3 b^{2}} + \frac{\sqrt{d + e x} \left (- 2 A a b e + 2 A b^{2} d + 2 B a^{2} e - 2 B a b d\right )}{b^{3}} - \frac{2 \left (- A b + B a\right ) \left (a e - b d\right )^{2} \left (\begin{cases} \frac{\operatorname{atan}{\left (\frac{\sqrt{d + e x}}{\sqrt{\frac{a e - b d}{b}}} \right )}}{b \sqrt{\frac{a e - b d}{b}}} & \text{for}\: \frac{a e - b d}{b} > 0 \\- \frac{\operatorname{acoth}{\left (\frac{\sqrt{d + e x}}{\sqrt{\frac{- a e + b d}{b}}} \right )}}{b \sqrt{\frac{- a e + b d}{b}}} & \text{for}\: d + e x > \frac{- a e + b d}{b} \wedge \frac{a e - b d}{b} < 0 \\- \frac{\operatorname{atanh}{\left (\frac{\sqrt{d + e x}}{\sqrt{\frac{- a e + b d}{b}}} \right )}}{b \sqrt{\frac{- a e + b d}{b}}} & \text{for}\: \frac{a e - b d}{b} < 0 \wedge d + e x < \frac{- a e + b d}{b} \end{cases}\right )}{b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*B*(d + e*x)**(5/2)/(5*b*e) + (d + e*x)**(3/2)*(2*A*b - 2*B*a)/(3*b**2) + sqrt(
d + e*x)*(-2*A*a*b*e + 2*A*b**2*d + 2*B*a**2*e - 2*B*a*b*d)/b**3 - 2*(-A*b + B*a
)*(a*e - b*d)**2*Piecewise((atan(sqrt(d + e*x)/sqrt((a*e - b*d)/b))/(b*sqrt((a*e
- b*d)/b)), (a*e - b*d)/b > 0), (-acoth(sqrt(d + e*x)/sqrt((-a*e + b*d)/b))/(b*
sqrt((-a*e + b*d)/b)), ((a*e - b*d)/b < 0) & (d + e*x > (-a*e + b*d)/b)), (-atan
h(sqrt(d + e*x)/sqrt((-a*e + b*d)/b))/(b*sqrt((-a*e + b*d)/b)), ((a*e - b*d)/b <
0) & (d + e*x < (-a*e + b*d)/b)))/b**3

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GIAC/XCAS [A]  time = 0.212768, size = 308, normalized size = 2.37 \[ -\frac{2 \,{\left (B a b^{2} d^{2} - A b^{3} d^{2} - 2 \, B a^{2} b d e + 2 \, A a b^{2} d e + B a^{3} e^{2} - A a^{2} b e^{2}\right )} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right )}{\sqrt{-b^{2} d + a b e} b^{3}} + \frac{2 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} B b^{4} e^{4} - 5 \,{\left (x e + d\right )}^{\frac{3}{2}} B a b^{3} e^{5} + 5 \,{\left (x e + d\right )}^{\frac{3}{2}} A b^{4} e^{5} - 15 \, \sqrt{x e + d} B a b^{3} d e^{5} + 15 \, \sqrt{x e + d} A b^{4} d e^{5} + 15 \, \sqrt{x e + d} B a^{2} b^{2} e^{6} - 15 \, \sqrt{x e + d} A a b^{3} e^{6}\right )} e^{\left (-5\right )}}{15 \, b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(B*a*b^2*d^2 - A*b^3*d^2 - 2*B*a^2*b*d*e + 2*A*a*b^2*d*e + B*a^3*e^2 - A*a^2*
b*e^2)*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))/(sqrt(-b^2*d + a*b*e)*b^3) +
2/15*(3*(x*e + d)^(5/2)*B*b^4*e^4 - 5*(x*e + d)^(3/2)*B*a*b^3*e^5 + 5*(x*e + d)
^(3/2)*A*b^4*e^5 - 15*sqrt(x*e + d)*B*a*b^3*d*e^5 + 15*sqrt(x*e + d)*A*b^4*d*e^5
+ 15*sqrt(x*e + d)*B*a^2*b^2*e^6 - 15*sqrt(x*e + d)*A*a*b^3*e^6)*e^(-5)/b^5