3.1739 \(\int \frac{(a+b x)^3 (A+B x)}{(d+e x)^{3/2}} \, dx\)

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)

________________________________________________________________________________________

Rubi [A]  time = 0.067525, 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, Rules used = {77} \[ -\frac{2 b^2 (d+e x)^{5/2} (-3 a B e-A b e+4 b B d)}{5 e^5}+\frac{2 b (d+e x)^{3/2} (b d-a e) (-a B e-A b e+2 b B d)}{e^5}-\frac{2 \sqrt{d+e x} (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{e^5}-\frac{2 (b d-a e)^3 (B d-A e)}{e^5 \sqrt{d+e x}}+\frac{2 b^3 B (d+e x)^{7/2}}{7 e^5} \]

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)

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

Mathematica [A]  time = 0.120638, size = 145, normalized size = 0.87 \[ \frac{2 \left (-7 b^2 (d+e x)^3 (-3 a B e-A b e+4 b B d)+35 b (d+e x)^2 (b d-a e) (-a B e-A b e+2 b B d)-35 (d+e x) (b d-a e)^2 (-a B e-3 A b e+4 b B d)-35 (b d-a e)^3 (B d-A e)+5 b^3 B (d+e x)^4\right )}{35 e^5 \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.005, size = 301, normalized size = 1.8 \begin{align*} -{\frac{-10\,{b}^{3}B{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}}}} \end{align*}

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

________________________________________________________________________________________

Maxima [A]  time = 1.80309, size = 369, normalized size = 2.21 \begin{align*} \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} \end{align*}

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, 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

________________________________________________________________________________________

Fricas [A]  time = 1.31212, size = 583, normalized size = 3.49 \begin{align*} \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 )} \sqrt{e x + d}}{35 \,{\left (e^{6} x + d 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)^(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^6*x + d*e^5)

________________________________________________________________________________________

Sympy [A]  time = 43.3904, size = 255, normalized size = 1.53 \begin{align*} \frac{2 B b^{3} \left (d + e x\right )^{\frac{7}{2}}}{7 e^{5}} + \frac{\left (d + e x\right )^{\frac{5}{2}} \left (2 A b^{3} e + 6 B a b^{2} e - 8 B b^{3} d\right )}{5 e^{5}} + \frac{\left (d + e x\right )^{\frac{3}{2}} \left (6 A a b^{2} e^{2} - 6 A b^{3} d e + 6 B a^{2} b e^{2} - 18 B a b^{2} d e + 12 B b^{3} d^{2}\right )}{3 e^{5}} + \frac{\sqrt{d + e x} \left (6 A a^{2} b e^{3} - 12 A a b^{2} d e^{2} + 6 A b^{3} d^{2} e + 2 B a^{3} e^{3} - 12 B a^{2} b d e^{2} + 18 B a b^{2} d^{2} e - 8 B b^{3} d^{3}\right )}{e^{5}} + \frac{2 \left (- A e + B d\right ) \left (a e - b d\right )^{3}}{e^{5} \sqrt{d + e x}} \end{align*}

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]

2*B*b**3*(d + e*x)**(7/2)/(7*e**5) + (d + e*x)**(5/2)*(2*A*b**3*e + 6*B*a*b**2*e - 8*B*b**3*d)/(5*e**5) + (d +
e*x)**(3/2)*(6*A*a*b**2*e**2 - 6*A*b**3*d*e + 6*B*a**2*b*e**2 - 18*B*a*b**2*d*e + 12*B*b**3*d**2)/(3*e**5) +
sqrt(d + e*x)*(6*A*a**2*b*e**3 - 12*A*a*b**2*d*e**2 + 6*A*b**3*d**2*e + 2*B*a**3*e**3 - 12*B*a**2*b*d*e**2 + 1
8*B*a*b**2*d**2*e - 8*B*b**3*d**3)/e**5 + 2*(-A*e + B*d)*(a*e - b*d)**3/(e**5*sqrt(d + e*x))

________________________________________________________________________________________

Giac [B]  time = 1.95955, size = 514, normalized size = 3.08 \begin{align*} \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}} \end{align*}

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, 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^31 - 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*sqrt(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)