3.1787 \(\int (A+B x) \sqrt{d+e x} \left (a^2+2 a b x+b^2 x^2\right ) \, dx\)

Optimal. Leaf size=128 \[ -\frac{2 b (d+e x)^{7/2} (-2 a B e-A b e+3 b B d)}{7 e^4}+\frac{2 (d+e x)^{5/2} (b d-a e) (-a B e-2 A b e+3 b B d)}{5 e^4}-\frac{2 (d+e x)^{3/2} (b d-a e)^2 (B d-A e)}{3 e^4}+\frac{2 b^2 B (d+e x)^{9/2}}{9 e^4} \]

[Out]

(-2*(b*d - a*e)^2*(B*d - A*e)*(d + e*x)^(3/2))/(3*e^4) + (2*(b*d - a*e)*(3*b*B*d
- 2*A*b*e - a*B*e)*(d + e*x)^(5/2))/(5*e^4) - (2*b*(3*b*B*d - A*b*e - 2*a*B*e)*
(d + e*x)^(7/2))/(7*e^4) + (2*b^2*B*(d + e*x)^(9/2))/(9*e^4)

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Rubi [A]  time = 0.155805, antiderivative size = 128, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065 \[ -\frac{2 b (d+e x)^{7/2} (-2 a B e-A b e+3 b B d)}{7 e^4}+\frac{2 (d+e x)^{5/2} (b d-a e) (-a B e-2 A b e+3 b B d)}{5 e^4}-\frac{2 (d+e x)^{3/2} (b d-a e)^2 (B d-A e)}{3 e^4}+\frac{2 b^2 B (d+e x)^{9/2}}{9 e^4} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(b*d - a*e)^2*(B*d - A*e)*(d + e*x)^(3/2))/(3*e^4) + (2*(b*d - a*e)*(3*b*B*d
- 2*A*b*e - a*B*e)*(d + e*x)^(5/2))/(5*e^4) - (2*b*(3*b*B*d - A*b*e - 2*a*B*e)*
(d + e*x)^(7/2))/(7*e^4) + (2*b^2*B*(d + e*x)^(9/2))/(9*e^4)

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Rubi in Sympy [A]  time = 54.0808, size = 126, normalized size = 0.98 \[ \frac{2 B b^{2} \left (d + e x\right )^{\frac{9}{2}}}{9 e^{4}} + \frac{2 b \left (d + e x\right )^{\frac{7}{2}} \left (A b e + 2 B a e - 3 B b d\right )}{7 e^{4}} + \frac{2 \left (d + e x\right )^{\frac{5}{2}} \left (a e - b d\right ) \left (2 A b e + B a e - 3 B b d\right )}{5 e^{4}} + \frac{2 \left (d + e x\right )^{\frac{3}{2}} \left (A e - B d\right ) \left (a e - b d\right )^{2}}{3 e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*B*b**2*(d + e*x)**(9/2)/(9*e**4) + 2*b*(d + e*x)**(7/2)*(A*b*e + 2*B*a*e - 3*B
*b*d)/(7*e**4) + 2*(d + e*x)**(5/2)*(a*e - b*d)*(2*A*b*e + B*a*e - 3*B*b*d)/(5*e
**4) + 2*(d + e*x)**(3/2)*(A*e - B*d)*(a*e - b*d)**2/(3*e**4)

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Mathematica [A]  time = 0.187455, size = 138, normalized size = 1.08 \[ \frac{2 (d+e x)^{3/2} \left (21 a^2 e^2 (5 A e-2 B d+3 B e x)+6 a b e \left (7 A e (3 e x-2 d)+B \left (8 d^2-12 d e x+15 e^2 x^2\right )\right )+b^2 \left (3 A e \left (8 d^2-12 d e x+15 e^2 x^2\right )+B \left (-16 d^3+24 d^2 e x-30 d e^2 x^2+35 e^3 x^3\right )\right )\right )}{315 e^4} \]

Antiderivative was successfully verified.

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

[Out]

(2*(d + e*x)^(3/2)*(21*a^2*e^2*(-2*B*d + 5*A*e + 3*B*e*x) + 6*a*b*e*(7*A*e*(-2*d
+ 3*e*x) + B*(8*d^2 - 12*d*e*x + 15*e^2*x^2)) + b^2*(3*A*e*(8*d^2 - 12*d*e*x +
15*e^2*x^2) + B*(-16*d^3 + 24*d^2*e*x - 30*d*e^2*x^2 + 35*e^3*x^3))))/(315*e^4)

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Maple [A]  time = 0.015, size = 169, normalized size = 1.3 \[{\frac{70\,B{x}^{3}{b}^{2}{e}^{3}+90\,A{b}^{2}{e}^{3}{x}^{2}+180\,Bab{e}^{3}{x}^{2}-60\,B{b}^{2}d{e}^{2}{x}^{2}+252\,Axab{e}^{3}-72\,Ax{b}^{2}d{e}^{2}+126\,Bx{a}^{2}{e}^{3}-144\,Bxabd{e}^{2}+48\,B{b}^{2}{d}^{2}ex+210\,A{a}^{2}{e}^{3}-168\,Aabd{e}^{2}+48\,A{b}^{2}{d}^{2}e-84\,Bd{e}^{2}{a}^{2}+96\,B{d}^{2}abe-32\,B{b}^{2}{d}^{3}}{315\,{e}^{4}} \left ( ex+d \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/315*(e*x+d)^(3/2)*(35*B*b^2*e^3*x^3+45*A*b^2*e^3*x^2+90*B*a*b*e^3*x^2-30*B*b^2
*d*e^2*x^2+126*A*a*b*e^3*x-36*A*b^2*d*e^2*x+63*B*a^2*e^3*x-72*B*a*b*d*e^2*x+24*B
*b^2*d^2*e*x+105*A*a^2*e^3-84*A*a*b*d*e^2+24*A*b^2*d^2*e-42*B*a^2*d*e^2+48*B*a*b
*d^2*e-16*B*b^2*d^3)/e^4

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Maxima [A]  time = 0.735892, size = 215, normalized size = 1.68 \[ \frac{2 \,{\left (35 \,{\left (e x + d\right )}^{\frac{9}{2}} B b^{2} - 45 \,{\left (3 \, B b^{2} d -{\left (2 \, B a b + A b^{2}\right )} e\right )}{\left (e x + d\right )}^{\frac{7}{2}} + 63 \,{\left (3 \, B b^{2} d^{2} - 2 \,{\left (2 \, B a b + A b^{2}\right )} d e +{\left (B a^{2} + 2 \, A a b\right )} e^{2}\right )}{\left (e x + d\right )}^{\frac{5}{2}} - 105 \,{\left (B b^{2} d^{3} - A a^{2} e^{3} -{\left (2 \, B a b + A b^{2}\right )} d^{2} e +{\left (B a^{2} + 2 \, A a b\right )} d e^{2}\right )}{\left (e x + d\right )}^{\frac{3}{2}}\right )}}{315 \, e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/315*(35*(e*x + d)^(9/2)*B*b^2 - 45*(3*B*b^2*d - (2*B*a*b + A*b^2)*e)*(e*x + d)
^(7/2) + 63*(3*B*b^2*d^2 - 2*(2*B*a*b + A*b^2)*d*e + (B*a^2 + 2*A*a*b)*e^2)*(e*x
+ d)^(5/2) - 105*(B*b^2*d^3 - A*a^2*e^3 - (2*B*a*b + A*b^2)*d^2*e + (B*a^2 + 2*
A*a*b)*d*e^2)*(e*x + d)^(3/2))/e^4

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Fricas [A]  time = 0.304369, size = 297, normalized size = 2.32 \[ \frac{2 \,{\left (35 \, B b^{2} e^{4} x^{4} - 16 \, B b^{2} d^{4} + 105 \, A a^{2} d e^{3} + 24 \,{\left (2 \, B a b + A b^{2}\right )} d^{3} e - 42 \,{\left (B a^{2} + 2 \, A a b\right )} d^{2} e^{2} + 5 \,{\left (B b^{2} d e^{3} + 9 \,{\left (2 \, B a b + A b^{2}\right )} e^{4}\right )} x^{3} - 3 \,{\left (2 \, B b^{2} d^{2} e^{2} - 3 \,{\left (2 \, B a b + A b^{2}\right )} d e^{3} - 21 \,{\left (B a^{2} + 2 \, A a b\right )} e^{4}\right )} x^{2} +{\left (8 \, B b^{2} d^{3} e + 105 \, A a^{2} e^{4} - 12 \,{\left (2 \, B a b + A b^{2}\right )} d^{2} e^{2} + 21 \,{\left (B a^{2} + 2 \, A a b\right )} d e^{3}\right )} x\right )} \sqrt{e x + d}}{315 \, e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/315*(35*B*b^2*e^4*x^4 - 16*B*b^2*d^4 + 105*A*a^2*d*e^3 + 24*(2*B*a*b + A*b^2)*
d^3*e - 42*(B*a^2 + 2*A*a*b)*d^2*e^2 + 5*(B*b^2*d*e^3 + 9*(2*B*a*b + A*b^2)*e^4)
*x^3 - 3*(2*B*b^2*d^2*e^2 - 3*(2*B*a*b + A*b^2)*d*e^3 - 21*(B*a^2 + 2*A*a*b)*e^4
)*x^2 + (8*B*b^2*d^3*e + 105*A*a^2*e^4 - 12*(2*B*a*b + A*b^2)*d^2*e^2 + 21*(B*a^
2 + 2*A*a*b)*d*e^3)*x)*sqrt(e*x + d)/e^4

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Sympy [A]  time = 8.42333, size = 201, normalized size = 1.57 \[ \frac{2 \left (\frac{B b^{2} \left (d + e x\right )^{\frac{9}{2}}}{9 e^{3}} + \frac{\left (d + e x\right )^{\frac{7}{2}} \left (A b^{2} e + 2 B a b e - 3 B b^{2} d\right )}{7 e^{3}} + \frac{\left (d + e x\right )^{\frac{5}{2}} \left (2 A a b e^{2} - 2 A b^{2} d e + B a^{2} e^{2} - 4 B a b d e + 3 B b^{2} d^{2}\right )}{5 e^{3}} + \frac{\left (d + e x\right )^{\frac{3}{2}} \left (A a^{2} e^{3} - 2 A a b d e^{2} + A b^{2} d^{2} e - B a^{2} d e^{2} + 2 B a b d^{2} e - B b^{2} d^{3}\right )}{3 e^{3}}\right )}{e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(B*b**2*(d + e*x)**(9/2)/(9*e**3) + (d + e*x)**(7/2)*(A*b**2*e + 2*B*a*b*e - 3
*B*b**2*d)/(7*e**3) + (d + e*x)**(5/2)*(2*A*a*b*e**2 - 2*A*b**2*d*e + B*a**2*e**
2 - 4*B*a*b*d*e + 3*B*b**2*d**2)/(5*e**3) + (d + e*x)**(3/2)*(A*a**2*e**3 - 2*A*
a*b*d*e**2 + A*b**2*d**2*e - B*a**2*d*e**2 + 2*B*a*b*d**2*e - B*b**2*d**3)/(3*e*
*3))/e

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GIAC/XCAS [A]  time = 0.289927, size = 321, normalized size = 2.51 \[ \frac{2}{315} \,{\left (21 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 5 \,{\left (x e + d\right )}^{\frac{3}{2}} d\right )} B a^{2} e^{\left (-1\right )} + 42 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 5 \,{\left (x e + d\right )}^{\frac{3}{2}} d\right )} A a b e^{\left (-1\right )} + 6 \,{\left (15 \,{\left (x e + d\right )}^{\frac{7}{2}} e^{12} - 42 \,{\left (x e + d\right )}^{\frac{5}{2}} d e^{12} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2} e^{12}\right )} B a b e^{\left (-14\right )} + 3 \,{\left (15 \,{\left (x e + d\right )}^{\frac{7}{2}} e^{12} - 42 \,{\left (x e + d\right )}^{\frac{5}{2}} d e^{12} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2} e^{12}\right )} A b^{2} e^{\left (-14\right )} +{\left (35 \,{\left (x e + d\right )}^{\frac{9}{2}} e^{24} - 135 \,{\left (x e + d\right )}^{\frac{7}{2}} d e^{24} + 189 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{2} e^{24} - 105 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{3} e^{24}\right )} B b^{2} e^{\left (-27\right )} + 105 \,{\left (x e + d\right )}^{\frac{3}{2}} A a^{2}\right )} e^{\left (-1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/315*(21*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*B*a^2*e^(-1) + 42*(3*(x*e +
d)^(5/2) - 5*(x*e + d)^(3/2)*d)*A*a*b*e^(-1) + 6*(15*(x*e + d)^(7/2)*e^12 - 42*(
x*e + d)^(5/2)*d*e^12 + 35*(x*e + d)^(3/2)*d^2*e^12)*B*a*b*e^(-14) + 3*(15*(x*e
+ d)^(7/2)*e^12 - 42*(x*e + d)^(5/2)*d*e^12 + 35*(x*e + d)^(3/2)*d^2*e^12)*A*b^2
*e^(-14) + (35*(x*e + d)^(9/2)*e^24 - 135*(x*e + d)^(7/2)*d*e^24 + 189*(x*e + d)
^(5/2)*d^2*e^24 - 105*(x*e + d)^(3/2)*d^3*e^24)*B*b^2*e^(-27) + 105*(x*e + d)^(3
/2)*A*a^2)*e^(-1)