3.1741 \(\int \frac{(a+b x)^3 (A+B x)}{(d+e x)^{7/2}} \, dx\)

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 = 0.0665374, 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, Rules used = {77} \[ -\frac{2 b^2 \sqrt{d+e x} (-3 a B e-A b e+4 b B d)}{e^5}-\frac{6 b (b d-a e) (-a B e-A b e+2 b B d)}{e^5 \sqrt{d+e x}}+\frac{2 (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{3 e^5 (d+e x)^{3/2}}-\frac{2 (b d-a e)^3 (B d-A e)}{5 e^5 (d+e x)^{5/2}}+\frac{2 b^3 B (d+e x)^{3/2}}{3 e^5} \]

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)

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

\begin{align*} \int \frac{(a+b x)^3 (A+B x)}{(d+e x)^{7/2}} \, dx &=\int \left (\frac{(-b d+a e)^3 (-B d+A e)}{e^4 (d+e x)^{7/2}}+\frac{(-b d+a e)^2 (-4 b B d+3 A b e+a B e)}{e^4 (d+e x)^{5/2}}-\frac{3 b (b d-a e) (-2 b B d+A b e+a B e)}{e^4 (d+e x)^{3/2}}+\frac{b^2 (-4 b B d+A b e+3 a B e)}{e^4 \sqrt{d+e x}}+\frac{b^3 B \sqrt{d+e x}}{e^4}\right ) \, dx\\ &=-\frac{2 (b d-a e)^3 (B d-A e)}{5 e^5 (d+e x)^{5/2}}+\frac{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}}-\frac{6 b (b d-a e) (2 b B d-A b e-a B e)}{e^5 \sqrt{d+e x}}-\frac{2 b^2 (4 b B d-A b e-3 a B e) \sqrt{d+e x}}{e^5}+\frac{2 b^3 B (d+e x)^{3/2}}{3 e^5}\\ \end{align*}

Mathematica [A]  time = 0.127863, size = 145, normalized size = 0.86 \[ \frac{2 \left (-15 b^2 (d+e x)^3 (-3 a B e-A b e+4 b B d)-45 b (d+e x)^2 (b d-a e) (-a B e-A b e+2 b B d)+5 (d+e x) (b d-a e)^2 (-a B e-3 A b e+4 b B d)-3 (b d-a e)^3 (B d-A e)+5 b^3 B (d+e x)^4\right )}{15 e^5 (d+e x)^{5/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.007, size = 301, normalized size = 1.8 \begin{align*} -{\frac{-10\,{b}^{3}B{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}}}} \end{align*}

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+24*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 = 1.06978, size = 369, normalized size = 2.18 \begin{align*} \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} \end{align*}

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, 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 = 1.3768, size = 622, normalized size = 3.68 \begin{align*} \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 )} \sqrt{e x + d}}{15 \,{\left (e^{8} x^{3} + 3 \, d e^{7} x^{2} + 3 \, d^{2} e^{6} x + d^{3} e^{5}\right )}} \end{align*}

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, 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)*sqrt(e*x + d)/(e^8*x^3 + 3*d*e^7*x^2 + 3*d
^2*e^6*x + d^3*e^5)

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Sympy [A]  time = 4.45605, size = 1654, normalized size = 9.79 \begin{align*} \text{result too large to display} \end{align*}

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)) - 12
0*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*sqrt(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 [B]  time = 1.71218, size = 491, normalized size = 2.91 \begin{align*} \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}}} \end{align*}

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, 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)