3.2183 \(\int \sqrt{a+b x} (A+B x) (d+e x)^{5/2} \, dx\)

Optimal. Leaf size=304 \[ \frac{(b d-a e)^4 (7 a B e-10 A b e+3 b B d) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{128 b^{9/2} e^{5/2}}-\frac{\sqrt{a+b x} \sqrt{d+e x} (b d-a e)^3 (7 a B e-10 A b e+3 b B d)}{128 b^4 e^2}-\frac{(a+b x)^{3/2} \sqrt{d+e x} (b d-a e)^2 (7 a B e-10 A b e+3 b B d)}{64 b^4 e}-\frac{(a+b x)^{3/2} (d+e x)^{3/2} (b d-a e) (7 a B e-10 A b e+3 b B d)}{48 b^3 e}-\frac{(a+b x)^{3/2} (d+e x)^{5/2} (7 a B e-10 A b e+3 b B d)}{40 b^2 e}+\frac{B (a+b x)^{3/2} (d+e x)^{7/2}}{5 b e} \]

[Out]

-((b*d - a*e)^3*(3*b*B*d - 10*A*b*e + 7*a*B*e)*Sqrt[a + b*x]*Sqrt[d + e*x])/(128
*b^4*e^2) - ((b*d - a*e)^2*(3*b*B*d - 10*A*b*e + 7*a*B*e)*(a + b*x)^(3/2)*Sqrt[d
+ e*x])/(64*b^4*e) - ((b*d - a*e)*(3*b*B*d - 10*A*b*e + 7*a*B*e)*(a + b*x)^(3/2
)*(d + e*x)^(3/2))/(48*b^3*e) - ((3*b*B*d - 10*A*b*e + 7*a*B*e)*(a + b*x)^(3/2)*
(d + e*x)^(5/2))/(40*b^2*e) + (B*(a + b*x)^(3/2)*(d + e*x)^(7/2))/(5*b*e) + ((b*
d - a*e)^4*(3*b*B*d - 10*A*b*e + 7*a*B*e)*ArcTanh[(Sqrt[e]*Sqrt[a + b*x])/(Sqrt[
b]*Sqrt[d + e*x])])/(128*b^(9/2)*e^(5/2))

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Rubi [A]  time = 0.681626, 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 \[ \frac{(b d-a e)^4 (7 a B e-10 A b e+3 b B d) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{128 b^{9/2} e^{5/2}}-\frac{\sqrt{a+b x} \sqrt{d+e x} (b d-a e)^3 (7 a B e-10 A b e+3 b B d)}{128 b^4 e^2}-\frac{(a+b x)^{3/2} \sqrt{d+e x} (b d-a e)^2 (7 a B e-10 A b e+3 b B d)}{64 b^4 e}-\frac{(a+b x)^{3/2} (d+e x)^{3/2} (b d-a e) (7 a B e-10 A b e+3 b B d)}{48 b^3 e}-\frac{(a+b x)^{3/2} (d+e x)^{5/2} (7 a B e-10 A b e+3 b B d)}{40 b^2 e}+\frac{B (a+b x)^{3/2} (d+e x)^{7/2}}{5 b e} \]

Antiderivative was successfully verified.

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

[Out]

-((b*d - a*e)^3*(3*b*B*d - 10*A*b*e + 7*a*B*e)*Sqrt[a + b*x]*Sqrt[d + e*x])/(128
*b^4*e^2) - ((b*d - a*e)^2*(3*b*B*d - 10*A*b*e + 7*a*B*e)*(a + b*x)^(3/2)*Sqrt[d
+ e*x])/(64*b^4*e) - ((b*d - a*e)*(3*b*B*d - 10*A*b*e + 7*a*B*e)*(a + b*x)^(3/2
)*(d + e*x)^(3/2))/(48*b^3*e) - ((3*b*B*d - 10*A*b*e + 7*a*B*e)*(a + b*x)^(3/2)*
(d + e*x)^(5/2))/(40*b^2*e) + (B*(a + b*x)^(3/2)*(d + e*x)^(7/2))/(5*b*e) + ((b*
d - a*e)^4*(3*b*B*d - 10*A*b*e + 7*a*B*e)*ArcTanh[(Sqrt[e]*Sqrt[a + b*x])/(Sqrt[
b]*Sqrt[d + e*x])])/(128*b^(9/2)*e^(5/2))

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Rubi in Sympy [A]  time = 56.5868, size = 298, normalized size = 0.98 \[ \frac{B \left (a + b x\right )^{\frac{3}{2}} \left (d + e x\right )^{\frac{7}{2}}}{5 b e} + \frac{\sqrt{a + b x} \left (d + e x\right )^{\frac{7}{2}} \left (10 A b e - 7 B a e - 3 B b d\right )}{40 b e^{2}} + \frac{\sqrt{a + b x} \left (d + e x\right )^{\frac{5}{2}} \left (a e - b d\right ) \left (10 A b e - 7 B a e - 3 B b d\right )}{240 b^{2} e^{2}} - \frac{\sqrt{a + b x} \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right )^{2} \left (10 A b e - 7 B a e - 3 B b d\right )}{192 b^{3} e^{2}} + \frac{\sqrt{a + b x} \sqrt{d + e x} \left (a e - b d\right )^{3} \left (10 A b e - 7 B a e - 3 B b d\right )}{128 b^{4} e^{2}} - \frac{\left (a e - b d\right )^{4} \left (10 A b e - 7 B a e - 3 B b d\right ) \operatorname{atanh}{\left (\frac{\sqrt{e} \sqrt{a + b x}}{\sqrt{b} \sqrt{d + e x}} \right )}}{128 b^{\frac{9}{2}} e^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(e*x+d)**(5/2)*(b*x+a)**(1/2),x)

[Out]

B*(a + b*x)**(3/2)*(d + e*x)**(7/2)/(5*b*e) + sqrt(a + b*x)*(d + e*x)**(7/2)*(10
*A*b*e - 7*B*a*e - 3*B*b*d)/(40*b*e**2) + sqrt(a + b*x)*(d + e*x)**(5/2)*(a*e -
b*d)*(10*A*b*e - 7*B*a*e - 3*B*b*d)/(240*b**2*e**2) - sqrt(a + b*x)*(d + e*x)**(
3/2)*(a*e - b*d)**2*(10*A*b*e - 7*B*a*e - 3*B*b*d)/(192*b**3*e**2) + sqrt(a + b*
x)*sqrt(d + e*x)*(a*e - b*d)**3*(10*A*b*e - 7*B*a*e - 3*B*b*d)/(128*b**4*e**2) -
(a*e - b*d)**4*(10*A*b*e - 7*B*a*e - 3*B*b*d)*atanh(sqrt(e)*sqrt(a + b*x)/(sqrt
(b)*sqrt(d + e*x)))/(128*b**(9/2)*e**(5/2))

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Mathematica [A]  time = 0.484386, size = 335, normalized size = 1.1 \[ \frac{\sqrt{a+b x} \sqrt{d+e x} \left (-105 a^4 B e^4+10 a^3 b e^3 (15 A e+34 B d+7 B e x)-2 a^2 b^2 e^2 \left (25 A e (11 d+2 e x)+B \left (173 d^2+111 d e x+28 e^2 x^2\right )\right )+2 a b^3 e \left (5 A e \left (73 d^2+36 d e x+8 e^2 x^2\right )+B \left (30 d^3+109 d^2 e x+88 d e^2 x^2+24 e^3 x^3\right )\right )+b^4 \left (10 A e \left (15 d^3+118 d^2 e x+136 d e^2 x^2+48 e^3 x^3\right )+B \left (-45 d^4+30 d^3 e x+744 d^2 e^2 x^2+1008 d e^3 x^3+384 e^4 x^4\right )\right )\right )}{1920 b^4 e^2}+\frac{(b d-a e)^4 (7 a B e-10 A b e+3 b B d) \log \left (2 \sqrt{b} \sqrt{e} \sqrt{a+b x} \sqrt{d+e x}+a e+b d+2 b e x\right )}{256 b^{9/2} e^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a + b*x]*Sqrt[d + e*x]*(-105*a^4*B*e^4 + 10*a^3*b*e^3*(34*B*d + 15*A*e + 7
*B*e*x) - 2*a^2*b^2*e^2*(25*A*e*(11*d + 2*e*x) + B*(173*d^2 + 111*d*e*x + 28*e^2
*x^2)) + 2*a*b^3*e*(5*A*e*(73*d^2 + 36*d*e*x + 8*e^2*x^2) + B*(30*d^3 + 109*d^2*
e*x + 88*d*e^2*x^2 + 24*e^3*x^3)) + b^4*(10*A*e*(15*d^3 + 118*d^2*e*x + 136*d*e^
2*x^2 + 48*e^3*x^3) + B*(-45*d^4 + 30*d^3*e*x + 744*d^2*e^2*x^2 + 1008*d*e^3*x^3
+ 384*e^4*x^4))))/(1920*b^4*e^2) + ((b*d - a*e)^4*(3*b*B*d - 10*A*b*e + 7*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^(9/2)*e^(5/2))

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Maple [B]  time = 0.038, 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)*(e*x+d)^(5/2)*(b*x+a)^(1/2),x)

[Out]

-1/3840*(e*x+d)^(1/2)*(b*x+a)^(1/2)*(-960*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)-120*(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-96*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)-2016*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)-160*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)-2720*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)+112*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)-1488*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)-436*(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^2-45*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-105*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+1100*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)+20
0*(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)-2360*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^2-140*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)-60*(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+150*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+444*(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)-720*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)-
352*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)+150*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+210*e^4*B*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*a^4*(b*e)^(1/2)+90*(b*e*x^2+a*e*
x+b*d*x+a*d)^(1/2)*d^4*B*b^4*(b*e)^(1/2)-768*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)-600*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+900*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^3-600*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+375*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-450*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+150*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+75*a*d^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))*B*b^4*e-300*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*a^3*e^4*A*b*
(b*e)^(1/2)-300*d^3*A*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*b^4*(b*e)^(1/2)*e+692*(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^2-1460*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^2-680*(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)/b^4/(b*e)^(1/2)/e^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)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.295755, 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)^(5/2),x, algorithm="fricas")

[Out]

[1/7680*(4*(384*B*b^4*e^4*x^4 - 45*B*b^4*d^4 + 30*(2*B*a*b^3 + 5*A*b^4)*d^3*e -
2*(173*B*a^2*b^2 - 365*A*a*b^3)*d^2*e^2 + 10*(34*B*a^3*b - 55*A*a^2*b^2)*d*e^3 -
15*(7*B*a^4 - 10*A*a^3*b)*e^4 + 48*(21*B*b^4*d*e^3 + (B*a*b^3 + 10*A*b^4)*e^4)*
x^3 + 8*(93*B*b^4*d^2*e^2 + 2*(11*B*a*b^3 + 85*A*b^4)*d*e^3 - (7*B*a^2*b^2 - 10*
A*a*b^3)*e^4)*x^2 + 2*(15*B*b^4*d^3*e + (109*B*a*b^3 + 590*A*b^4)*d^2*e^2 - 3*(3
7*B*a^2*b^2 - 60*A*a*b^3)*d*e^3 + 5*(7*B*a^3*b - 10*A*a^2*b^2)*e^4)*x)*sqrt(b*e)
*sqrt(b*x + a)*sqrt(e*x + d) - 15*(3*B*b^5*d^5 - 5*(B*a*b^4 + 2*A*b^5)*d^4*e - 1
0*(B*a^2*b^3 - 4*A*a*b^4)*d^3*e^2 + 30*(B*a^3*b^2 - 2*A*a^2*b^3)*d^2*e^3 - 5*(5*
B*a^4*b - 8*A*a^3*b^2)*d*e^4 + (7*B*a^5 - 10*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^4*e^2), 1/3
840*(2*(384*B*b^4*e^4*x^4 - 45*B*b^4*d^4 + 30*(2*B*a*b^3 + 5*A*b^4)*d^3*e - 2*(1
73*B*a^2*b^2 - 365*A*a*b^3)*d^2*e^2 + 10*(34*B*a^3*b - 55*A*a^2*b^2)*d*e^3 - 15*
(7*B*a^4 - 10*A*a^3*b)*e^4 + 48*(21*B*b^4*d*e^3 + (B*a*b^3 + 10*A*b^4)*e^4)*x^3
+ 8*(93*B*b^4*d^2*e^2 + 2*(11*B*a*b^3 + 85*A*b^4)*d*e^3 - (7*B*a^2*b^2 - 10*A*a*
b^3)*e^4)*x^2 + 2*(15*B*b^4*d^3*e + (109*B*a*b^3 + 590*A*b^4)*d^2*e^2 - 3*(37*B*
a^2*b^2 - 60*A*a*b^3)*d*e^3 + 5*(7*B*a^3*b - 10*A*a^2*b^2)*e^4)*x)*sqrt(-b*e)*sq
rt(b*x + a)*sqrt(e*x + d) + 15*(3*B*b^5*d^5 - 5*(B*a*b^4 + 2*A*b^5)*d^4*e - 10*(
B*a^2*b^3 - 4*A*a*b^4)*d^3*e^2 + 30*(B*a^3*b^2 - 2*A*a^2*b^3)*d^2*e^3 - 5*(5*B*a
^4*b - 8*A*a^3*b^2)*d*e^4 + (7*B*a^5 - 10*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^4*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)**(5/2)*(b*x+a)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.36242, 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)^(5/2),x, algorithm="giac")

[Out]

Done