3.1721 \(\int \frac{(a+b x)^3 (A+B x)}{\sqrt{d+e x}} \, dx\)

Optimal. Leaf size=171 \[ -\frac{2 b^2 (d+e x)^{7/2} (-3 a B e-A b e+4 b B d)}{7 e^5}+\frac{6 b (d+e x)^{5/2} (b d-a e) (-a B e-A b e+2 b B d)}{5 e^5}-\frac{2 (d+e x)^{3/2} (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{3 e^5}+\frac{2 \sqrt{d+e x} (b d-a e)^3 (B d-A e)}{e^5}+\frac{2 b^3 B (d+e x)^{9/2}}{9 e^5} \]

[Out]

(2*(b*d - a*e)^3*(B*d - A*e)*Sqrt[d + e*x])/e^5 - (2*(b*d - a*e)^2*(4*b*B*d - 3*
A*b*e - a*B*e)*(d + e*x)^(3/2))/(3*e^5) + (6*b*(b*d - a*e)*(2*b*B*d - A*b*e - a*
B*e)*(d + e*x)^(5/2))/(5*e^5) - (2*b^2*(4*b*B*d - A*b*e - 3*a*B*e)*(d + e*x)^(7/
2))/(7*e^5) + (2*b^3*B*(d + e*x)^(9/2))/(9*e^5)

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Rubi [A]  time = 0.209443, antiderivative size = 171, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045 \[ -\frac{2 b^2 (d+e x)^{7/2} (-3 a B e-A b e+4 b B d)}{7 e^5}+\frac{6 b (d+e x)^{5/2} (b d-a e) (-a B e-A b e+2 b B d)}{5 e^5}-\frac{2 (d+e x)^{3/2} (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{3 e^5}+\frac{2 \sqrt{d+e x} (b d-a e)^3 (B d-A e)}{e^5}+\frac{2 b^3 B (d+e x)^{9/2}}{9 e^5} \]

Antiderivative was successfully verified.

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

[Out]

(2*(b*d - a*e)^3*(B*d - A*e)*Sqrt[d + e*x])/e^5 - (2*(b*d - a*e)^2*(4*b*B*d - 3*
A*b*e - a*B*e)*(d + e*x)^(3/2))/(3*e^5) + (6*b*(b*d - a*e)*(2*b*B*d - A*b*e - a*
B*e)*(d + e*x)^(5/2))/(5*e^5) - (2*b^2*(4*b*B*d - A*b*e - 3*a*B*e)*(d + e*x)^(7/
2))/(7*e^5) + (2*b^3*B*(d + e*x)^(9/2))/(9*e^5)

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Rubi in Sympy [A]  time = 43.7818, size = 168, normalized size = 0.98 \[ \frac{2 B b^{3} \left (d + e x\right )^{\frac{9}{2}}}{9 e^{5}} + \frac{2 b^{2} \left (d + e x\right )^{\frac{7}{2}} \left (A b e + 3 B a e - 4 B b d\right )}{7 e^{5}} + \frac{6 b \left (d + e x\right )^{\frac{5}{2}} \left (a e - b d\right ) \left (A b e + B a e - 2 B b d\right )}{5 e^{5}} + \frac{2 \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right )^{2} \left (3 A b e + B a e - 4 B b d\right )}{3 e^{5}} + \frac{2 \sqrt{d + e x} \left (A e - B d\right ) \left (a e - b d\right )^{3}}{e^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**3*(B*x+A)/(e*x+d)**(1/2),x)

[Out]

2*B*b**3*(d + e*x)**(9/2)/(9*e**5) + 2*b**2*(d + e*x)**(7/2)*(A*b*e + 3*B*a*e -
4*B*b*d)/(7*e**5) + 6*b*(d + e*x)**(5/2)*(a*e - b*d)*(A*b*e + B*a*e - 2*B*b*d)/(
5*e**5) + 2*(d + e*x)**(3/2)*(a*e - b*d)**2*(3*A*b*e + B*a*e - 4*B*b*d)/(3*e**5)
+ 2*sqrt(d + e*x)*(A*e - B*d)*(a*e - b*d)**3/e**5

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Mathematica [A]  time = 0.233443, size = 226, normalized size = 1.32 \[ \frac{2 \sqrt{d+e x} \left (105 a^3 e^3 (3 A e-2 B d+B e x)+63 a^2 b e^2 \left (5 A e (e x-2 d)+B \left (8 d^2-4 d e x+3 e^2 x^2\right )\right )-9 a b^2 e \left (3 B \left (16 d^3-8 d^2 e x+6 d e^2 x^2-5 e^3 x^3\right )-7 A e \left (8 d^2-4 d e x+3 e^2 x^2\right )\right )+b^3 \left (9 A e \left (-16 d^3+8 d^2 e x-6 d e^2 x^2+5 e^3 x^3\right )+B \left (128 d^4-64 d^3 e x+48 d^2 e^2 x^2-40 d e^3 x^3+35 e^4 x^4\right )\right )\right )}{315 e^5} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[d + e*x]*(105*a^3*e^3*(-2*B*d + 3*A*e + B*e*x) + 63*a^2*b*e^2*(5*A*e*(-2
*d + e*x) + B*(8*d^2 - 4*d*e*x + 3*e^2*x^2)) - 9*a*b^2*e*(-7*A*e*(8*d^2 - 4*d*e*
x + 3*e^2*x^2) + 3*B*(16*d^3 - 8*d^2*e*x + 6*d*e^2*x^2 - 5*e^3*x^3)) + b^3*(9*A*
e*(-16*d^3 + 8*d^2*e*x - 6*d*e^2*x^2 + 5*e^3*x^3) + B*(128*d^4 - 64*d^3*e*x + 48
*d^2*e^2*x^2 - 40*d*e^3*x^3 + 35*e^4*x^4))))/(315*e^5)

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Maple [A]  time = 0.011, size = 301, normalized size = 1.8 \[{\frac{70\,B{b}^{3}{x}^{4}{e}^{4}+90\,A{b}^{3}{e}^{4}{x}^{3}+270\,Ba{b}^{2}{e}^{4}{x}^{3}-80\,B{b}^{3}d{e}^{3}{x}^{3}+378\,Aa{b}^{2}{e}^{4}{x}^{2}-108\,A{b}^{3}d{e}^{3}{x}^{2}+378\,B{a}^{2}b{e}^{4}{x}^{2}-324\,Ba{b}^{2}d{e}^{3}{x}^{2}+96\,B{b}^{3}{d}^{2}{e}^{2}{x}^{2}+630\,A{a}^{2}b{e}^{4}x-504\,Aa{b}^{2}d{e}^{3}x+144\,A{b}^{3}{d}^{2}{e}^{2}x+210\,B{a}^{3}{e}^{4}x-504\,B{a}^{2}bd{e}^{3}x+432\,Ba{b}^{2}{d}^{2}{e}^{2}x-128\,B{b}^{3}{d}^{3}ex+630\,{a}^{3}A{e}^{4}-1260\,A{a}^{2}bd{e}^{3}+1008\,Aa{b}^{2}{d}^{2}{e}^{2}-288\,A{b}^{3}{d}^{3}e-420\,B{a}^{3}d{e}^{3}+1008\,B{a}^{2}b{d}^{2}{e}^{2}-864\,Ba{b}^{2}{d}^{3}e+256\,B{b}^{3}{d}^{4}}{315\,{e}^{5}}\sqrt{ex+d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^3*(B*x+A)/(e*x+d)^(1/2),x)

[Out]

2/315*(e*x+d)^(1/2)*(35*B*b^3*e^4*x^4+45*A*b^3*e^4*x^3+135*B*a*b^2*e^4*x^3-40*B*
b^3*d*e^3*x^3+189*A*a*b^2*e^4*x^2-54*A*b^3*d*e^3*x^2+189*B*a^2*b*e^4*x^2-162*B*a
*b^2*d*e^3*x^2+48*B*b^3*d^2*e^2*x^2+315*A*a^2*b*e^4*x-252*A*a*b^2*d*e^3*x+72*A*b
^3*d^2*e^2*x+105*B*a^3*e^4*x-252*B*a^2*b*d*e^3*x+216*B*a*b^2*d^2*e^2*x-64*B*b^3*
d^3*e*x+315*A*a^3*e^4-630*A*a^2*b*d*e^3+504*A*a*b^2*d^2*e^2-144*A*b^3*d^3*e-210*
B*a^3*d*e^3+504*B*a^2*b*d^2*e^2-432*B*a*b^2*d^3*e+128*B*b^3*d^4)/e^5

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Maxima [A]  time = 1.37529, size = 358, normalized size = 2.09 \[ \frac{2 \,{\left (35 \,{\left (e x + d\right )}^{\frac{9}{2}} B b^{3} - 45 \,{\left (4 \, B b^{3} d -{\left (3 \, B a b^{2} + A b^{3}\right )} e\right )}{\left (e x + d\right )}^{\frac{7}{2}} + 189 \,{\left (2 \, B b^{3} d^{2} -{\left (3 \, B a b^{2} + A b^{3}\right )} d e +{\left (B a^{2} b + A a b^{2}\right )} e^{2}\right )}{\left (e x + d\right )}^{\frac{5}{2}} - 105 \,{\left (4 \, B b^{3} d^{3} - 3 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e + 6 \,{\left (B a^{2} b + A a b^{2}\right )} d e^{2} -{\left (B a^{3} + 3 \, A a^{2} b\right )} e^{3}\right )}{\left (e x + d\right )}^{\frac{3}{2}} + 315 \,{\left (B b^{3} d^{4} + A a^{3} e^{4} -{\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e + 3 \,{\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} -{\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3}\right )} \sqrt{e x + d}\right )}}{315 \, e^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/315*(35*(e*x + d)^(9/2)*B*b^3 - 45*(4*B*b^3*d - (3*B*a*b^2 + A*b^3)*e)*(e*x +
d)^(7/2) + 189*(2*B*b^3*d^2 - (3*B*a*b^2 + A*b^3)*d*e + (B*a^2*b + A*a*b^2)*e^2)
*(e*x + d)^(5/2) - 105*(4*B*b^3*d^3 - 3*(3*B*a*b^2 + A*b^3)*d^2*e + 6*(B*a^2*b +
A*a*b^2)*d*e^2 - (B*a^3 + 3*A*a^2*b)*e^3)*(e*x + d)^(3/2) + 315*(B*b^3*d^4 + A*
a^3*e^4 - (3*B*a*b^2 + A*b^3)*d^3*e + 3*(B*a^2*b + A*a*b^2)*d^2*e^2 - (B*a^3 + 3
*A*a^2*b)*d*e^3)*sqrt(e*x + d))/e^5

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Fricas [A]  time = 0.23031, size = 355, normalized size = 2.08 \[ \frac{2 \,{\left (35 \, B b^{3} e^{4} x^{4} + 128 \, B b^{3} d^{4} + 315 \, A a^{3} e^{4} - 144 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e + 504 \,{\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} - 210 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3} - 5 \,{\left (8 \, B b^{3} d e^{3} - 9 \,{\left (3 \, B a b^{2} + A b^{3}\right )} e^{4}\right )} x^{3} + 3 \,{\left (16 \, B b^{3} d^{2} e^{2} - 18 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d e^{3} + 63 \,{\left (B a^{2} b + A a b^{2}\right )} e^{4}\right )} x^{2} -{\left (64 \, B b^{3} d^{3} e - 72 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{2} + 252 \,{\left (B a^{2} b + A a b^{2}\right )} d e^{3} - 105 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} e^{4}\right )} x\right )} \sqrt{e x + d}}{315 \, e^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/315*(35*B*b^3*e^4*x^4 + 128*B*b^3*d^4 + 315*A*a^3*e^4 - 144*(3*B*a*b^2 + A*b^3
)*d^3*e + 504*(B*a^2*b + A*a*b^2)*d^2*e^2 - 210*(B*a^3 + 3*A*a^2*b)*d*e^3 - 5*(8
*B*b^3*d*e^3 - 9*(3*B*a*b^2 + A*b^3)*e^4)*x^3 + 3*(16*B*b^3*d^2*e^2 - 18*(3*B*a*
b^2 + A*b^3)*d*e^3 + 63*(B*a^2*b + A*a*b^2)*e^4)*x^2 - (64*B*b^3*d^3*e - 72*(3*B
*a*b^2 + A*b^3)*d^2*e^2 + 252*(B*a^2*b + A*a*b^2)*d*e^3 - 105*(B*a^3 + 3*A*a^2*b
)*e^4)*x)*sqrt(e*x + d)/e^5

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Sympy [A]  time = 58.8297, size = 916, normalized size = 5.36 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**3*(B*x+A)/(e*x+d)**(1/2),x)

[Out]

Piecewise((-(2*A*a**3*d/sqrt(d + e*x) + 2*A*a**3*(-d/sqrt(d + e*x) - sqrt(d + e*
x)) + 6*A*a**2*b*d*(-d/sqrt(d + e*x) - sqrt(d + e*x))/e + 6*A*a**2*b*(d**2/sqrt(
d + e*x) + 2*d*sqrt(d + e*x) - (d + e*x)**(3/2)/3)/e + 6*A*a*b**2*d*(d**2/sqrt(d
+ e*x) + 2*d*sqrt(d + e*x) - (d + e*x)**(3/2)/3)/e**2 + 6*A*a*b**2*(-d**3/sqrt(
d + e*x) - 3*d**2*sqrt(d + e*x) + d*(d + e*x)**(3/2) - (d + e*x)**(5/2)/5)/e**2
+ 2*A*b**3*d*(-d**3/sqrt(d + e*x) - 3*d**2*sqrt(d + e*x) + d*(d + e*x)**(3/2) -
(d + e*x)**(5/2)/5)/e**3 + 2*A*b**3*(d**4/sqrt(d + e*x) + 4*d**3*sqrt(d + e*x) -
2*d**2*(d + e*x)**(3/2) + 4*d*(d + e*x)**(5/2)/5 - (d + e*x)**(7/2)/7)/e**3 + 2
*B*a**3*d*(-d/sqrt(d + e*x) - sqrt(d + e*x))/e + 2*B*a**3*(d**2/sqrt(d + e*x) +
2*d*sqrt(d + e*x) - (d + e*x)**(3/2)/3)/e + 6*B*a**2*b*d*(d**2/sqrt(d + e*x) + 2
*d*sqrt(d + e*x) - (d + e*x)**(3/2)/3)/e**2 + 6*B*a**2*b*(-d**3/sqrt(d + e*x) -
3*d**2*sqrt(d + e*x) + d*(d + e*x)**(3/2) - (d + e*x)**(5/2)/5)/e**2 + 6*B*a*b**
2*d*(-d**3/sqrt(d + e*x) - 3*d**2*sqrt(d + e*x) + d*(d + e*x)**(3/2) - (d + e*x)
**(5/2)/5)/e**3 + 6*B*a*b**2*(d**4/sqrt(d + e*x) + 4*d**3*sqrt(d + e*x) - 2*d**2
*(d + e*x)**(3/2) + 4*d*(d + e*x)**(5/2)/5 - (d + e*x)**(7/2)/7)/e**3 + 2*B*b**3
*d*(d**4/sqrt(d + e*x) + 4*d**3*sqrt(d + e*x) - 2*d**2*(d + e*x)**(3/2) + 4*d*(d
+ e*x)**(5/2)/5 - (d + e*x)**(7/2)/7)/e**4 + 2*B*b**3*(-d**5/sqrt(d + e*x) - 5*
d**4*sqrt(d + e*x) + 10*d**3*(d + e*x)**(3/2)/3 - 2*d**2*(d + e*x)**(5/2) + 5*d*
(d + e*x)**(7/2)/7 - (d + e*x)**(9/2)/9)/e**4)/e, Ne(e, 0)), ((A*a**3*x + B*b**3
*x**5/5 + x**4*(A*b**3 + 3*B*a*b**2)/4 + x**3*(3*A*a*b**2 + 3*B*a**2*b)/3 + x**2
*(3*A*a**2*b + B*a**3)/2)/sqrt(d), True))

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GIAC/XCAS [A]  time = 0.2154, size = 517, normalized size = 3.02 \[ \frac{2}{315} \,{\left (105 \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} - 3 \, \sqrt{x e + d} d\right )} B a^{3} e^{\left (-1\right )} + 315 \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} - 3 \, \sqrt{x e + d} d\right )} A a^{2} b e^{\left (-1\right )} + 63 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} e^{8} - 10 \,{\left (x e + d\right )}^{\frac{3}{2}} d e^{8} + 15 \, \sqrt{x e + d} d^{2} e^{8}\right )} B a^{2} b e^{\left (-10\right )} + 63 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} e^{8} - 10 \,{\left (x e + d\right )}^{\frac{3}{2}} d e^{8} + 15 \, \sqrt{x e + d} d^{2} e^{8}\right )} A a b^{2} e^{\left (-10\right )} + 27 \,{\left (5 \,{\left (x e + d\right )}^{\frac{7}{2}} e^{18} - 21 \,{\left (x e + d\right )}^{\frac{5}{2}} d e^{18} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2} e^{18} - 35 \, \sqrt{x e + d} d^{3} e^{18}\right )} B a b^{2} e^{\left (-21\right )} + 9 \,{\left (5 \,{\left (x e + d\right )}^{\frac{7}{2}} e^{18} - 21 \,{\left (x e + d\right )}^{\frac{5}{2}} d e^{18} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2} e^{18} - 35 \, \sqrt{x e + d} d^{3} e^{18}\right )} A b^{3} e^{\left (-21\right )} +{\left (35 \,{\left (x e + d\right )}^{\frac{9}{2}} e^{32} - 180 \,{\left (x e + d\right )}^{\frac{7}{2}} d e^{32} + 378 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{2} e^{32} - 420 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{3} e^{32} + 315 \, \sqrt{x e + d} d^{4} e^{32}\right )} B b^{3} e^{\left (-36\right )} + 315 \, \sqrt{x e + d} A a^{3}\right )} e^{\left (-1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/315*(105*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*B*a^3*e^(-1) + 315*((x*e + d)^(
3/2) - 3*sqrt(x*e + d)*d)*A*a^2*b*e^(-1) + 63*(3*(x*e + d)^(5/2)*e^8 - 10*(x*e +
d)^(3/2)*d*e^8 + 15*sqrt(x*e + d)*d^2*e^8)*B*a^2*b*e^(-10) + 63*(3*(x*e + d)^(5
/2)*e^8 - 10*(x*e + d)^(3/2)*d*e^8 + 15*sqrt(x*e + d)*d^2*e^8)*A*a*b^2*e^(-10) +
27*(5*(x*e + d)^(7/2)*e^18 - 21*(x*e + d)^(5/2)*d*e^18 + 35*(x*e + d)^(3/2)*d^2
*e^18 - 35*sqrt(x*e + d)*d^3*e^18)*B*a*b^2*e^(-21) + 9*(5*(x*e + d)^(7/2)*e^18 -
21*(x*e + d)^(5/2)*d*e^18 + 35*(x*e + d)^(3/2)*d^2*e^18 - 35*sqrt(x*e + d)*d^3*
e^18)*A*b^3*e^(-21) + (35*(x*e + d)^(9/2)*e^32 - 180*(x*e + d)^(7/2)*d*e^32 + 37
8*(x*e + d)^(5/2)*d^2*e^32 - 420*(x*e + d)^(3/2)*d^3*e^32 + 315*sqrt(x*e + d)*d^
4*e^32)*B*b^3*e^(-36) + 315*sqrt(x*e + d)*A*a^3)*e^(-1)