3.1703 \(\int (a+b x) (A+B x) (d+e x)^{3/2} \, dx\)

Optimal. Leaf size=83 \[ -\frac{2 (d+e x)^{7/2} (-a B e-A b e+2 b B d)}{7 e^3}+\frac{2 (d+e x)^{5/2} (b d-a e) (B d-A e)}{5 e^3}+\frac{2 b B (d+e x)^{9/2}}{9 e^3} \]

[Out]

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

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Rubi [A]  time = 0.0990722, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05 \[ -\frac{2 (d+e x)^{7/2} (-a B e-A b e+2 b B d)}{7 e^3}+\frac{2 (d+e x)^{5/2} (b d-a e) (B d-A e)}{5 e^3}+\frac{2 b B (d+e x)^{9/2}}{9 e^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 16.9556, size = 78, normalized size = 0.94 \[ \frac{2 B b \left (d + e x\right )^{\frac{9}{2}}}{9 e^{3}} + \frac{2 \left (d + e x\right )^{\frac{7}{2}} \left (A b e + B a e - 2 B b d\right )}{7 e^{3}} + \frac{2 \left (d + e x\right )^{\frac{5}{2}} \left (A e - B d\right ) \left (a e - b d\right )}{5 e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0989525, size = 70, normalized size = 0.84 \[ \frac{2 (d+e x)^{5/2} \left (9 a e (7 A e-2 B d+5 B e x)+9 A b e (5 e x-2 d)+b B \left (8 d^2-20 d e x+35 e^2 x^2\right )\right )}{315 e^3} \]

Antiderivative was successfully verified.

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

[Out]

(2*(d + e*x)^(5/2)*(9*A*b*e*(-2*d + 5*e*x) + 9*a*e*(-2*B*d + 7*A*e + 5*B*e*x) +
b*B*(8*d^2 - 20*d*e*x + 35*e^2*x^2)))/(315*e^3)

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Maple [A]  time = 0.005, size = 73, normalized size = 0.9 \[{\frac{70\,bB{x}^{2}{e}^{2}+90\,Ab{e}^{2}x+90\,Ba{e}^{2}x-40\,Bbdex+126\,aA{e}^{2}-36\,Abde-36\,Bade+16\,bB{d}^{2}}{315\,{e}^{3}} \left ( ex+d \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/315*(e*x+d)^(5/2)*(35*B*b*e^2*x^2+45*A*b*e^2*x+45*B*a*e^2*x-20*B*b*d*e*x+63*A*
a*e^2-18*A*b*d*e-18*B*a*d*e+8*B*b*d^2)/e^3

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Maxima [A]  time = 1.35445, size = 101, normalized size = 1.22 \[ \frac{2 \,{\left (35 \,{\left (e x + d\right )}^{\frac{9}{2}} B b - 45 \,{\left (2 \, B b d -{\left (B a + A b\right )} e\right )}{\left (e x + d\right )}^{\frac{7}{2}} + 63 \,{\left (B b d^{2} + A a e^{2} -{\left (B a + A b\right )} d e\right )}{\left (e x + d\right )}^{\frac{5}{2}}\right )}}{315 \, e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/315*(35*(e*x + d)^(9/2)*B*b - 45*(2*B*b*d - (B*a + A*b)*e)*(e*x + d)^(7/2) + 6
3*(B*b*d^2 + A*a*e^2 - (B*a + A*b)*d*e)*(e*x + d)^(5/2))/e^3

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Fricas [A]  time = 0.220094, size = 201, normalized size = 2.42 \[ \frac{2 \,{\left (35 \, B b e^{4} x^{4} + 8 \, B b d^{4} + 63 \, A a d^{2} e^{2} - 18 \,{\left (B a + A b\right )} d^{3} e + 5 \,{\left (10 \, B b d e^{3} + 9 \,{\left (B a + A b\right )} e^{4}\right )} x^{3} + 3 \,{\left (B b d^{2} e^{2} + 21 \, A a e^{4} + 24 \,{\left (B a + A b\right )} d e^{3}\right )} x^{2} -{\left (4 \, B b d^{3} e - 126 \, A a d e^{3} - 9 \,{\left (B a + A b\right )} d^{2} e^{2}\right )} x\right )} \sqrt{e x + d}}{315 \, e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/315*(35*B*b*e^4*x^4 + 8*B*b*d^4 + 63*A*a*d^2*e^2 - 18*(B*a + A*b)*d^3*e + 5*(1
0*B*b*d*e^3 + 9*(B*a + A*b)*e^4)*x^3 + 3*(B*b*d^2*e^2 + 21*A*a*e^4 + 24*(B*a + A
*b)*d*e^3)*x^2 - (4*B*b*d^3*e - 126*A*a*d*e^3 - 9*(B*a + A*b)*d^2*e^2)*x)*sqrt(e
*x + d)/e^3

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Sympy [A]  time = 8.04398, size = 318, normalized size = 3.83 \[ A a d \left (\begin{cases} \sqrt{d} x & \text{for}\: e = 0 \\\frac{2 \left (d + e x\right )^{\frac{3}{2}}}{3 e} & \text{otherwise} \end{cases}\right ) + \frac{2 A a \left (- \frac{d \left (d + e x\right )^{\frac{3}{2}}}{3} + \frac{\left (d + e x\right )^{\frac{5}{2}}}{5}\right )}{e} + \frac{2 A b d \left (- \frac{d \left (d + e x\right )^{\frac{3}{2}}}{3} + \frac{\left (d + e x\right )^{\frac{5}{2}}}{5}\right )}{e^{2}} + \frac{2 A b \left (\frac{d^{2} \left (d + e x\right )^{\frac{3}{2}}}{3} - \frac{2 d \left (d + e x\right )^{\frac{5}{2}}}{5} + \frac{\left (d + e x\right )^{\frac{7}{2}}}{7}\right )}{e^{2}} + \frac{2 B a d \left (- \frac{d \left (d + e x\right )^{\frac{3}{2}}}{3} + \frac{\left (d + e x\right )^{\frac{5}{2}}}{5}\right )}{e^{2}} + \frac{2 B a \left (\frac{d^{2} \left (d + e x\right )^{\frac{3}{2}}}{3} - \frac{2 d \left (d + e x\right )^{\frac{5}{2}}}{5} + \frac{\left (d + e x\right )^{\frac{7}{2}}}{7}\right )}{e^{2}} + \frac{2 B b d \left (\frac{d^{2} \left (d + e x\right )^{\frac{3}{2}}}{3} - \frac{2 d \left (d + e x\right )^{\frac{5}{2}}}{5} + \frac{\left (d + e x\right )^{\frac{7}{2}}}{7}\right )}{e^{3}} + \frac{2 B b \left (- \frac{d^{3} \left (d + e x\right )^{\frac{3}{2}}}{3} + \frac{3 d^{2} \left (d + e x\right )^{\frac{5}{2}}}{5} - \frac{3 d \left (d + e x\right )^{\frac{7}{2}}}{7} + \frac{\left (d + e x\right )^{\frac{9}{2}}}{9}\right )}{e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*a*d*Piecewise((sqrt(d)*x, Eq(e, 0)), (2*(d + e*x)**(3/2)/(3*e), True)) + 2*A*a
*(-d*(d + e*x)**(3/2)/3 + (d + e*x)**(5/2)/5)/e + 2*A*b*d*(-d*(d + e*x)**(3/2)/3
+ (d + e*x)**(5/2)/5)/e**2 + 2*A*b*(d**2*(d + e*x)**(3/2)/3 - 2*d*(d + e*x)**(5
/2)/5 + (d + e*x)**(7/2)/7)/e**2 + 2*B*a*d*(-d*(d + e*x)**(3/2)/3 + (d + e*x)**(
5/2)/5)/e**2 + 2*B*a*(d**2*(d + e*x)**(3/2)/3 - 2*d*(d + e*x)**(5/2)/5 + (d + e*
x)**(7/2)/7)/e**2 + 2*B*b*d*(d**2*(d + e*x)**(3/2)/3 - 2*d*(d + e*x)**(5/2)/5 +
(d + e*x)**(7/2)/7)/e**3 + 2*B*b*(-d**3*(d + e*x)**(3/2)/3 + 3*d**2*(d + e*x)**(
5/2)/5 - 3*d*(d + e*x)**(7/2)/7 + (d + e*x)**(9/2)/9)/e**3

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GIAC/XCAS [A]  time = 0.216147, size = 412, normalized size = 4.96 \[ \frac{2}{315} \,{\left (21 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 5 \,{\left (x e + d\right )}^{\frac{3}{2}} d\right )} B a d e^{\left (-1\right )} + 21 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 5 \,{\left (x e + d\right )}^{\frac{3}{2}} d\right )} A b d e^{\left (-1\right )} + 3 \,{\left (15 \,{\left (x e + d\right )}^{\frac{7}{2}} e^{12} - 42 \,{\left (x e + d\right )}^{\frac{5}{2}} d e^{12} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2} e^{12}\right )} B b d e^{\left (-14\right )} + 105 \,{\left (x e + d\right )}^{\frac{3}{2}} A a d + 3 \,{\left (15 \,{\left (x e + d\right )}^{\frac{7}{2}} e^{12} - 42 \,{\left (x e + d\right )}^{\frac{5}{2}} d e^{12} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2} e^{12}\right )} B a e^{\left (-13\right )} + 3 \,{\left (15 \,{\left (x e + d\right )}^{\frac{7}{2}} e^{12} - 42 \,{\left (x e + d\right )}^{\frac{5}{2}} d e^{12} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2} e^{12}\right )} A b e^{\left (-13\right )} +{\left (35 \,{\left (x e + d\right )}^{\frac{9}{2}} e^{24} - 135 \,{\left (x e + d\right )}^{\frac{7}{2}} d e^{24} + 189 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{2} e^{24} - 105 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{3} e^{24}\right )} B b e^{\left (-26\right )} + 21 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 5 \,{\left (x e + d\right )}^{\frac{3}{2}} d\right )} A a\right )} e^{\left (-1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/315*(21*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*B*a*d*e^(-1) + 21*(3*(x*e +
d)^(5/2) - 5*(x*e + d)^(3/2)*d)*A*b*d*e^(-1) + 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)*B*b*d*e^(-14) + 105*(x*e +
d)^(3/2)*A*a*d + 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)*B*a*e^(-13) + 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*e^(-13) + (35*(x*e + d)^(9/2)*e^2
4 - 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*e^(-26) + 21*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*A*a)*e^
(-1)