3.1722 \(\int \frac{(a+b x)^3 (A+B x)}{(d+e x)^{3/2}} \, dx\)

Optimal. Leaf size=167 \[ -\frac{2 b^2 (d+e x)^{5/2} (-3 a B e-A b e+4 b B d)}{5 e^5}+\frac{2 b (d+e x)^{3/2} (b d-a e) (-a B e-A b e+2 b B d)}{e^5}-\frac{2 \sqrt{d+e x} (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{e^5}-\frac{2 (b d-a e)^3 (B d-A e)}{e^5 \sqrt{d+e x}}+\frac{2 b^3 B (d+e x)^{7/2}}{7 e^5} \]

[Out]

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

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Rubi [A]  time = 0.217848, antiderivative size = 167, 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 \[ -\frac{2 b^2 (d+e x)^{5/2} (-3 a B e-A b e+4 b B d)}{5 e^5}+\frac{2 b (d+e x)^{3/2} (b d-a e) (-a B e-A b e+2 b B d)}{e^5}-\frac{2 \sqrt{d+e x} (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{e^5}-\frac{2 (b d-a e)^3 (B d-A e)}{e^5 \sqrt{d+e x}}+\frac{2 b^3 B (d+e x)^{7/2}}{7 e^5} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 43.1489, size = 165, normalized size = 0.99 \[ \frac{2 B b^{3} \left (d + e x\right )^{\frac{7}{2}}}{7 e^{5}} + \frac{2 b^{2} \left (d + e x\right )^{\frac{5}{2}} \left (A b e + 3 B a e - 4 B b d\right )}{5 e^{5}} + \frac{2 b \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right ) \left (A b e + B a e - 2 B b d\right )}{e^{5}} + \frac{2 \sqrt{d + e x} \left (a e - b d\right )^{2} \left (3 A b e + B a e - 4 B b d\right )}{e^{5}} - \frac{2 \left (A e - B d\right ) \left (a e - b d\right )^{3}}{e^{5} \sqrt{d + e x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.247825, size = 222, normalized size = 1.33 \[ \frac{2 \left (35 a^3 e^3 (-A e+2 B d+B e x)+35 a^2 b e^2 \left (3 A e (2 d+e x)+B \left (-8 d^2-4 d e x+e^2 x^2\right )\right )+7 a b^2 e \left (5 A e \left (-8 d^2-4 d e x+e^2 x^2\right )+3 B \left (16 d^3+8 d^2 e x-2 d e^2 x^2+e^3 x^3\right )\right )+b^3 \left (7 A e \left (16 d^3+8 d^2 e x-2 d e^2 x^2+e^3 x^3\right )+B \left (-128 d^4-64 d^3 e x+16 d^2 e^2 x^2-8 d e^3 x^3+5 e^4 x^4\right )\right )\right )}{35 e^5 \sqrt{d+e x}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(35*a^3*e^3*(2*B*d - A*e + B*e*x) + 35*a^2*b*e^2*(3*A*e*(2*d + e*x) + B*(-8*d
^2 - 4*d*e*x + e^2*x^2)) + 7*a*b^2*e*(5*A*e*(-8*d^2 - 4*d*e*x + e^2*x^2) + 3*B*(
16*d^3 + 8*d^2*e*x - 2*d*e^2*x^2 + e^3*x^3)) + b^3*(7*A*e*(16*d^3 + 8*d^2*e*x -
2*d*e^2*x^2 + e^3*x^3) + B*(-128*d^4 - 64*d^3*e*x + 16*d^2*e^2*x^2 - 8*d*e^3*x^3
+ 5*e^4*x^4))))/(35*e^5*Sqrt[d + e*x])

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Maple [A]  time = 0.013, size = 301, normalized size = 1.8 \[ -{\frac{-10\,B{b}^{3}{x}^{4}{e}^{4}-14\,A{b}^{3}{e}^{4}{x}^{3}-42\,Ba{b}^{2}{e}^{4}{x}^{3}+16\,B{b}^{3}d{e}^{3}{x}^{3}-70\,Aa{b}^{2}{e}^{4}{x}^{2}+28\,A{b}^{3}d{e}^{3}{x}^{2}-70\,B{a}^{2}b{e}^{4}{x}^{2}+84\,Ba{b}^{2}d{e}^{3}{x}^{2}-32\,B{b}^{3}{d}^{2}{e}^{2}{x}^{2}-210\,A{a}^{2}b{e}^{4}x+280\,Aa{b}^{2}d{e}^{3}x-112\,A{b}^{3}{d}^{2}{e}^{2}x-70\,B{a}^{3}{e}^{4}x+280\,B{a}^{2}bd{e}^{3}x-336\,Ba{b}^{2}{d}^{2}{e}^{2}x+128\,B{b}^{3}{d}^{3}ex+70\,{a}^{3}A{e}^{4}-420\,A{a}^{2}bd{e}^{3}+560\,Aa{b}^{2}{d}^{2}{e}^{2}-224\,A{b}^{3}{d}^{3}e-140\,B{a}^{3}d{e}^{3}+560\,B{a}^{2}b{d}^{2}{e}^{2}-672\,Ba{b}^{2}{d}^{3}e+256\,B{b}^{3}{d}^{4}}{35\,{e}^{5}}{\frac{1}{\sqrt{ex+d}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/35/(e*x+d)^(1/2)*(-5*B*b^3*e^4*x^4-7*A*b^3*e^4*x^3-21*B*a*b^2*e^4*x^3+8*B*b^3
*d*e^3*x^3-35*A*a*b^2*e^4*x^2+14*A*b^3*d*e^3*x^2-35*B*a^2*b*e^4*x^2+42*B*a*b^2*d
*e^3*x^2-16*B*b^3*d^2*e^2*x^2-105*A*a^2*b*e^4*x+140*A*a*b^2*d*e^3*x-56*A*b^3*d^2
*e^2*x-35*B*a^3*e^4*x+140*B*a^2*b*d*e^3*x-168*B*a*b^2*d^2*e^2*x+64*B*b^3*d^3*e*x
+35*A*a^3*e^4-210*A*a^2*b*d*e^3+280*A*a*b^2*d^2*e^2-112*A*b^3*d^3*e-70*B*a^3*d*e
^3+280*B*a^2*b*d^2*e^2-336*B*a*b^2*d^3*e+128*B*b^3*d^4)/e^5

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Maxima [A]  time = 1.36191, size = 369, normalized size = 2.21 \[ \frac{2 \,{\left (\frac{5 \,{\left (e x + d\right )}^{\frac{7}{2}} B b^{3} - 7 \,{\left (4 \, B b^{3} d -{\left (3 \, B a b^{2} + A b^{3}\right )} e\right )}{\left (e x + d\right )}^{\frac{5}{2}} + 35 \,{\left (2 \, B b^{3} d^{2} -{\left (3 \, B a b^{2} + A b^{3}\right )} d e +{\left (B a^{2} b + A a b^{2}\right )} e^{2}\right )}{\left (e x + d\right )}^{\frac{3}{2}} - 35 \,{\left (4 \, B b^{3} d^{3} - 3 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e + 6 \,{\left (B a^{2} b + A a b^{2}\right )} d e^{2} -{\left (B a^{3} + 3 \, A a^{2} b\right )} e^{3}\right )} \sqrt{e x + d}}{e^{4}} - \frac{35 \,{\left (B b^{3} d^{4} + A a^{3} e^{4} -{\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e + 3 \,{\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} -{\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3}\right )}}{\sqrt{e x + d} e^{4}}\right )}}{35 \, e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/35*((5*(e*x + d)^(7/2)*B*b^3 - 7*(4*B*b^3*d - (3*B*a*b^2 + A*b^3)*e)*(e*x + d)
^(5/2) + 35*(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)^(3/2) - 35*(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)*sqrt(e*x + d))/e^4 - 35*(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^4))/e

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Fricas [A]  time = 0.224013, size = 354, normalized size = 2.12 \[ \frac{2 \,{\left (5 \, B b^{3} e^{4} x^{4} - 128 \, B b^{3} d^{4} - 35 \, A a^{3} e^{4} + 112 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e - 280 \,{\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} + 70 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3} -{\left (8 \, B b^{3} d e^{3} - 7 \,{\left (3 \, B a b^{2} + A b^{3}\right )} e^{4}\right )} x^{3} +{\left (16 \, B b^{3} d^{2} e^{2} - 14 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d e^{3} + 35 \,{\left (B a^{2} b + A a b^{2}\right )} e^{4}\right )} x^{2} -{\left (64 \, B b^{3} d^{3} e - 56 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{2} + 140 \,{\left (B a^{2} b + A a b^{2}\right )} d e^{3} - 35 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} e^{4}\right )} x\right )}}{35 \, \sqrt{e x + d} e^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/35*(5*B*b^3*e^4*x^4 - 128*B*b^3*d^4 - 35*A*a^3*e^4 + 112*(3*B*a*b^2 + A*b^3)*d
^3*e - 280*(B*a^2*b + A*a*b^2)*d^2*e^2 + 70*(B*a^3 + 3*A*a^2*b)*d*e^3 - (8*B*b^3
*d*e^3 - 7*(3*B*a*b^2 + A*b^3)*e^4)*x^3 + (16*B*b^3*d^2*e^2 - 14*(3*B*a*b^2 + A*
b^3)*d*e^3 + 35*(B*a^2*b + A*a*b^2)*e^4)*x^2 - (64*B*b^3*d^3*e - 56*(3*B*a*b^2 +
A*b^3)*d^2*e^2 + 140*(B*a^2*b + A*a*b^2)*d*e^3 - 35*(B*a^3 + 3*A*a^2*b)*e^4)*x)
/(sqrt(e*x + d)*e^5)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (A + B x\right ) \left (a + b x\right )^{3}}{\left (d + e x\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((A + B*x)*(a + b*x)**3/(d + e*x)**(3/2), x)

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GIAC/XCAS [A]  time = 0.215658, size = 514, normalized size = 3.08 \[ \frac{2}{35} \,{\left (5 \,{\left (x e + d\right )}^{\frac{7}{2}} B b^{3} e^{30} - 28 \,{\left (x e + d\right )}^{\frac{5}{2}} B b^{3} d e^{30} + 70 \,{\left (x e + d\right )}^{\frac{3}{2}} B b^{3} d^{2} e^{30} - 140 \, \sqrt{x e + d} B b^{3} d^{3} e^{30} + 21 \,{\left (x e + d\right )}^{\frac{5}{2}} B a b^{2} e^{31} + 7 \,{\left (x e + d\right )}^{\frac{5}{2}} A b^{3} e^{31} - 105 \,{\left (x e + d\right )}^{\frac{3}{2}} B a b^{2} d e^{31} - 35 \,{\left (x e + d\right )}^{\frac{3}{2}} A b^{3} d e^{31} + 315 \, \sqrt{x e + d} B a b^{2} d^{2} e^{31} + 105 \, \sqrt{x e + d} A b^{3} d^{2} e^{31} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} B a^{2} b e^{32} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} A a b^{2} e^{32} - 210 \, \sqrt{x e + d} B a^{2} b d e^{32} - 210 \, \sqrt{x e + d} A a b^{2} d e^{32} + 35 \, \sqrt{x e + d} B a^{3} e^{33} + 105 \, \sqrt{x e + d} A a^{2} b e^{33}\right )} e^{\left (-35\right )} - \frac{2 \,{\left (B b^{3} d^{4} - 3 \, B a b^{2} d^{3} e - A b^{3} d^{3} e + 3 \, B a^{2} b d^{2} e^{2} + 3 \, A a b^{2} d^{2} e^{2} - B a^{3} d e^{3} - 3 \, A a^{2} b d e^{3} + A a^{3} e^{4}\right )} e^{\left (-5\right )}}{\sqrt{x e + d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/35*(5*(x*e + d)^(7/2)*B*b^3*e^30 - 28*(x*e + d)^(5/2)*B*b^3*d*e^30 + 70*(x*e +
d)^(3/2)*B*b^3*d^2*e^30 - 140*sqrt(x*e + d)*B*b^3*d^3*e^30 + 21*(x*e + d)^(5/2)
*B*a*b^2*e^31 + 7*(x*e + d)^(5/2)*A*b^3*e^31 - 105*(x*e + d)^(3/2)*B*a*b^2*d*e^3
1 - 35*(x*e + d)^(3/2)*A*b^3*d*e^31 + 315*sqrt(x*e + d)*B*a*b^2*d^2*e^31 + 105*s
qrt(x*e + d)*A*b^3*d^2*e^31 + 35*(x*e + d)^(3/2)*B*a^2*b*e^32 + 35*(x*e + d)^(3/
2)*A*a*b^2*e^32 - 210*sqrt(x*e + d)*B*a^2*b*d*e^32 - 210*sqrt(x*e + d)*A*a*b^2*d
*e^32 + 35*sqrt(x*e + d)*B*a^3*e^33 + 105*sqrt(x*e + d)*A*a^2*b*e^33)*e^(-35) -
2*(B*b^3*d^4 - 3*B*a*b^2*d^3*e - A*b^3*d^3*e + 3*B*a^2*b*d^2*e^2 + 3*A*a*b^2*d^2
*e^2 - B*a^3*d*e^3 - 3*A*a^2*b*d*e^3 + A*a^3*e^4)*e^(-5)/sqrt(x*e + d)