3.1724 \(\int \frac{(a+b x)^3 (A+B x)}{(d+e x)^{7/2}} \, dx\)

Optimal. Leaf size=169 \[ -\frac{2 b^2 \sqrt{d+e x} (-3 a B e-A b e+4 b B d)}{e^5}-\frac{6 b (b d-a e) (-a B e-A b e+2 b B d)}{e^5 \sqrt{d+e x}}+\frac{2 (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{3 e^5 (d+e x)^{3/2}}-\frac{2 (b d-a e)^3 (B d-A e)}{5 e^5 (d+e x)^{5/2}}+\frac{2 b^3 B (d+e x)^{3/2}}{3 e^5} \]

[Out]

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

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Rubi [A]  time = 0.204597, antiderivative size = 169, 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 \[ -\frac{2 b^2 \sqrt{d+e x} (-3 a B e-A b e+4 b B d)}{e^5}-\frac{6 b (b d-a e) (-a B e-A b e+2 b B d)}{e^5 \sqrt{d+e x}}+\frac{2 (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{3 e^5 (d+e x)^{3/2}}-\frac{2 (b d-a e)^3 (B d-A e)}{5 e^5 (d+e x)^{5/2}}+\frac{2 b^3 B (d+e x)^{3/2}}{3 e^5} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)^3*(A + B*x))/(d + e*x)^(7/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 45.8058, size = 167, normalized size = 0.99 \[ \frac{2 B b^{3} \left (d + e x\right )^{\frac{3}{2}}}{3 e^{5}} + \frac{2 b^{2} \sqrt{d + e x} \left (A b e + 3 B a e - 4 B b d\right )}{e^{5}} - \frac{6 b \left (a e - b d\right ) \left (A b e + B a e - 2 B b d\right )}{e^{5} \sqrt{d + e x}} - \frac{2 \left (a e - b d\right )^{2} \left (3 A b e + B a e - 4 B b d\right )}{3 e^{5} \left (d + e x\right )^{\frac{3}{2}}} - \frac{2 \left (A e - B d\right ) \left (a e - b d\right )^{3}}{5 e^{5} \left (d + e x\right )^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**3*(B*x+A)/(e*x+d)**(7/2),x)

[Out]

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

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Mathematica [A]  time = 0.366143, size = 139, normalized size = 0.82 \[ \frac{2 \sqrt{d+e x} \left (-5 b^2 (-9 a B e-3 A b e+11 b B d)+\frac{45 b (b d-a e) (a B e+A b e-2 b B d)}{d+e x}-\frac{5 (b d-a e)^2 (a B e+3 A b e-4 b B d)}{(d+e x)^2}-\frac{3 (b d-a e)^3 (B d-A e)}{(d+e x)^3}+5 b^3 B e x\right )}{15 e^5} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)^3*(A + B*x))/(d + e*x)^(7/2),x]

[Out]

(2*Sqrt[d + e*x]*(-5*b^2*(11*b*B*d - 3*A*b*e - 9*a*B*e) + 5*b^3*B*e*x - (3*(b*d
- a*e)^3*(B*d - A*e))/(d + e*x)^3 - (5*(b*d - a*e)^2*(-4*b*B*d + 3*A*b*e + a*B*e
))/(d + e*x)^2 + (45*b*(b*d - a*e)*(-2*b*B*d + A*b*e + a*B*e))/(d + e*x)))/(15*e
^5)

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Maple [A]  time = 0.01, size = 301, normalized size = 1.8 \[ -{\frac{-10\,B{b}^{3}{x}^{4}{e}^{4}-30\,A{b}^{3}{e}^{4}{x}^{3}-90\,Ba{b}^{2}{e}^{4}{x}^{3}+80\,B{b}^{3}d{e}^{3}{x}^{3}+90\,Aa{b}^{2}{e}^{4}{x}^{2}-180\,A{b}^{3}d{e}^{3}{x}^{2}+90\,B{a}^{2}b{e}^{4}{x}^{2}-540\,Ba{b}^{2}d{e}^{3}{x}^{2}+480\,B{b}^{3}{d}^{2}{e}^{2}{x}^{2}+30\,A{a}^{2}b{e}^{4}x+120\,Aa{b}^{2}d{e}^{3}x-240\,A{b}^{3}{d}^{2}{e}^{2}x+10\,B{a}^{3}{e}^{4}x+120\,B{a}^{2}bd{e}^{3}x-720\,Ba{b}^{2}{d}^{2}{e}^{2}x+640\,B{b}^{3}{d}^{3}ex+6\,{a}^{3}A{e}^{4}+12\,A{a}^{2}bd{e}^{3}+48\,Aa{b}^{2}{d}^{2}{e}^{2}-96\,A{b}^{3}{d}^{3}e+4\,B{a}^{3}d{e}^{3}+48\,B{a}^{2}b{d}^{2}{e}^{2}-288\,Ba{b}^{2}{d}^{3}e+256\,B{b}^{3}{d}^{4}}{15\,{e}^{5}} \left ( ex+d \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^3*(B*x+A)/(e*x+d)^(7/2),x)

[Out]

-2/15/(e*x+d)^(5/2)*(-5*B*b^3*e^4*x^4-15*A*b^3*e^4*x^3-45*B*a*b^2*e^4*x^3+40*B*b
^3*d*e^3*x^3+45*A*a*b^2*e^4*x^2-90*A*b^3*d*e^3*x^2+45*B*a^2*b*e^4*x^2-270*B*a*b^
2*d*e^3*x^2+240*B*b^3*d^2*e^2*x^2+15*A*a^2*b*e^4*x+60*A*a*b^2*d*e^3*x-120*A*b^3*
d^2*e^2*x+5*B*a^3*e^4*x+60*B*a^2*b*d*e^3*x-360*B*a*b^2*d^2*e^2*x+320*B*b^3*d^3*e
*x+3*A*a^3*e^4+6*A*a^2*b*d*e^3+24*A*a*b^2*d^2*e^2-48*A*b^3*d^3*e+2*B*a^3*d*e^3+2
4*B*a^2*b*d^2*e^2-144*B*a*b^2*d^3*e+128*B*b^3*d^4)/e^5

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Maxima [A]  time = 1.36751, size = 369, normalized size = 2.18 \[ \frac{2 \,{\left (\frac{5 \,{\left ({\left (e x + d\right )}^{\frac{3}{2}} B b^{3} - 3 \,{\left (4 \, B b^{3} d -{\left (3 \, B a b^{2} + A b^{3}\right )} e\right )} \sqrt{e x + d}\right )}}{e^{4}} - \frac{3 \, B b^{3} d^{4} + 3 \, A a^{3} e^{4} - 3 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e + 9 \,{\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} - 3 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3} + 45 \,{\left (2 \, B b^{3} d^{2} -{\left (3 \, B a b^{2} + A b^{3}\right )} d e +{\left (B a^{2} b + A a b^{2}\right )} e^{2}\right )}{\left (e x + d\right )}^{2} - 5 \,{\left (4 \, B b^{3} d^{3} - 3 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e + 6 \,{\left (B a^{2} b + A a b^{2}\right )} d e^{2} -{\left (B a^{3} + 3 \, A a^{2} b\right )} e^{3}\right )}{\left (e x + d\right )}}{{\left (e x + d\right )}^{\frac{5}{2}} e^{4}}\right )}}{15 \, e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^3/(e*x + d)^(7/2),x, algorithm="maxima")

[Out]

2/15*(5*((e*x + d)^(3/2)*B*b^3 - 3*(4*B*b^3*d - (3*B*a*b^2 + A*b^3)*e)*sqrt(e*x
+ d))/e^4 - (3*B*b^3*d^4 + 3*A*a^3*e^4 - 3*(3*B*a*b^2 + A*b^3)*d^3*e + 9*(B*a^2*
b + A*a*b^2)*d^2*e^2 - 3*(B*a^3 + 3*A*a^2*b)*d*e^3 + 45*(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)^2 - 5*(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))/((e*x + d)^(5/2)*e^4))/e

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Fricas [A]  time = 0.232319, size = 382, normalized size = 2.26 \[ \frac{2 \,{\left (5 \, B b^{3} e^{4} x^{4} - 128 \, B b^{3} d^{4} - 3 \, A a^{3} e^{4} + 48 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e - 24 \,{\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} - 2 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3} - 5 \,{\left (8 \, B b^{3} d e^{3} - 3 \,{\left (3 \, B a b^{2} + A b^{3}\right )} e^{4}\right )} x^{3} - 15 \,{\left (16 \, B b^{3} d^{2} e^{2} - 6 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d e^{3} + 3 \,{\left (B a^{2} b + A a b^{2}\right )} e^{4}\right )} x^{2} - 5 \,{\left (64 \, B b^{3} d^{3} e - 24 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{2} + 12 \,{\left (B a^{2} b + A a b^{2}\right )} d e^{3} +{\left (B a^{3} + 3 \, A a^{2} b\right )} e^{4}\right )} x\right )}}{15 \,{\left (e^{7} x^{2} + 2 \, d e^{6} x + d^{2} e^{5}\right )} \sqrt{e x + d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^3/(e*x + d)^(7/2),x, algorithm="fricas")

[Out]

2/15*(5*B*b^3*e^4*x^4 - 128*B*b^3*d^4 - 3*A*a^3*e^4 + 48*(3*B*a*b^2 + A*b^3)*d^3
*e - 24*(B*a^2*b + A*a*b^2)*d^2*e^2 - 2*(B*a^3 + 3*A*a^2*b)*d*e^3 - 5*(8*B*b^3*d
*e^3 - 3*(3*B*a*b^2 + A*b^3)*e^4)*x^3 - 15*(16*B*b^3*d^2*e^2 - 6*(3*B*a*b^2 + A*
b^3)*d*e^3 + 3*(B*a^2*b + A*a*b^2)*e^4)*x^2 - 5*(64*B*b^3*d^3*e - 24*(3*B*a*b^2
+ A*b^3)*d^2*e^2 + 12*(B*a^2*b + A*a*b^2)*d*e^3 + (B*a^3 + 3*A*a^2*b)*e^4)*x)/((
e^7*x^2 + 2*d*e^6*x + d^2*e^5)*sqrt(e*x + d))

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Sympy [A]  time = 13.338, size = 1654, normalized size = 9.79 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**3*(B*x+A)/(e*x+d)**(7/2),x)

[Out]

Piecewise((-6*A*a**3*e**4/(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x
) + 15*e**7*x**2*sqrt(d + e*x)) - 12*A*a**2*b*d*e**3/(15*d**2*e**5*sqrt(d + e*x)
+ 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) - 30*A*a**2*b*e**4*x/
(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d +
e*x)) - 48*A*a*b**2*d**2*e**2/(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d +
e*x) + 15*e**7*x**2*sqrt(d + e*x)) - 120*A*a*b**2*d*e**3*x/(15*d**2*e**5*sqrt(d
+ e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) - 90*A*a*b**2*
e**4*x**2/(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2
*sqrt(d + e*x)) + 96*A*b**3*d**3*e/(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqr
t(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) + 240*A*b**3*d**2*e**2*x/(15*d**2*e**5*
sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) + 180*A*
b**3*d*e**3*x**2/(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e*
*7*x**2*sqrt(d + e*x)) + 30*A*b**3*e**4*x**3/(15*d**2*e**5*sqrt(d + e*x) + 30*d*
e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) - 4*B*a**3*d*e**3/(15*d**2*e*
*5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) - 10*
B*a**3*e**4*x/(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*
x**2*sqrt(d + e*x)) - 48*B*a**2*b*d**2*e**2/(15*d**2*e**5*sqrt(d + e*x) + 30*d*e
**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) - 120*B*a**2*b*d*e**3*x/(15*d*
*2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x))
- 90*B*a**2*b*e**4*x**2/(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x)
+ 15*e**7*x**2*sqrt(d + e*x)) + 288*B*a*b**2*d**3*e/(15*d**2*e**5*sqrt(d + e*x)
+ 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) + 720*B*a*b**2*d**2*e*
*2*x/(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt
(d + e*x)) + 540*B*a*b**2*d*e**3*x**2/(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*
sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) + 90*B*a*b**2*e**4*x**3/(15*d**2*e**
5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) - 256*
B*b**3*d**4/(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x*
*2*sqrt(d + e*x)) - 640*B*b**3*d**3*e*x/(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*
x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) - 480*B*b**3*d**2*e**2*x**2/(15*d*
*2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x))
- 80*B*b**3*d*e**3*x**3/(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x)
+ 15*e**7*x**2*sqrt(d + e*x)) + 10*B*b**3*e**4*x**4/(15*d**2*e**5*sqrt(d + e*x)
+ 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)), Ne(e, 0)), ((A*a**3*x
+ 3*A*a**2*b*x**2/2 + A*a*b**2*x**3 + A*b**3*x**4/4 + B*a**3*x**2/2 + B*a**2*b*
x**3 + 3*B*a*b**2*x**4/4 + B*b**3*x**5/5)/d**(7/2), True))

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GIAC/XCAS [A]  time = 0.215547, size = 491, normalized size = 2.91 \[ \frac{2}{3} \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} B b^{3} e^{10} - 12 \, \sqrt{x e + d} B b^{3} d e^{10} + 9 \, \sqrt{x e + d} B a b^{2} e^{11} + 3 \, \sqrt{x e + d} A b^{3} e^{11}\right )} e^{\left (-15\right )} - \frac{2 \,{\left (90 \,{\left (x e + d\right )}^{2} B b^{3} d^{2} - 20 \,{\left (x e + d\right )} B b^{3} d^{3} + 3 \, B b^{3} d^{4} - 135 \,{\left (x e + d\right )}^{2} B a b^{2} d e - 45 \,{\left (x e + d\right )}^{2} A b^{3} d e + 45 \,{\left (x e + d\right )} B a b^{2} d^{2} e + 15 \,{\left (x e + d\right )} A b^{3} d^{2} e - 9 \, B a b^{2} d^{3} e - 3 \, A b^{3} d^{3} e + 45 \,{\left (x e + d\right )}^{2} B a^{2} b e^{2} + 45 \,{\left (x e + d\right )}^{2} A a b^{2} e^{2} - 30 \,{\left (x e + d\right )} B a^{2} b d e^{2} - 30 \,{\left (x e + d\right )} A a b^{2} d e^{2} + 9 \, B a^{2} b d^{2} e^{2} + 9 \, A a b^{2} d^{2} e^{2} + 5 \,{\left (x e + d\right )} B a^{3} e^{3} + 15 \,{\left (x e + d\right )} A a^{2} b e^{3} - 3 \, B a^{3} d e^{3} - 9 \, A a^{2} b d e^{3} + 3 \, A a^{3} e^{4}\right )} e^{\left (-5\right )}}{15 \,{\left (x e + d\right )}^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^3/(e*x + d)^(7/2),x, algorithm="giac")

[Out]

2/3*((x*e + d)^(3/2)*B*b^3*e^10 - 12*sqrt(x*e + d)*B*b^3*d*e^10 + 9*sqrt(x*e + d
)*B*a*b^2*e^11 + 3*sqrt(x*e + d)*A*b^3*e^11)*e^(-15) - 2/15*(90*(x*e + d)^2*B*b^
3*d^2 - 20*(x*e + d)*B*b^3*d^3 + 3*B*b^3*d^4 - 135*(x*e + d)^2*B*a*b^2*d*e - 45*
(x*e + d)^2*A*b^3*d*e + 45*(x*e + d)*B*a*b^2*d^2*e + 15*(x*e + d)*A*b^3*d^2*e -
9*B*a*b^2*d^3*e - 3*A*b^3*d^3*e + 45*(x*e + d)^2*B*a^2*b*e^2 + 45*(x*e + d)^2*A*
a*b^2*e^2 - 30*(x*e + d)*B*a^2*b*d*e^2 - 30*(x*e + d)*A*a*b^2*d*e^2 + 9*B*a^2*b*
d^2*e^2 + 9*A*a*b^2*d^2*e^2 + 5*(x*e + d)*B*a^3*e^3 + 15*(x*e + d)*A*a^2*b*e^3 -
3*B*a^3*d*e^3 - 9*A*a^2*b*d*e^3 + 3*A*a^3*e^4)*e^(-5)/(x*e + d)^(5/2)