3.2219 \(\int \frac{(A+B x) (d+e x)^{5/2}}{\sqrt{a+b x}} \, dx\)

Optimal. Leaf size=246 \[ -\frac{5 (b d-a e)^3 (7 a B e-8 A b e+b B d) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{64 b^{9/2} e^{3/2}}-\frac{5 \sqrt{a+b x} \sqrt{d+e x} (b d-a e)^2 (7 a B e-8 A b e+b B d)}{64 b^4 e}-\frac{5 \sqrt{a+b x} (d+e x)^{3/2} (b d-a e) (7 a B e-8 A b e+b B d)}{96 b^3 e}-\frac{\sqrt{a+b x} (d+e x)^{5/2} (7 a B e-8 A b e+b B d)}{24 b^2 e}+\frac{B \sqrt{a+b x} (d+e x)^{7/2}}{4 b e} \]

[Out]

(-5*(b*d - a*e)^2*(b*B*d - 8*A*b*e + 7*a*B*e)*Sqrt[a + b*x]*Sqrt[d + e*x])/(64*b
^4*e) - (5*(b*d - a*e)*(b*B*d - 8*A*b*e + 7*a*B*e)*Sqrt[a + b*x]*(d + e*x)^(3/2)
)/(96*b^3*e) - ((b*B*d - 8*A*b*e + 7*a*B*e)*Sqrt[a + b*x]*(d + e*x)^(5/2))/(24*b
^2*e) + (B*Sqrt[a + b*x]*(d + e*x)^(7/2))/(4*b*e) - (5*(b*d - a*e)^3*(b*B*d - 8*
A*b*e + 7*a*B*e)*ArcTanh[(Sqrt[e]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[d + e*x])])/(64*b
^(9/2)*e^(3/2))

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Rubi [A]  time = 0.522734, antiderivative size = 246, 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 \[ -\frac{5 (b d-a e)^3 (7 a B e-8 A b e+b B d) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{64 b^{9/2} e^{3/2}}-\frac{5 \sqrt{a+b x} \sqrt{d+e x} (b d-a e)^2 (7 a B e-8 A b e+b B d)}{64 b^4 e}-\frac{5 \sqrt{a+b x} (d+e x)^{3/2} (b d-a e) (7 a B e-8 A b e+b B d)}{96 b^3 e}-\frac{\sqrt{a+b x} (d+e x)^{5/2} (7 a B e-8 A b e+b B d)}{24 b^2 e}+\frac{B \sqrt{a+b x} (d+e x)^{7/2}}{4 b e} \]

Antiderivative was successfully verified.

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

[Out]

(-5*(b*d - a*e)^2*(b*B*d - 8*A*b*e + 7*a*B*e)*Sqrt[a + b*x]*Sqrt[d + e*x])/(64*b
^4*e) - (5*(b*d - a*e)*(b*B*d - 8*A*b*e + 7*a*B*e)*Sqrt[a + b*x]*(d + e*x)^(3/2)
)/(96*b^3*e) - ((b*B*d - 8*A*b*e + 7*a*B*e)*Sqrt[a + b*x]*(d + e*x)^(5/2))/(24*b
^2*e) + (B*Sqrt[a + b*x]*(d + e*x)^(7/2))/(4*b*e) - (5*(b*d - a*e)^3*(b*B*d - 8*
A*b*e + 7*a*B*e)*ArcTanh[(Sqrt[e]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[d + e*x])])/(64*b
^(9/2)*e^(3/2))

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Rubi in Sympy [A]  time = 40.617, size = 238, normalized size = 0.97 \[ \frac{B \sqrt{a + b x} \left (d + e x\right )^{\frac{7}{2}}}{4 b e} + \frac{\sqrt{a + b x} \left (d + e x\right )^{\frac{5}{2}} \left (8 A b e - 7 B a e - B b d\right )}{24 b^{2} e} - \frac{5 \sqrt{a + b x} \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right ) \left (8 A b e - 7 B a e - B b d\right )}{96 b^{3} e} + \frac{5 \sqrt{a + b x} \sqrt{d + e x} \left (a e - b d\right )^{2} \left (8 A b e - 7 B a e - B b d\right )}{64 b^{4} e} - \frac{5 \left (a e - b d\right )^{3} \left (8 A b e - 7 B a e - B b d\right ) \operatorname{atanh}{\left (\frac{\sqrt{e} \sqrt{a + b x}}{\sqrt{b} \sqrt{d + e x}} \right )}}{64 b^{\frac{9}{2}} e^{\frac{3}{2}}} \]

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*sqrt(a + b*x)*(d + e*x)**(7/2)/(4*b*e) + sqrt(a + b*x)*(d + e*x)**(5/2)*(8*A*b
*e - 7*B*a*e - B*b*d)/(24*b**2*e) - 5*sqrt(a + b*x)*(d + e*x)**(3/2)*(a*e - b*d)
*(8*A*b*e - 7*B*a*e - B*b*d)/(96*b**3*e) + 5*sqrt(a + b*x)*sqrt(d + e*x)*(a*e -
b*d)**2*(8*A*b*e - 7*B*a*e - B*b*d)/(64*b**4*e) - 5*(a*e - b*d)**3*(8*A*b*e - 7*
B*a*e - B*b*d)*atanh(sqrt(e)*sqrt(a + b*x)/(sqrt(b)*sqrt(d + e*x)))/(64*b**(9/2)
*e**(3/2))

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Mathematica [A]  time = 0.411643, size = 244, normalized size = 0.99 \[ \frac{\sqrt{a+b x} \sqrt{d+e x} \left (-105 a^3 B e^3+5 a^2 b e^2 (24 A e+53 B d+14 B e x)-a b^2 e \left (80 A e (4 d+e x)+B \left (191 d^2+172 d e x+56 e^2 x^2\right )\right )+b^3 \left (8 A e \left (33 d^2+26 d e x+8 e^2 x^2\right )+B \left (15 d^3+118 d^2 e x+136 d e^2 x^2+48 e^3 x^3\right )\right )\right )}{192 b^4 e}-\frac{5 (b d-a e)^3 (7 a B e-8 A b e+b B d) \log \left (2 \sqrt{b} \sqrt{e} \sqrt{a+b x} \sqrt{d+e x}+a e+b d+2 b e x\right )}{128 b^{9/2} e^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a + b*x]*Sqrt[d + e*x]*(-105*a^3*B*e^3 + 5*a^2*b*e^2*(53*B*d + 24*A*e + 14
*B*e*x) - a*b^2*e*(80*A*e*(4*d + e*x) + B*(191*d^2 + 172*d*e*x + 56*e^2*x^2)) +
b^3*(8*A*e*(33*d^2 + 26*d*e*x + 8*e^2*x^2) + B*(15*d^3 + 118*d^2*e*x + 136*d*e^2
*x^2 + 48*e^3*x^3))))/(192*b^4*e) - (5*(b*d - a*e)^3*(b*B*d - 8*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]])/(128*
b^(9/2)*e^(3/2))

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Maple [B]  time = 0.036, size = 968, normalized size = 3.9 \[ \text{result too large to display} \]

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]

-1/384*(e*x+d)^(1/2)*(b*x+a)^(1/2)*(-105*e^4*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^4+15*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))*b^4*d^4*B-360*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*A*b^2*d*e^3+344*((b*
x+a)*(e*x+d))^(1/2)*x*a*d*B*b^2*e^2*(b*e)^(1/2)-120*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))*b^4*d^3*A*e-30*((b*x+a)*(e*x+d))
^(1/2)*B*b^3*d^3*(b*e)^(1/2)+120*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*A*e^4*b+210*e^3*B*((b*x+a)*(e*x+d))^(1/2)*a^3*(
b*e)^(1/2)+360*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*A*b^3*d^2*e^2+300*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*B*d*e^3*b-270*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*b^2*d^2*e^2+60*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*b^3*d^3*e-240*((b*
x+a)*(e*x+d))^(1/2)*A*a^2*e^3*(b*e)^(1/2)*b-528*((b*x+a)*(e*x+d))^(1/2)*A*b^3*d^
2*e*(b*e)^(1/2)-96*B*x^3*b^3*e^3*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)-128*A*x^2*b
^3*e^3*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)-140*e^3*B*((b*x+a)*(e*x+d))^(1/2)*x*a
^2*b*(b*e)^(1/2)-236*((b*x+a)*(e*x+d))^(1/2)*x*d^2*B*b^3*e*(b*e)^(1/2)+160*((b*x
+a)*(e*x+d))^(1/2)*x*a*A*e^3*b^2*(b*e)^(1/2)-416*d*A*((b*x+a)*(e*x+d))^(1/2)*x*b
^3*e^2*(b*e)^(1/2)+112*B*x^2*a*b^2*e^3*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)-272*B
*x^2*b^3*d*e^2*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)+640*((b*x+a)*(e*x+d))^(1/2)*A
*a*b^2*d*e^2*(b*e)^(1/2)-530*((b*x+a)*(e*x+d))^(1/2)*B*a^2*d*e^2*(b*e)^(1/2)*b+3
82*((b*x+a)*(e*x+d))^(1/2)*B*a*b^2*d^2*e*(b*e)^(1/2))/((b*x+a)*(e*x+d))^(1/2)/b^
4/(b*e)^(1/2)/e

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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)^(5/2)/sqrt(b*x + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.627192, 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)*(e*x + d)^(5/2)/sqrt(b*x + a),x, algorithm="fricas")

[Out]

[1/768*(4*(48*B*b^3*e^3*x^3 + 15*B*b^3*d^3 - (191*B*a*b^2 - 264*A*b^3)*d^2*e + 5
*(53*B*a^2*b - 64*A*a*b^2)*d*e^2 - 15*(7*B*a^3 - 8*A*a^2*b)*e^3 + 8*(17*B*b^3*d*
e^2 - (7*B*a*b^2 - 8*A*b^3)*e^3)*x^2 + 2*(59*B*b^3*d^2*e - 2*(43*B*a*b^2 - 52*A*
b^3)*d*e^2 + 5*(7*B*a^2*b - 8*A*a*b^2)*e^3)*x)*sqrt(b*e)*sqrt(b*x + a)*sqrt(e*x
+ d) - 15*(B*b^4*d^4 + 4*(B*a*b^3 - 2*A*b^4)*d^3*e - 6*(3*B*a^2*b^2 - 4*A*a*b^3)
*d^2*e^2 + 4*(5*B*a^3*b - 6*A*a^2*b^2)*d*e^3 - (7*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^4*e), 1/384*(2*(48*B*b^3*e^3*x^3 + 15*B*b^3*d^3 - (191*B*a*b^2 - 264*A*b^3)*d
^2*e + 5*(53*B*a^2*b - 64*A*a*b^2)*d*e^2 - 15*(7*B*a^3 - 8*A*a^2*b)*e^3 + 8*(17*
B*b^3*d*e^2 - (7*B*a*b^2 - 8*A*b^3)*e^3)*x^2 + 2*(59*B*b^3*d^2*e - 2*(43*B*a*b^2
- 52*A*b^3)*d*e^2 + 5*(7*B*a^2*b - 8*A*a*b^2)*e^3)*x)*sqrt(-b*e)*sqrt(b*x + a)*
sqrt(e*x + d) - 15*(B*b^4*d^4 + 4*(B*a*b^3 - 2*A*b^4)*d^3*e - 6*(3*B*a^2*b^2 - 4
*A*a*b^3)*d^2*e^2 + 4*(5*B*a^3*b - 6*A*a^2*b^2)*d*e^3 - (7*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^4*e)]

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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 = 0.368254, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(e*x + d)^(5/2)/sqrt(b*x + a),x, algorithm="giac")

[Out]

Done