3.2238 \(\int \frac{(A+B x) (d+e x)^{3/2}}{(a+b x)^{5/2}} \, dx\)

Optimal. Leaf size=201 \[ \frac{\sqrt{e} (-5 a B e+2 A b e+3 b B d) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{b^{7/2}}+\frac{e \sqrt{a+b x} \sqrt{d+e x} (-5 a B e+2 A b e+3 b B d)}{b^3 (b d-a e)}-\frac{2 (d+e x)^{3/2} (-5 a B e+2 A b e+3 b B d)}{3 b^2 \sqrt{a+b x} (b d-a e)}-\frac{2 (d+e x)^{5/2} (A b-a B)}{3 b (a+b x)^{3/2} (b d-a e)} \]

[Out]

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

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Rubi [A]  time = 0.395576, antiderivative size = 201, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ \frac{\sqrt{e} (-5 a B e+2 A b e+3 b B d) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{b^{7/2}}+\frac{e \sqrt{a+b x} \sqrt{d+e x} (-5 a B e+2 A b e+3 b B d)}{b^3 (b d-a e)}-\frac{2 (d+e x)^{3/2} (-5 a B e+2 A b e+3 b B d)}{3 b^2 \sqrt{a+b x} (b d-a e)}-\frac{2 (d+e x)^{5/2} (A b-a B)}{3 b (a+b x)^{3/2} (b d-a e)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 37.7, size = 194, normalized size = 0.97 \[ \frac{2 \left (d + e x\right )^{\frac{5}{2}} \left (A b - B a\right )}{3 b \left (a + b x\right )^{\frac{3}{2}} \left (a e - b d\right )} + \frac{2 \left (d + e x\right )^{\frac{3}{2}} \left (2 A b e - 5 B a e + 3 B b d\right )}{3 b^{2} \sqrt{a + b x} \left (a e - b d\right )} - \frac{e \sqrt{a + b x} \sqrt{d + e x} \left (2 A b e - 5 B a e + 3 B b d\right )}{b^{3} \left (a e - b d\right )} + \frac{\sqrt{e} \left (2 A b e - 5 B a e + 3 B b d\right ) \operatorname{atanh}{\left (\frac{\sqrt{e} \sqrt{a + b x}}{\sqrt{b} \sqrt{d + e x}} \right )}}{b^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.296008, size = 149, normalized size = 0.74 \[ \frac{\sqrt{e} (-5 a B e+2 A b e+3 b B d) \log \left (2 \sqrt{b} \sqrt{e} \sqrt{a+b x} \sqrt{d+e x}+a e+b d+2 b e x\right )}{2 b^{7/2}}-\frac{\sqrt{d+e x} \left (B \left (-15 a^2 e+4 a b (d-5 e x)-3 b^2 x (e x-2 d)\right )+2 A b (3 a e+b (d+4 e x))\right )}{3 b^3 (a+b x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.033, size = 698, normalized size = 3.5 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*(e*x+d)^(1/2)*(6*A*ln(1/2*(2*b*x*e+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e
+b*d)/(b*e)^(1/2))*x^2*b^3*e^2-15*B*ln(1/2*(2*b*x*e+2*((b*x+a)*(e*x+d))^(1/2)*(b
*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*x^2*a*b^2*e^2+9*B*ln(1/2*(2*b*x*e+2*((b*x+a)*(e*
x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*x^2*b^3*d*e+12*A*ln(1/2*(2*b*x*e+2
*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*x*a*b^2*e^2-30*B*ln(1
/2*(2*b*x*e+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*x*a^2*b*
e^2+18*B*ln(1/2*(2*b*x*e+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1
/2))*x*a*b^2*d*e+6*B*x^2*b^2*e*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+6*A*ln(1/2*(2
*b*x*e+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a^2*b*e^2-16*
A*x*b^2*e*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-15*B*ln(1/2*(2*b*x*e+2*((b*x+a)*(e
*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a^3*e^2+9*B*ln(1/2*(2*b*x*e+2*((b
*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a^2*b*d*e+40*B*x*a*b*e*((
b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-12*B*x*b^2*d*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/
2)-12*A*a*b*e*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-4*A*b^2*d*((b*x+a)*(e*x+d))^(1
/2)*(b*e)^(1/2)+30*B*a^2*e*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-8*B*a*b*d*((b*x+a
)*(e*x+d))^(1/2)*(b*e)^(1/2))/(b*e)^(1/2)/((b*x+a)*(e*x+d))^(1/2)/b^3/(b*x+a)^(3
/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)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.833445, size = 1, normalized size = 0. \[ \left [\frac{3 \,{\left (3 \, B a^{2} b d +{\left (3 \, B b^{3} d -{\left (5 \, B a b^{2} - 2 \, A b^{3}\right )} e\right )} x^{2} -{\left (5 \, B a^{3} - 2 \, A a^{2} b\right )} e + 2 \,{\left (3 \, B a b^{2} d -{\left (5 \, B a^{2} b - 2 \, A a b^{2}\right )} e\right )} x\right )} \sqrt{\frac{e}{b}} \log \left (8 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + 6 \, a b d e + a^{2} e^{2} + 4 \,{\left (2 \, b^{2} e x + b^{2} d + a b e\right )} \sqrt{b x + a} \sqrt{e x + d} \sqrt{\frac{e}{b}} + 8 \,{\left (b^{2} d e + a b e^{2}\right )} x\right ) + 4 \,{\left (3 \, B b^{2} e x^{2} - 2 \,{\left (2 \, B a b + A b^{2}\right )} d + 3 \,{\left (5 \, B a^{2} - 2 \, A a b\right )} e - 2 \,{\left (3 \, B b^{2} d - 2 \,{\left (5 \, B a b - 2 \, A b^{2}\right )} e\right )} x\right )} \sqrt{b x + a} \sqrt{e x + d}}{12 \,{\left (b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{3}\right )}}, \frac{3 \,{\left (3 \, B a^{2} b d +{\left (3 \, B b^{3} d -{\left (5 \, B a b^{2} - 2 \, A b^{3}\right )} e\right )} x^{2} -{\left (5 \, B a^{3} - 2 \, A a^{2} b\right )} e + 2 \,{\left (3 \, B a b^{2} d -{\left (5 \, B a^{2} b - 2 \, A a b^{2}\right )} e\right )} x\right )} \sqrt{-\frac{e}{b}} \arctan \left (\frac{2 \, b e x + b d + a e}{2 \, \sqrt{b x + a} \sqrt{e x + d} b \sqrt{-\frac{e}{b}}}\right ) + 2 \,{\left (3 \, B b^{2} e x^{2} - 2 \,{\left (2 \, B a b + A b^{2}\right )} d + 3 \,{\left (5 \, B a^{2} - 2 \, A a b\right )} e - 2 \,{\left (3 \, B b^{2} d - 2 \,{\left (5 \, B a b - 2 \, A b^{2}\right )} e\right )} x\right )} \sqrt{b x + a} \sqrt{e x + d}}{6 \,{\left (b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{3}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/12*(3*(3*B*a^2*b*d + (3*B*b^3*d - (5*B*a*b^2 - 2*A*b^3)*e)*x^2 - (5*B*a^3 - 2
*A*a^2*b)*e + 2*(3*B*a*b^2*d - (5*B*a^2*b - 2*A*a*b^2)*e)*x)*sqrt(e/b)*log(8*b^2
*e^2*x^2 + b^2*d^2 + 6*a*b*d*e + a^2*e^2 + 4*(2*b^2*e*x + b^2*d + a*b*e)*sqrt(b*
x + a)*sqrt(e*x + d)*sqrt(e/b) + 8*(b^2*d*e + a*b*e^2)*x) + 4*(3*B*b^2*e*x^2 - 2
*(2*B*a*b + A*b^2)*d + 3*(5*B*a^2 - 2*A*a*b)*e - 2*(3*B*b^2*d - 2*(5*B*a*b - 2*A
*b^2)*e)*x)*sqrt(b*x + a)*sqrt(e*x + d))/(b^5*x^2 + 2*a*b^4*x + a^2*b^3), 1/6*(3
*(3*B*a^2*b*d + (3*B*b^3*d - (5*B*a*b^2 - 2*A*b^3)*e)*x^2 - (5*B*a^3 - 2*A*a^2*b
)*e + 2*(3*B*a*b^2*d - (5*B*a^2*b - 2*A*a*b^2)*e)*x)*sqrt(-e/b)*arctan(1/2*(2*b*
e*x + b*d + a*e)/(sqrt(b*x + a)*sqrt(e*x + d)*b*sqrt(-e/b))) + 2*(3*B*b^2*e*x^2
- 2*(2*B*a*b + A*b^2)*d + 3*(5*B*a^2 - 2*A*a*b)*e - 2*(3*B*b^2*d - 2*(5*B*a*b -
2*A*b^2)*e)*x)*sqrt(b*x + a)*sqrt(e*x + d))/(b^5*x^2 + 2*a*b^4*x + a^2*b^3)]

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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)**(3/2)/(b*x+a)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.633909, size = 4, normalized size = 0.02 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x