3.2195 \(\int (a+b x)^{3/2} (A+B x) (d+e x)^{3/2} \, dx\)

Optimal. Leaf size=304 \[ \frac{3 (b d-a e)^4 (2 A b e-B (a e+b d)) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{128 b^{7/2} e^{7/2}}-\frac{3 \sqrt{a+b x} \sqrt{d+e x} (b d-a e)^3 (2 A b e-B (a e+b d))}{128 b^3 e^3}+\frac{(a+b x)^{3/2} \sqrt{d+e x} (b d-a e)^2 (2 A b e-B (a e+b d))}{64 b^3 e^2}+\frac{(a+b x)^{5/2} \sqrt{d+e x} (b d-a e) (2 A b e-B (a e+b d))}{16 b^3 e}+\frac{(a+b x)^{5/2} (d+e x)^{3/2} (2 A b e-B (a e+b d))}{8 b^2 e}+\frac{B (a+b x)^{5/2} (d+e x)^{5/2}}{5 b e} \]

[Out]

(-3*(b*d - a*e)^3*(2*A*b*e - B*(b*d + a*e))*Sqrt[a + b*x]*Sqrt[d + e*x])/(128*b^
3*e^3) + ((b*d - a*e)^2*(2*A*b*e - B*(b*d + a*e))*(a + b*x)^(3/2)*Sqrt[d + e*x])
/(64*b^3*e^2) + ((b*d - a*e)*(2*A*b*e - B*(b*d + a*e))*(a + b*x)^(5/2)*Sqrt[d +
e*x])/(16*b^3*e) + ((2*A*b*e - B*(b*d + a*e))*(a + b*x)^(5/2)*(d + e*x)^(3/2))/(
8*b^2*e) + (B*(a + b*x)^(5/2)*(d + e*x)^(5/2))/(5*b*e) + (3*(b*d - a*e)^4*(2*A*b
*e - B*(b*d + a*e))*ArcTanh[(Sqrt[e]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[d + e*x])])/(1
28*b^(7/2)*e^(7/2))

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Rubi [A]  time = 0.636911, antiderivative size = 304, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{3 (b d-a e)^4 (2 A b e-B (a e+b d)) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{128 b^{7/2} e^{7/2}}-\frac{3 \sqrt{a+b x} \sqrt{d+e x} (b d-a e)^3 (2 A b e-B (a e+b d))}{128 b^3 e^3}+\frac{(a+b x)^{3/2} \sqrt{d+e x} (b d-a e)^2 (2 A b e-B (a e+b d))}{64 b^3 e^2}+\frac{(a+b x)^{5/2} \sqrt{d+e x} (b d-a e) (2 A b e-B (a e+b d))}{16 b^3 e}+\frac{(a+b x)^{5/2} (d+e x)^{3/2} (2 A b e-B (a e+b d))}{8 b^2 e}+\frac{B (a+b x)^{5/2} (d+e x)^{5/2}}{5 b e} \]

Antiderivative was successfully verified.

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

[Out]

(-3*(b*d - a*e)^3*(2*A*b*e - B*(b*d + a*e))*Sqrt[a + b*x]*Sqrt[d + e*x])/(128*b^
3*e^3) + ((b*d - a*e)^2*(2*A*b*e - B*(b*d + a*e))*(a + b*x)^(3/2)*Sqrt[d + e*x])
/(64*b^3*e^2) + ((b*d - a*e)*(2*A*b*e - B*(b*d + a*e))*(a + b*x)^(5/2)*Sqrt[d +
e*x])/(16*b^3*e) + ((2*A*b*e - B*(b*d + a*e))*(a + b*x)^(5/2)*(d + e*x)^(3/2))/(
8*b^2*e) + (B*(a + b*x)^(5/2)*(d + e*x)^(5/2))/(5*b*e) + (3*(b*d - a*e)^4*(2*A*b
*e - B*(b*d + a*e))*ArcTanh[(Sqrt[e]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[d + e*x])])/(1
28*b^(7/2)*e^(7/2))

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Rubi in Sympy [A]  time = 56.8905, size = 275, normalized size = 0.9 \[ \frac{B \left (a + b x\right )^{\frac{5}{2}} \left (d + e x\right )^{\frac{5}{2}}}{5 b e} - \frac{\left (a + b x\right )^{\frac{3}{2}} \left (d + e x\right )^{\frac{5}{2}} \left (- A b e + \frac{B \left (a e + b d\right )}{2}\right )}{4 b e^{2}} + \frac{\sqrt{a + b x} \left (d + e x\right )^{\frac{5}{2}} \left (a e - b d\right ) \left (2 A b e - B a e - B b d\right )}{16 b e^{3}} - \frac{\sqrt{a + b x} \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right )^{2} \left (- A b e + \frac{B \left (a e + b d\right )}{2}\right )}{32 b^{2} e^{3}} + \frac{3 \sqrt{a + b x} \sqrt{d + e x} \left (a e - b d\right )^{3} \left (- A b e + \frac{B \left (a e + b d\right )}{2}\right )}{64 b^{3} e^{3}} - \frac{3 \left (a e - b d\right )^{4} \left (- A b e + \frac{B \left (a e + b d\right )}{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{e} \sqrt{a + b x}}{\sqrt{b} \sqrt{d + e x}} \right )}}{64 b^{\frac{7}{2}} e^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

B*(a + b*x)**(5/2)*(d + e*x)**(5/2)/(5*b*e) - (a + b*x)**(3/2)*(d + e*x)**(5/2)*
(-A*b*e + B*(a*e + b*d)/2)/(4*b*e**2) + sqrt(a + b*x)*(d + e*x)**(5/2)*(a*e - b*
d)*(2*A*b*e - B*a*e - B*b*d)/(16*b*e**3) - sqrt(a + b*x)*(d + e*x)**(3/2)*(a*e -
b*d)**2*(-A*b*e + B*(a*e + b*d)/2)/(32*b**2*e**3) + 3*sqrt(a + b*x)*sqrt(d + e*
x)*(a*e - b*d)**3*(-A*b*e + B*(a*e + b*d)/2)/(64*b**3*e**3) - 3*(a*e - b*d)**4*(
-A*b*e + B*(a*e + b*d)/2)*atanh(sqrt(e)*sqrt(a + b*x)/(sqrt(b)*sqrt(d + e*x)))/(
64*b**(7/2)*e**(7/2))

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Mathematica [A]  time = 0.538166, size = 332, normalized size = 1.09 \[ \frac{\sqrt{a+b x} \sqrt{d+e x} \left (15 a^4 B e^4-10 a^3 b e^3 (3 A e+4 B d+B e x)+2 a^2 b^2 e^2 \left (5 A e (11 d+2 e x)+B \left (9 d^2+13 d e x+4 e^2 x^2\right )\right )+2 a b^3 e \left (5 A e \left (11 d^2+44 d e x+24 e^2 x^2\right )+B \left (-20 d^3+13 d^2 e x+136 d e^2 x^2+88 e^3 x^3\right )\right )+b^4 \left (10 A e \left (-3 d^3+2 d^2 e x+24 d e^2 x^2+16 e^3 x^3\right )+B \left (15 d^4-10 d^3 e x+8 d^2 e^2 x^2+176 d e^3 x^3+128 e^4 x^4\right )\right )\right )}{640 b^3 e^3}-\frac{3 (b d-a e)^4 (a B e-2 A b e+b B d) \log \left (2 \sqrt{b} \sqrt{e} \sqrt{a+b x} \sqrt{d+e x}+a e+b d+2 b e x\right )}{256 b^{7/2} e^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a + b*x]*Sqrt[d + e*x]*(15*a^4*B*e^4 - 10*a^3*b*e^3*(4*B*d + 3*A*e + B*e*x
) + 2*a^2*b^2*e^2*(5*A*e*(11*d + 2*e*x) + B*(9*d^2 + 13*d*e*x + 4*e^2*x^2)) + 2*
a*b^3*e*(5*A*e*(11*d^2 + 44*d*e*x + 24*e^2*x^2) + B*(-20*d^3 + 13*d^2*e*x + 136*
d*e^2*x^2 + 88*e^3*x^3)) + b^4*(10*A*e*(-3*d^3 + 2*d^2*e*x + 24*d*e^2*x^2 + 16*e
^3*x^3) + B*(15*d^4 - 10*d^3*e*x + 8*d^2*e^2*x^2 + 176*d*e^3*x^3 + 128*e^4*x^4))
))/(640*b^3*e^3) - (3*(b*d - a*e)^4*(b*B*d - 2*A*b*e + 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]])/(256*b^(7/2)*e^(7/2))

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Maple [B]  time = 0.024, size = 1631, normalized size = 5.4 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/1280*(b*x+a)^(1/2)*(e*x+d)^(1/2)*(320*A*x^3*b^4*e^4*(b*e)^(1/2)*(b*e*x^2+a*e*x
+b*d*x+a*d)^(1/2)-80*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*a*d^3*B*b^3*(b*e)^(1/2)*e+3
52*B*x^3*a*b^3*e^4*(b*e)^(1/2)*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)+352*B*x^3*b^4*d*e
^3*(b*e)^(1/2)*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)+480*A*x^2*a*b^3*e^4*(b*e)^(1/2)*(
b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)+480*A*x^2*b^4*d*e^3*(b*e)^(1/2)*(b*e*x^2+a*e*x+b*
d*x+a*d)^(1/2)+16*B*x^2*a^2*b^2*e^4*(b*e)^(1/2)*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)+
16*B*x^2*b^4*d^2*e^2*(b*e)^(1/2)*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)+52*(b*e*x^2+a*e
*x+b*d*x+a*d)^(1/2)*x*a*d^2*B*b^3*(b*e)^(1/2)*e^2-15*b^5*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^5*B-15*e^5*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^5+220*e^3*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*a^2*d*A*b^2*(b*e)^(1/2)+40*(b*e*x^2+
a*e*x+b*d*x+a*d)^(1/2)*x*a^2*e^4*A*b^2*(b*e)^(1/2)+40*d^2*A*(b*e*x^2+a*e*x+b*d*x
+a*d)^(1/2)*x*b^4*(b*e)^(1/2)*e^2-20*e^4*B*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*x*a^3
*b*(b*e)^(1/2)-20*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*x*d^3*B*b^4*(b*e)^(1/2)*e+30*e
^5*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*A*b+52*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*x*a^2*d*e^3*B*b^2*(b*e)^(1/2)+
880*e^3*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*x*a*d*A*b^3*(b*e)^(1/2)+544*B*x^2*a*b^3*
d*e^3*(b*e)^(1/2)*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)+30*d^4*A*b^5*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+30*e^4*B*(b
*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*a^4*(b*e)^(1/2)+30*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)
*d^4*B*b^4*(b*e)^(1/2)+256*B*x^4*b^4*e^4*(b*e)^(1/2)*(b*e*x^2+a*e*x+b*d*x+a*d)^(
1/2)-120*a^3*d*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^4*A*b^2+180*d^2*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^3-120*d^3*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^4*e^2+4
5*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^4*d*B*b-30*a^3*d^2*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^3*B*b^2-30*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^3*B*b^3*e^2+45*a*d^4*l
n(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
))*B*b^4*e-60*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*a^3*e^4*A*b*(b*e)^(1/2)-60*d^3*A*(
b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*b^4*(b*e)^(1/2)*e+36*(b*e*x^2+a*e*x+b*d*x+a*d)^(1
/2)*a^2*d^2*B*b^2*(b*e)^(1/2)*e^2+220*d^2*A*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*a*b^
3*(b*e)^(1/2)*e^2-80*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*a^3*d*e^3*B*b*(b*e)^(1/2))/
(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)/e^3/b^3/(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)*(b*x + a)^(3/2)*(e*x + d)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.304649, 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)*(e*x + d)^(3/2),x, algorithm="fricas")

[Out]

[1/2560*(4*(128*B*b^4*e^4*x^4 + 15*B*b^4*d^4 - 10*(4*B*a*b^3 + 3*A*b^4)*d^3*e +
2*(9*B*a^2*b^2 + 55*A*a*b^3)*d^2*e^2 - 10*(4*B*a^3*b - 11*A*a^2*b^2)*d*e^3 + 15*
(B*a^4 - 2*A*a^3*b)*e^4 + 16*(11*B*b^4*d*e^3 + (11*B*a*b^3 + 10*A*b^4)*e^4)*x^3
+ 8*(B*b^4*d^2*e^2 + 2*(17*B*a*b^3 + 15*A*b^4)*d*e^3 + (B*a^2*b^2 + 30*A*a*b^3)*
e^4)*x^2 - 2*(5*B*b^4*d^3*e - (13*B*a*b^3 + 10*A*b^4)*d^2*e^2 - (13*B*a^2*b^2 +
220*A*a*b^3)*d*e^3 + 5*(B*a^3*b - 2*A*a^2*b^2)*e^4)*x)*sqrt(b*e)*sqrt(b*x + a)*s
qrt(e*x + d) + 15*(B*b^5*d^5 - (3*B*a*b^4 + 2*A*b^5)*d^4*e + 2*(B*a^2*b^3 + 4*A*
a*b^4)*d^3*e^2 + 2*(B*a^3*b^2 - 6*A*a^2*b^3)*d^2*e^3 - (3*B*a^4*b - 8*A*a^3*b^2)
*d*e^4 + (B*a^5 - 2*A*a^4*b)*e^5)*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^3), 1/1280*(2*(128*B*b^4*e^4*x^4
+ 15*B*b^4*d^4 - 10*(4*B*a*b^3 + 3*A*b^4)*d^3*e + 2*(9*B*a^2*b^2 + 55*A*a*b^3)*
d^2*e^2 - 10*(4*B*a^3*b - 11*A*a^2*b^2)*d*e^3 + 15*(B*a^4 - 2*A*a^3*b)*e^4 + 16*
(11*B*b^4*d*e^3 + (11*B*a*b^3 + 10*A*b^4)*e^4)*x^3 + 8*(B*b^4*d^2*e^2 + 2*(17*B*
a*b^3 + 15*A*b^4)*d*e^3 + (B*a^2*b^2 + 30*A*a*b^3)*e^4)*x^2 - 2*(5*B*b^4*d^3*e -
(13*B*a*b^3 + 10*A*b^4)*d^2*e^2 - (13*B*a^2*b^2 + 220*A*a*b^3)*d*e^3 + 5*(B*a^3
*b - 2*A*a^2*b^2)*e^4)*x)*sqrt(-b*e)*sqrt(b*x + a)*sqrt(e*x + d) - 15*(B*b^5*d^5
- (3*B*a*b^4 + 2*A*b^5)*d^4*e + 2*(B*a^2*b^3 + 4*A*a*b^4)*d^3*e^2 + 2*(B*a^3*b^
2 - 6*A*a^2*b^3)*d^2*e^3 - (3*B*a^4*b - 8*A*a^3*b^2)*d*e^4 + (B*a^5 - 2*A*a^4*b)
*e^5)*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^3)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.406087, 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)*(e*x + d)^(3/2),x, algorithm="giac")

[Out]

Done