3.2184 \(\int \sqrt{a+b x} (A+B x) (d+e x)^{3/2} \, dx\)

Optimal. Leaf size=250 \[ \frac{(b d-a e)^3 (5 a B e-8 A b e+3 b B d) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{64 b^{7/2} e^{5/2}}-\frac{\sqrt{a+b x} \sqrt{d+e x} (b d-a e)^2 (5 a B e-8 A b e+3 b B d)}{64 b^3 e^2}-\frac{(a+b x)^{3/2} \sqrt{d+e x} (b d-a e) (5 a B e-8 A b e+3 b B d)}{32 b^3 e}-\frac{(a+b x)^{3/2} (d+e x)^{3/2} (5 a B e-8 A b e+3 b B d)}{24 b^2 e}+\frac{B (a+b x)^{3/2} (d+e x)^{5/2}}{4 b e} \]

[Out]

-((b*d - a*e)^2*(3*b*B*d - 8*A*b*e + 5*a*B*e)*Sqrt[a + b*x]*Sqrt[d + e*x])/(64*b
^3*e^2) - ((b*d - a*e)*(3*b*B*d - 8*A*b*e + 5*a*B*e)*(a + b*x)^(3/2)*Sqrt[d + e*
x])/(32*b^3*e) - ((3*b*B*d - 8*A*b*e + 5*a*B*e)*(a + b*x)^(3/2)*(d + e*x)^(3/2))
/(24*b^2*e) + (B*(a + b*x)^(3/2)*(d + e*x)^(5/2))/(4*b*e) + ((b*d - a*e)^3*(3*b*
B*d - 8*A*b*e + 5*a*B*e)*ArcTanh[(Sqrt[e]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[d + e*x])
])/(64*b^(7/2)*e^(5/2))

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Rubi [A]  time = 0.512281, antiderivative size = 250, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{(b d-a e)^3 (5 a B e-8 A b e+3 b B d) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{64 b^{7/2} e^{5/2}}-\frac{\sqrt{a+b x} \sqrt{d+e x} (b d-a e)^2 (5 a B e-8 A b e+3 b B d)}{64 b^3 e^2}-\frac{(a+b x)^{3/2} \sqrt{d+e x} (b d-a e) (5 a B e-8 A b e+3 b B d)}{32 b^3 e}-\frac{(a+b x)^{3/2} (d+e x)^{3/2} (5 a B e-8 A b e+3 b B d)}{24 b^2 e}+\frac{B (a+b x)^{3/2} (d+e x)^{5/2}}{4 b e} \]

Antiderivative was successfully verified.

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

[Out]

-((b*d - a*e)^2*(3*b*B*d - 8*A*b*e + 5*a*B*e)*Sqrt[a + b*x]*Sqrt[d + e*x])/(64*b
^3*e^2) - ((b*d - a*e)*(3*b*B*d - 8*A*b*e + 5*a*B*e)*(a + b*x)^(3/2)*Sqrt[d + e*
x])/(32*b^3*e) - ((3*b*B*d - 8*A*b*e + 5*a*B*e)*(a + b*x)^(3/2)*(d + e*x)^(3/2))
/(24*b^2*e) + (B*(a + b*x)^(3/2)*(d + e*x)^(5/2))/(4*b*e) + ((b*d - a*e)^3*(3*b*
B*d - 8*A*b*e + 5*a*B*e)*ArcTanh[(Sqrt[e]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[d + e*x])
])/(64*b^(7/2)*e^(5/2))

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Rubi in Sympy [A]  time = 42.3028, size = 243, normalized size = 0.97 \[ \frac{B \left (a + b x\right )^{\frac{3}{2}} \left (d + e x\right )^{\frac{5}{2}}}{4 b e} + \frac{\sqrt{a + b x} \left (d + e x\right )^{\frac{5}{2}} \left (8 A b e - 5 B a e - 3 B b d\right )}{24 b e^{2}} + \frac{\sqrt{a + b x} \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right ) \left (8 A b e - 5 B a e - 3 B b d\right )}{96 b^{2} e^{2}} - \frac{\sqrt{a + b x} \sqrt{d + e x} \left (a e - b d\right )^{2} \left (8 A b e - 5 B a e - 3 B b d\right )}{64 b^{3} e^{2}} + \frac{\left (a e - b d\right )^{3} \left (8 A b e - 5 B a e - 3 B b d\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{d + e x}}{\sqrt{e} \sqrt{a + b x}} \right )}}{64 b^{\frac{7}{2}} e^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(e*x+d)**(3/2)*(b*x+a)**(1/2),x)

[Out]

B*(a + b*x)**(3/2)*(d + e*x)**(5/2)/(4*b*e) + sqrt(a + b*x)*(d + e*x)**(5/2)*(8*
A*b*e - 5*B*a*e - 3*B*b*d)/(24*b*e**2) + sqrt(a + b*x)*(d + e*x)**(3/2)*(a*e - b
*d)*(8*A*b*e - 5*B*a*e - 3*B*b*d)/(96*b**2*e**2) - sqrt(a + b*x)*sqrt(d + e*x)*(
a*e - b*d)**2*(8*A*b*e - 5*B*a*e - 3*B*b*d)/(64*b**3*e**2) + (a*e - b*d)**3*(8*A
*b*e - 5*B*a*e - 3*B*b*d)*atanh(sqrt(b)*sqrt(d + e*x)/(sqrt(e)*sqrt(a + b*x)))/(
64*b**(7/2)*e**(5/2))

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Mathematica [A]  time = 0.361099, size = 248, normalized size = 0.99 \[ \frac{3 (b d-a e)^3 (5 a B e-8 A b e+3 b B d) \log \left (2 \sqrt{b} \sqrt{e} \sqrt{a+b x} \sqrt{d+e x}+a e+b d+2 b e x\right )-2 \sqrt{b} \sqrt{e} \sqrt{a+b x} \sqrt{d+e x} \left (-15 a^3 B e^3+a^2 b e^2 (24 A e+31 B d+10 B e x)-a b^2 e \left (16 A e (4 d+e x)+B \left (9 d^2+20 d e x+8 e^2 x^2\right )\right )+b^3 \left (B \left (9 d^3-6 d^2 e x-72 d e^2 x^2-48 e^3 x^3\right )-8 A e \left (3 d^2+14 d e x+8 e^2 x^2\right )\right )\right )}{384 b^{7/2} e^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[b]*Sqrt[e]*Sqrt[a + b*x]*Sqrt[d + e*x]*(-15*a^3*B*e^3 + a^2*b*e^2*(31*B
*d + 24*A*e + 10*B*e*x) - a*b^2*e*(16*A*e*(4*d + e*x) + B*(9*d^2 + 20*d*e*x + 8*
e^2*x^2)) + b^3*(-8*A*e*(3*d^2 + 14*d*e*x + 8*e^2*x^2) + B*(9*d^3 - 6*d^2*e*x -
72*d*e^2*x^2 - 48*e^3*x^3))) + 3*(b*d - a*e)^3*(3*b*B*d - 8*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]])/(384*b^(7
/2)*e^(5/2))

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Maple [B]  time = 0.023, size = 1150, normalized size = 4.6 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(e*x+d)^(3/2)*(b*x+a)^(1/2),x)

[Out]

1/384*(e*x+d)^(1/2)*(b*x+a)^(1/2)*(48*d^2*A*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*b^3*
e*(b*e)^(1/2)+40*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*x*a*d*B*b^2*e^2*(b*e)^(1/2)-15*
e^4*B*ln(1/2*(2*b*x*e+2*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*
e)^(1/2))*a^4+9*b^4*ln(1/2*(2*b*x*e+2*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2
)+a*e+b*d)/(b*e)^(1/2))*d^4*B-18*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*d^3*B*b^3*(b*e)
^(1/2)-72*d*A*e^3*ln(1/2*(2*b*x*e+2*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)+
a*e+b*d)/(b*e)^(1/2))*a^2*b^2+72*d^2*A*ln(1/2*(2*b*x*e+2*(b*e*x^2+a*e*x+b*d*x+a*
d)^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a*b^3*e^2+36*e^3*ln(1/2*(2*b*x*e+2*(b
*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a^3*d*B*b-18*ln(
1/2*(2*b*x*e+2*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))
*a^2*d^2*B*b^2*e^2-12*ln(1/2*(2*b*x*e+2*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1
/2)+a*e+b*d)/(b*e)^(1/2))*a*d^3*B*b^3*e-48*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*a^2*A
*e^3*b*(b*e)^(1/2)+96*B*x^3*b^3*e^3*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)+
128*A*x^2*b^3*e^3*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)+30*e^3*B*(b*e*x^2+
a*e*x+b*d*x+a*d)^(1/2)*a^3*(b*e)^(1/2)-24*d^3*A*b^4*ln(1/2*(2*b*x*e+2*(b*e*x^2+a
*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*e+24*e^4*ln(1/2*(2*b*x*e
+2*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a^3*A*b-20*
e^3*B*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*x*a^2*b*(b*e)^(1/2)+12*(b*e*x^2+a*e*x+b*d*
x+a*d)^(1/2)*x*d^2*B*b^3*e*(b*e)^(1/2)-62*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*a^2*d*
B*b*e^2*(b*e)^(1/2)+18*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*a*d^2*B*b^2*e*(b*e)^(1/2)
+32*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*x*a*A*e^3*b^2*(b*e)^(1/2)+224*d*A*(b*e*x^2+a
*e*x+b*d*x+a*d)^(1/2)*x*b^3*e^2*(b*e)^(1/2)+16*B*x^2*a*b^2*e^3*(b*e*x^2+a*e*x+b*
d*x+a*d)^(1/2)*(b*e)^(1/2)+144*B*x^2*b^3*d*e^2*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(
b*e)^(1/2)+128*A*a*b^2*d*e^2*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2))/(b*e*x
^2+a*e*x+b*d*x+a*d)^(1/2)/b^3/e^2/(b*e)^(1/2)

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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)*sqrt(b*x + a)*(e*x + d)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.283945, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/768*(4*(48*B*b^3*e^3*x^3 - 9*B*b^3*d^3 + 3*(3*B*a*b^2 + 8*A*b^3)*d^2*e - (31*
B*a^2*b - 64*A*a*b^2)*d*e^2 + 3*(5*B*a^3 - 8*A*a^2*b)*e^3 + 8*(9*B*b^3*d*e^2 + (
B*a*b^2 + 8*A*b^3)*e^3)*x^2 + 2*(3*B*b^3*d^2*e + 2*(5*B*a*b^2 + 28*A*b^3)*d*e^2
- (5*B*a^2*b - 8*A*a*b^2)*e^3)*x)*sqrt(b*e)*sqrt(b*x + a)*sqrt(e*x + d) + 3*(3*B
*b^4*d^4 - 4*(B*a*b^3 + 2*A*b^4)*d^3*e - 6*(B*a^2*b^2 - 4*A*a*b^3)*d^2*e^2 + 12*
(B*a^3*b - 2*A*a^2*b^2)*d*e^3 - (5*B*a^4 - 8*A*a^3*b)*e^4)*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)))/(sqrt(b*e)*b^3*e^2), 1/38
4*(2*(48*B*b^3*e^3*x^3 - 9*B*b^3*d^3 + 3*(3*B*a*b^2 + 8*A*b^3)*d^2*e - (31*B*a^2
*b - 64*A*a*b^2)*d*e^2 + 3*(5*B*a^3 - 8*A*a^2*b)*e^3 + 8*(9*B*b^3*d*e^2 + (B*a*b
^2 + 8*A*b^3)*e^3)*x^2 + 2*(3*B*b^3*d^2*e + 2*(5*B*a*b^2 + 28*A*b^3)*d*e^2 - (5*
B*a^2*b - 8*A*a*b^2)*e^3)*x)*sqrt(-b*e)*sqrt(b*x + a)*sqrt(e*x + d) + 3*(3*B*b^4
*d^4 - 4*(B*a*b^3 + 2*A*b^4)*d^3*e - 6*(B*a^2*b^2 - 4*A*a*b^3)*d^2*e^2 + 12*(B*a
^3*b - 2*A*a^2*b^2)*d*e^3 - (5*B*a^4 - 8*A*a^3*b)*e^4)*arctan(1/2*(2*b*e*x + b*d
+ a*e)*sqrt(-b*e)/(sqrt(b*x + a)*sqrt(e*x + d)*b*e)))/(sqrt(-b*e)*b^3*e^2)]

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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)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.311694, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done