3.2229 \(\int \frac{(A+B x) (d+e x)^{3/2}}{(a+b x)^{3/2}} \, dx\)

Optimal. Leaf size=199 \[ \frac{3 (b d-a e) (-5 a B e+4 A b e+b B d) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{4 b^{7/2} \sqrt{e}}+\frac{3 \sqrt{a+b x} \sqrt{d+e x} (-5 a B e+4 A b e+b B d)}{4 b^3}+\frac{\sqrt{a+b x} (d+e x)^{3/2} (-5 a B e+4 A b e+b B d)}{2 b^2 (b d-a e)}-\frac{2 (d+e x)^{5/2} (A b-a B)}{b \sqrt{a+b x} (b d-a e)} \]

[Out]

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

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Rubi [A]  time = 0.394206, antiderivative size = 199, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{3 (b d-a e) (-5 a B e+4 A b e+b B d) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{4 b^{7/2} \sqrt{e}}+\frac{3 \sqrt{a+b x} \sqrt{d+e x} (-5 a B e+4 A b e+b B d)}{4 b^3}+\frac{\sqrt{a+b x} (d+e x)^{3/2} (-5 a B e+4 A b e+b B d)}{2 b^2 (b d-a e)}-\frac{2 (d+e x)^{5/2} (A b-a B)}{b \sqrt{a+b x} (b d-a e)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 37.1938, size = 190, normalized size = 0.95 \[ \frac{2 \left (d + e x\right )^{\frac{5}{2}} \left (A b - B a\right )}{b \sqrt{a + b x} \left (a e - b d\right )} - \frac{\sqrt{a + b x} \left (d + e x\right )^{\frac{3}{2}} \left (4 A b e - 5 B a e + B b d\right )}{2 b^{2} \left (a e - b d\right )} + \frac{3 \sqrt{a + b x} \sqrt{d + e x} \left (4 A b e - 5 B a e + B b d\right )}{4 b^{3}} - \frac{3 \left (a e - b d\right ) \left (4 A b e - 5 B a e + B b d\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{d + e x}}{\sqrt{e} \sqrt{a + b x}} \right )}}{4 b^{\frac{7}{2}} \sqrt{e}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.256384, size = 157, normalized size = 0.79 \[ \frac{\sqrt{d+e x} \left (B \left (-15 a^2 e+a b (13 d-5 e x)+b^2 x (5 d+2 e x)\right )+4 A b (3 a e-2 b d+b e x)\right )}{4 b^3 \sqrt{a+b x}}+\frac{3 (b d-a e) (-5 a B e+4 A b e+b B d) \log \left (2 \sqrt{b} \sqrt{e} \sqrt{a+b x} \sqrt{d+e x}+a e+b d+2 b e x\right )}{8 b^{7/2} \sqrt{e}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[d + e*x]*(4*A*b*(-2*b*d + 3*a*e + b*e*x) + B*(-15*a^2*e + a*b*(13*d - 5*e*
x) + b^2*x*(5*d + 2*e*x))))/(4*b^3*Sqrt[a + b*x]) + (3*(b*d - a*e)*(b*B*d + 4*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]])/(8*b^(7/2)*Sqrt[e])

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Maple [B]  time = 0.033, size = 740, normalized size = 3.7 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/8*(e*x+d)^(1/2)*(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^2-12*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*b^3*d*e-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*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-3*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*b^3*d^2-
4*B*x^2*b^2*e*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+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))*a^2*b*e^2-12*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))*a*b^2*d*e-8*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+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))*a^2*b*d*e-3*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*b^2*d^2+10*B*x*a*b
*e*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-10*B*x*b^2*d*((b*x+a)*(e*x+d))^(1/2)*(b*e
)^(1/2)-24*A*a*b*e*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+16*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)-26*B*a*b*d*
((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2))/((b*x+a)*(e*x+d))^(1/2)/(b*e)^(1/2)/(b*x+a)
^(1/2)/b^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)^(3/2)/(b*x + a)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.733484, size = 1, normalized size = 0.01 \[ \left [\frac{4 \,{\left (2 \, B b^{2} e x^{2} +{\left (13 \, B a b - 8 \, A b^{2}\right )} d - 3 \,{\left (5 \, B a^{2} - 4 \, A a b\right )} e +{\left (5 \, B b^{2} d -{\left (5 \, B a b - 4 \, A b^{2}\right )} e\right )} x\right )} \sqrt{b e} \sqrt{b x + a} \sqrt{e x + d} + 3 \,{\left (B a b^{2} d^{2} - 2 \,{\left (3 \, B a^{2} b - 2 \, A a b^{2}\right )} d e +{\left (5 \, B a^{3} - 4 \, A a^{2} b\right )} e^{2} +{\left (B b^{3} d^{2} - 2 \,{\left (3 \, B a b^{2} - 2 \, A b^{3}\right )} d e +{\left (5 \, B a^{2} b - 4 \, A a b^{2}\right )} e^{2}\right )} x\right )} \log \left (4 \,{\left (2 \, b^{2} e^{2} x + b^{2} d e + a b e^{2}\right )} \sqrt{b x + a} \sqrt{e x + d} +{\left (8 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + 6 \, a b d e + a^{2} e^{2} + 8 \,{\left (b^{2} d e + a b e^{2}\right )} x\right )} \sqrt{b e}\right )}{16 \,{\left (b^{4} x + a b^{3}\right )} \sqrt{b e}}, \frac{2 \,{\left (2 \, B b^{2} e x^{2} +{\left (13 \, B a b - 8 \, A b^{2}\right )} d - 3 \,{\left (5 \, B a^{2} - 4 \, A a b\right )} e +{\left (5 \, B b^{2} d -{\left (5 \, B a b - 4 \, A b^{2}\right )} e\right )} x\right )} \sqrt{-b e} \sqrt{b x + a} \sqrt{e x + d} + 3 \,{\left (B a b^{2} d^{2} - 2 \,{\left (3 \, B a^{2} b - 2 \, A a b^{2}\right )} d e +{\left (5 \, B a^{3} - 4 \, A a^{2} b\right )} e^{2} +{\left (B b^{3} d^{2} - 2 \,{\left (3 \, B a b^{2} - 2 \, A b^{3}\right )} d e +{\left (5 \, B a^{2} b - 4 \, A a b^{2}\right )} e^{2}\right )} x\right )} \arctan \left (\frac{{\left (2 \, b e x + b d + a e\right )} \sqrt{-b e}}{2 \, \sqrt{b x + a} \sqrt{e x + d} b e}\right )}{8 \,{\left (b^{4} x + a b^{3}\right )} \sqrt{-b e}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/16*(4*(2*B*b^2*e*x^2 + (13*B*a*b - 8*A*b^2)*d - 3*(5*B*a^2 - 4*A*a*b)*e + (5*
B*b^2*d - (5*B*a*b - 4*A*b^2)*e)*x)*sqrt(b*e)*sqrt(b*x + a)*sqrt(e*x + d) + 3*(B
*a*b^2*d^2 - 2*(3*B*a^2*b - 2*A*a*b^2)*d*e + (5*B*a^3 - 4*A*a^2*b)*e^2 + (B*b^3*
d^2 - 2*(3*B*a*b^2 - 2*A*b^3)*d*e + (5*B*a^2*b - 4*A*a*b^2)*e^2)*x)*log(4*(2*b^2
*e^2*x + b^2*d*e + a*b*e^2)*sqrt(b*x + a)*sqrt(e*x + d) + (8*b^2*e^2*x^2 + b^2*d
^2 + 6*a*b*d*e + a^2*e^2 + 8*(b^2*d*e + a*b*e^2)*x)*sqrt(b*e)))/((b^4*x + a*b^3)
*sqrt(b*e)), 1/8*(2*(2*B*b^2*e*x^2 + (13*B*a*b - 8*A*b^2)*d - 3*(5*B*a^2 - 4*A*a
*b)*e + (5*B*b^2*d - (5*B*a*b - 4*A*b^2)*e)*x)*sqrt(-b*e)*sqrt(b*x + a)*sqrt(e*x
+ d) + 3*(B*a*b^2*d^2 - 2*(3*B*a^2*b - 2*A*a*b^2)*d*e + (5*B*a^3 - 4*A*a^2*b)*e
^2 + (B*b^3*d^2 - 2*(3*B*a*b^2 - 2*A*b^3)*d*e + (5*B*a^2*b - 4*A*a*b^2)*e^2)*x)*
arctan(1/2*(2*b*e*x + b*d + a*e)*sqrt(-b*e)/(sqrt(b*x + a)*sqrt(e*x + d)*b*e)))/
((b^4*x + a*b^3)*sqrt(-b*e))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (A + B x\right ) \left (d + e x\right )^{\frac{3}{2}}}{\left (a + b x\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((A + B*x)*(d + e*x)**(3/2)/(a + b*x)**(3/2), x)

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GIAC/XCAS [A]  time = 0.587874, 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)^(3/2),x, algorithm="giac")

[Out]

sage0*x