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*(-a*e+b*d)*(-A*b*e-B*a*e+2*B*b*d)*(e*x+d)^(3/2)/e^5-2/5*b^2*(-A*b*e-3*B*a*e+4*B*b*d)*(e*x+d)^(5/2)/e^5+2/7
*b^3*B*(e*x+d)^(7/2)/e^5-2*(-a*e+b*d)^3*(-A*e+B*d)/e^5/(e*x+d)^(1/2)-2*(-a*e+b*d)^2*(-3*A*b*e-B*a*e+4*B*b*d)*(
e*x+d)^(1/2)/e^5

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Rubi [A]  time = 0.07, antiderivative size = 167, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.16, 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])

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fricas [A]  time = 0.60, size = 272, normalized size = 1.63 \[ \frac {2 \, {\left (5 \, B b^{3} e^{4} x^{4} - 128 \, B b^{3} d^{4} - 35 \, A a^{3} e^{4} + 112 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e - 280 \, {\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} + 70 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3} - {\left (8 \, B b^{3} d e^{3} - 7 \, {\left (3 \, B a b^{2} + A b^{3}\right )} e^{4}\right )} x^{3} + {\left (16 \, B b^{3} d^{2} e^{2} - 14 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d e^{3} + 35 \, {\left (B a^{2} b + A a b^{2}\right )} e^{4}\right )} x^{2} - {\left (64 \, B b^{3} d^{3} e - 56 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{2} + 140 \, {\left (B a^{2} b + A a b^{2}\right )} d e^{3} - 35 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} e^{4}\right )} x\right )} \sqrt {e x + d}}{35 \, {\left (e^{6} x + d e^{5}\right )}} \]

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)

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giac [B]  time = 1.35, size = 381, normalized size = 2.28 \[ \frac {2}{35} \, {\left (5 \, {\left (x e + d\right )}^{\frac {7}{2}} B b^{3} e^{30} - 28 \, {\left (x e + d\right )}^{\frac {5}{2}} B b^{3} d e^{30} + 70 \, {\left (x e + d\right )}^{\frac {3}{2}} B b^{3} d^{2} e^{30} - 140 \, \sqrt {x e + d} B b^{3} d^{3} e^{30} + 21 \, {\left (x e + d\right )}^{\frac {5}{2}} B a b^{2} e^{31} + 7 \, {\left (x e + d\right )}^{\frac {5}{2}} A b^{3} e^{31} - 105 \, {\left (x e + d\right )}^{\frac {3}{2}} B a b^{2} d e^{31} - 35 \, {\left (x e + d\right )}^{\frac {3}{2}} A b^{3} d e^{31} + 315 \, \sqrt {x e + d} B a b^{2} d^{2} e^{31} + 105 \, \sqrt {x e + d} A b^{3} d^{2} e^{31} + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} B a^{2} b e^{32} + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} A a b^{2} e^{32} - 210 \, \sqrt {x e + d} B a^{2} b d e^{32} - 210 \, \sqrt {x e + d} A a b^{2} d e^{32} + 35 \, \sqrt {x e + d} B a^{3} e^{33} + 105 \, \sqrt {x e + d} A a^{2} b e^{33}\right )} e^{\left (-35\right )} - \frac {2 \, {\left (B b^{3} d^{4} - 3 \, B a b^{2} d^{3} e - A b^{3} d^{3} e + 3 \, B a^{2} b d^{2} e^{2} + 3 \, A a b^{2} d^{2} e^{2} - B a^{3} d e^{3} - 3 \, A a^{2} b d e^{3} + A a^{3} e^{4}\right )} e^{\left (-5\right )}}{\sqrt {x e + d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^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)

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maple [A]  time = 0.01, size = 301, normalized size = 1.80 \[ -\frac {2 \left (-5 B \,b^{3} x^{4} e^{4}-7 A \,b^{3} e^{4} x^{3}-21 B a \,b^{2} e^{4} x^{3}+8 B \,b^{3} d \,e^{3} x^{3}-35 A a \,b^{2} e^{4} x^{2}+14 A \,b^{3} d \,e^{3} x^{2}-35 B \,a^{2} b \,e^{4} x^{2}+42 B a \,b^{2} d \,e^{3} x^{2}-16 B \,b^{3} d^{2} e^{2} x^{2}-105 A \,a^{2} b \,e^{4} x +140 A a \,b^{2} d \,e^{3} x -56 A \,b^{3} d^{2} e^{2} x -35 B \,a^{3} e^{4} x +140 B \,a^{2} b d \,e^{3} x -168 B a \,b^{2} d^{2} e^{2} x +64 B \,b^{3} d^{3} e x +35 a^{3} A \,e^{4}-210 A \,a^{2} b d \,e^{3}+280 A a \,b^{2} d^{2} e^{2}-112 A \,b^{3} d^{3} e -70 B \,a^{3} d \,e^{3}+280 B \,a^{2} b \,d^{2} e^{2}-336 B a \,b^{2} d^{3} e +128 B \,b^{3} d^{4}\right )}{35 \sqrt {e x +d}\, e^{5}} \]

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 = 0.51, size = 273, normalized size = 1.63 \[ \frac {2 \, {\left (\frac {5 \, {\left (e x + d\right )}^{\frac {7}{2}} B b^{3} - 7 \, {\left (4 \, B b^{3} d - {\left (3 \, B a b^{2} + A b^{3}\right )} e\right )} {\left (e x + d\right )}^{\frac {5}{2}} + 35 \, {\left (2 \, B b^{3} d^{2} - {\left (3 \, B a b^{2} + A b^{3}\right )} d e + {\left (B a^{2} b + A a b^{2}\right )} e^{2}\right )} {\left (e x + d\right )}^{\frac {3}{2}} - 35 \, {\left (4 \, B b^{3} d^{3} - 3 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e + 6 \, {\left (B a^{2} b + A a b^{2}\right )} d e^{2} - {\left (B a^{3} + 3 \, A a^{2} b\right )} e^{3}\right )} \sqrt {e x + d}}{e^{4}} - \frac {35 \, {\left (B b^{3} d^{4} + A a^{3} e^{4} - {\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e + 3 \, {\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} - {\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3}\right )}}{\sqrt {e x + d} e^{4}}\right )}}{35 \, e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^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

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mupad [B]  time = 0.06, size = 223, normalized size = 1.34 \[ \frac {{\left (d+e\,x\right )}^{5/2}\,\left (2\,A\,b^3\,e-8\,B\,b^3\,d+6\,B\,a\,b^2\,e\right )}{5\,e^5}-\frac {-2\,B\,a^3\,d\,e^3+2\,A\,a^3\,e^4+6\,B\,a^2\,b\,d^2\,e^2-6\,A\,a^2\,b\,d\,e^3-6\,B\,a\,b^2\,d^3\,e+6\,A\,a\,b^2\,d^2\,e^2+2\,B\,b^3\,d^4-2\,A\,b^3\,d^3\,e}{e^5\,\sqrt {d+e\,x}}+\frac {2\,{\left (a\,e-b\,d\right )}^2\,\sqrt {d+e\,x}\,\left (3\,A\,b\,e+B\,a\,e-4\,B\,b\,d\right )}{e^5}+\frac {2\,B\,b^3\,{\left (d+e\,x\right )}^{7/2}}{7\,e^5}+\frac {2\,b\,\left (a\,e-b\,d\right )\,{\left (d+e\,x\right )}^{3/2}\,\left (A\,b\,e+B\,a\,e-2\,B\,b\,d\right )}{e^5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((d + e*x)^(5/2)*(2*A*b^3*e - 8*B*b^3*d + 6*B*a*b^2*e))/(5*e^5) - (2*A*a^3*e^4 + 2*B*b^3*d^4 - 2*A*b^3*d^3*e -
2*B*a^3*d*e^3 + 6*A*a*b^2*d^2*e^2 + 6*B*a^2*b*d^2*e^2 - 6*A*a^2*b*d*e^3 - 6*B*a*b^2*d^3*e)/(e^5*(d + e*x)^(1/
2)) + (2*(a*e - b*d)^2*(d + e*x)^(1/2)*(3*A*b*e + B*a*e - 4*B*b*d))/e^5 + (2*B*b^3*(d + e*x)^(7/2))/(7*e^5) +
(2*b*(a*e - b*d)*(d + e*x)^(3/2)*(A*b*e + B*a*e - 2*B*b*d))/e^5

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sympy [A]  time = 62.68, size = 255, normalized size = 1.53 \[ \frac {2 B b^{3} \left (d + e x\right )^{\frac {7}{2}}}{7 e^{5}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \left (2 A b^{3} e + 6 B a b^{2} e - 8 B b^{3} d\right )}{5 e^{5}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \left (6 A a b^{2} e^{2} - 6 A b^{3} d e + 6 B a^{2} b e^{2} - 18 B a b^{2} d e + 12 B b^{3} d^{2}\right )}{3 e^{5}} + \frac {\sqrt {d + e x} \left (6 A a^{2} b e^{3} - 12 A a b^{2} d e^{2} + 6 A b^{3} d^{2} e + 2 B a^{3} e^{3} - 12 B a^{2} b d e^{2} + 18 B a b^{2} d^{2} e - 8 B b^{3} d^{3}\right )}{e^{5}} + \frac {2 \left (- A e + B d\right ) \left (a e - b d\right )^{3}}{e^{5} \sqrt {d + e x}} \]

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))

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