3.2196 \(\int (a+b x)^{3/2} (A+B x) \sqrt{d+e x} \, dx\)

Optimal. Leaf size=250 \[ -\frac{(b d-a e)^3 (3 a B e-8 A b e+5 b B d) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{64 b^{5/2} e^{7/2}}+\frac{\sqrt{a+b x} \sqrt{d+e x} (b d-a e)^2 (3 a B e-8 A b e+5 b B d)}{64 b^2 e^3}-\frac{(a+b x)^{3/2} \sqrt{d+e x} (b d-a e) (3 a B e-8 A b e+5 b B d)}{96 b^2 e^2}-\frac{(a+b x)^{5/2} \sqrt{d+e x} (3 a B e-8 A b e+5 b B d)}{24 b^2 e}+\frac{B (a+b x)^{5/2} (d+e x)^{3/2}}{4 b e} \]

[Out]

((b*d - a*e)^2*(5*b*B*d - 8*A*b*e + 3*a*B*e)*Sqrt[a + b*x]*Sqrt[d + e*x])/(64*b^
2*e^3) - ((b*d - a*e)*(5*b*B*d - 8*A*b*e + 3*a*B*e)*(a + b*x)^(3/2)*Sqrt[d + e*x
])/(96*b^2*e^2) - ((5*b*B*d - 8*A*b*e + 3*a*B*e)*(a + b*x)^(5/2)*Sqrt[d + e*x])/
(24*b^2*e) + (B*(a + b*x)^(5/2)*(d + e*x)^(3/2))/(4*b*e) - ((b*d - a*e)^3*(5*b*B
*d - 8*A*b*e + 3*a*B*e)*ArcTanh[(Sqrt[e]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[d + e*x])]
)/(64*b^(5/2)*e^(7/2))

_______________________________________________________________________________________

Rubi [A]  time = 0.500813, 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 \[ -\frac{(b d-a e)^3 (3 a B e-8 A b e+5 b B d) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{64 b^{5/2} e^{7/2}}+\frac{\sqrt{a+b x} \sqrt{d+e x} (b d-a e)^2 (3 a B e-8 A b e+5 b B d)}{64 b^2 e^3}-\frac{(a+b x)^{3/2} \sqrt{d+e x} (b d-a e) (3 a B e-8 A b e+5 b B d)}{96 b^2 e^2}-\frac{(a+b x)^{5/2} \sqrt{d+e x} (3 a B e-8 A b e+5 b B d)}{24 b^2 e}+\frac{B (a+b x)^{5/2} (d+e x)^{3/2}}{4 b e} \]

Antiderivative was successfully verified.

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

[Out]

((b*d - a*e)^2*(5*b*B*d - 8*A*b*e + 3*a*B*e)*Sqrt[a + b*x]*Sqrt[d + e*x])/(64*b^
2*e^3) - ((b*d - a*e)*(5*b*B*d - 8*A*b*e + 3*a*B*e)*(a + b*x)^(3/2)*Sqrt[d + e*x
])/(96*b^2*e^2) - ((5*b*B*d - 8*A*b*e + 3*a*B*e)*(a + b*x)^(5/2)*Sqrt[d + e*x])/
(24*b^2*e) + (B*(a + b*x)^(5/2)*(d + e*x)^(3/2))/(4*b*e) - ((b*d - a*e)^3*(5*b*B
*d - 8*A*b*e + 3*a*B*e)*ArcTanh[(Sqrt[e]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[d + e*x])]
)/(64*b^(5/2)*e^(7/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 42.0466, size = 241, normalized size = 0.96 \[ \frac{B \left (a + b x\right )^{\frac{5}{2}} \left (d + e x\right )^{\frac{3}{2}}}{4 b e} + \frac{\left (a + b x\right )^{\frac{3}{2}} \left (d + e x\right )^{\frac{3}{2}} \left (8 A b e - 3 B a e - 5 B b d\right )}{24 b e^{2}} + \frac{\sqrt{a + b x} \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right ) \left (8 A b e - 3 B a e - 5 B b d\right )}{32 b e^{3}} + \frac{\sqrt{a + b x} \sqrt{d + e x} \left (a e - b d\right )^{2} \left (8 A b e - 3 B a e - 5 B b d\right )}{64 b^{2} e^{3}} - \frac{\left (a e - b d\right )^{3} \left (8 A b e - 3 B a e - 5 B b d\right ) \operatorname{atanh}{\left (\frac{\sqrt{e} \sqrt{a + b x}}{\sqrt{b} \sqrt{d + e x}} \right )}}{64 b^{\frac{5}{2}} e^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

B*(a + b*x)**(5/2)*(d + e*x)**(3/2)/(4*b*e) + (a + b*x)**(3/2)*(d + e*x)**(3/2)*
(8*A*b*e - 3*B*a*e - 5*B*b*d)/(24*b*e**2) + sqrt(a + b*x)*(d + e*x)**(3/2)*(a*e
- b*d)*(8*A*b*e - 3*B*a*e - 5*B*b*d)/(32*b*e**3) + sqrt(a + b*x)*sqrt(d + e*x)*(
a*e - b*d)**2*(8*A*b*e - 3*B*a*e - 5*B*b*d)/(64*b**2*e**3) - (a*e - b*d)**3*(8*A
*b*e - 3*B*a*e - 5*B*b*d)*atanh(sqrt(e)*sqrt(a + b*x)/(sqrt(b)*sqrt(d + e*x)))/(
64*b**(5/2)*e**(7/2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.355982, size = 249, normalized size = 1. \[ \frac{2 \sqrt{b} \sqrt{e} \sqrt{a+b x} \sqrt{d+e x} \left (-9 a^3 B e^3+3 a^2 b e^2 (8 A e+3 B d+2 B e x)+a b^2 e \left (16 A e (4 d+7 e x)+B \left (-31 d^2+20 d e x+72 e^2 x^2\right )\right )+b^3 \left (8 A e \left (-3 d^2+2 d e x+8 e^2 x^2\right )+B \left (15 d^3-10 d^2 e x+8 d e^2 x^2+48 e^3 x^3\right )\right )\right )-3 (b d-a e)^3 (3 a B e-8 A b e+5 b B d) \log \left (2 \sqrt{b} \sqrt{e} \sqrt{a+b x} \sqrt{d+e x}+a e+b d+2 b e x\right )}{384 b^{5/2} e^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [B]  time = 0.021, 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)^(3/2)*(B*x+A)*(e*x+d)^(1/2),x)

[Out]

-1/384*(b*x+a)^(1/2)*(e*x+d)^(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)-9*
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+15*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-30*(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+12*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^2-36*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)+18*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-12
*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)+20*(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)-18*(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)+62*(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
)-224*(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)-32*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)-144*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)-16*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^2/(b*e)^(1/2)/e^3

_______________________________________________________________________________________

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

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.271801, 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)*(b*x + a)^(3/2)*sqrt(e*x + d),x, algorithm="fricas")

[Out]

[1/768*(4*(48*B*b^3*e^3*x^3 + 15*B*b^3*d^3 - (31*B*a*b^2 + 24*A*b^3)*d^2*e + (9*
B*a^2*b + 64*A*a*b^2)*d*e^2 - 3*(3*B*a^3 - 8*A*a^2*b)*e^3 + 8*(B*b^3*d*e^2 + (9*
B*a*b^2 + 8*A*b^3)*e^3)*x^2 - 2*(5*B*b^3*d^2*e - 2*(5*B*a*b^2 + 4*A*b^3)*d*e^2 -
(3*B*a^2*b + 56*A*a*b^2)*e^3)*x)*sqrt(b*e)*sqrt(b*x + a)*sqrt(e*x + d) + 3*(5*B
*b^4*d^4 - 4*(3*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 + 4
*(B*a^3*b - 6*A*a^2*b^2)*d*e^3 - (3*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^2*e^3), 1/
384*(2*(48*B*b^3*e^3*x^3 + 15*B*b^3*d^3 - (31*B*a*b^2 + 24*A*b^3)*d^2*e + (9*B*a
^2*b + 64*A*a*b^2)*d*e^2 - 3*(3*B*a^3 - 8*A*a^2*b)*e^3 + 8*(B*b^3*d*e^2 + (9*B*a
*b^2 + 8*A*b^3)*e^3)*x^2 - 2*(5*B*b^3*d^2*e - 2*(5*B*a*b^2 + 4*A*b^3)*d*e^2 - (3
*B*a^2*b + 56*A*a*b^2)*e^3)*x)*sqrt(-b*e)*sqrt(b*x + a)*sqrt(e*x + d) - 3*(5*B*b
^4*d^4 - 4*(3*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 + 4*(
B*a^3*b - 6*A*a^2*b^2)*d*e^3 - (3*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^2*e^3)]

_______________________________________________________________________________________

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)**(3/2)*(B*x+A)*(e*x+d)**(1/2),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.30049, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done