\(\int \frac {x^3 (A+B x)}{(a+b x)^{3/2}} \, dx\) [435]

Optimal result

Integrand size = 18, antiderivative size = 116 \[ \int \frac {x^3 (A+B x)}{(a+b x)^{3/2}} \, dx=\frac {2 a^3 (A b-a B)}{b^5 \sqrt {a+b x}}+\frac {2 a^2 (3 A b-4 a B) \sqrt {a+b x}}{b^5}-\frac {2 a (A b-2 a B) (a+b x)^{3/2}}{b^5}+\frac {2 (A b-4 a B) (a+b x)^{5/2}}{5 b^5}+\frac {2 B (a+b x)^{7/2}}{7 b^5} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {78} \[ \int \frac {x^3 (A+B x)}{(a+b x)^{3/2}} \, dx=\frac {2 a^3 (A b-a B)}{b^5 \sqrt {a+b x}}+\frac {2 a^2 \sqrt {a+b x} (3 A b-4 a B)}{b^5}+\frac {2 (a+b x)^{5/2} (A b-4 a B)}{5 b^5}-\frac {2 a (a+b x)^{3/2} (A b-2 a B)}{b^5}+\frac {2 B (a+b x)^{7/2}}{7 b^5} \]

[In]

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

[Out]

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

Rule 78

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

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.74 \[ \int \frac {x^3 (A+B x)}{(a+b x)^{3/2}} \, dx=\frac {2 \left (-128 a^4 B+16 a^3 b (7 A-4 B x)+8 a^2 b^2 x (7 A+2 B x)-2 a b^3 x^2 (7 A+4 B x)+b^4 x^3 (7 A+5 B x)\right )}{35 b^5 \sqrt {a+b x}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.52 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.68

method	result	size
pseudoelliptic	\(\frac {\left (10 x^{4} B +14 A \,x^{3}\right ) b^{4}-28 x^{2} \left (\frac {4 B x}{7}+A \right ) a \,b^{3}+112 x \,a^{2} \left (\frac {2 B x}{7}+A \right ) b^{2}+224 a^{3} \left (-\frac {4 B x}{7}+A \right ) b -256 B \,a^{4}}{35 \sqrt {b x +a}\, b^{5}}\)	\(79\)
gosper	\(\frac {\frac {2}{7} B \,x^{4} b^{4}+\frac {2}{5} A \,x^{3} b^{4}-\frac {16}{35} B \,x^{3} a \,b^{3}-\frac {4}{5} A \,x^{2} a \,b^{3}+\frac {32}{35} B \,x^{2} a^{2} b^{2}+\frac {16}{5} A x \,a^{2} b^{2}-\frac {128}{35} B x \,a^{3} b +\frac {32}{5} A \,a^{3} b -\frac {256}{35} B \,a^{4}}{\sqrt {b x +a}\, b^{5}}\)	\(95\)
trager	\(\frac {\frac {2}{7} B \,x^{4} b^{4}+\frac {2}{5} A \,x^{3} b^{4}-\frac {16}{35} B \,x^{3} a \,b^{3}-\frac {4}{5} A \,x^{2} a \,b^{3}+\frac {32}{35} B \,x^{2} a^{2} b^{2}+\frac {16}{5} A x \,a^{2} b^{2}-\frac {128}{35} B x \,a^{3} b +\frac {32}{5} A \,a^{3} b -\frac {256}{35} B \,a^{4}}{\sqrt {b x +a}\, b^{5}}\)	\(95\)
risch	\(\frac {2 \left (5 b^{3} B \,x^{3}+7 A \,b^{3} x^{2}-13 B a \,b^{2} x^{2}-21 a \,b^{2} A x +29 a^{2} b B x +77 a^{2} b A -93 a^{3} B \right ) \sqrt {b x +a}}{35 b^{5}}+\frac {2 a^{3} \left (A b -B a \right )}{b^{5} \sqrt {b x +a}}\)	\(95\)
derivativedivides	\(\frac {\frac {2 B \left (b x +a \right )^{\frac {7}{2}}}{7}+\frac {2 A b \left (b x +a \right )^{\frac {5}{2}}}{5}-\frac {8 B a \left (b x +a \right )^{\frac {5}{2}}}{5}-2 A b a \left (b x +a \right )^{\frac {3}{2}}+4 B \,a^{2} \left (b x +a \right )^{\frac {3}{2}}+6 A \,a^{2} b \sqrt {b x +a}-8 B \,a^{3} \sqrt {b x +a}+\frac {2 a^{3} \left (A b -B a \right )}{\sqrt {b x +a}}}{b^{5}}\)	\(110\)
default	\(\frac {\frac {2 B \left (b x +a \right )^{\frac {7}{2}}}{7}+\frac {2 A b \left (b x +a \right )^{\frac {5}{2}}}{5}-\frac {8 B a \left (b x +a \right )^{\frac {5}{2}}}{5}-2 A b a \left (b x +a \right )^{\frac {3}{2}}+4 B \,a^{2} \left (b x +a \right )^{\frac {3}{2}}+6 A \,a^{2} b \sqrt {b x +a}-8 B \,a^{3} \sqrt {b x +a}+\frac {2 a^{3} \left (A b -B a \right )}{\sqrt {b x +a}}}{b^{5}}\)	\(110\)

[In]

int(x^3*(B*x+A)/(b*x+a)^(3/2),x,method=_RETURNVERBOSE)

[Out]

1/35*((10*B*x^4+14*A*x^3)*b^4-28*x^2*(4/7*B*x+A)*a*b^3+112*x*a^2*(2/7*B*x+A)*b^2+224*a^3*(-4/7*B*x+A)*b-256*B*
a^4)/(b*x+a)^(1/2)/b^5

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.91 \[ \int \frac {x^3 (A+B x)}{(a+b x)^{3/2}} \, dx=\frac {2 \, {\left (5 \, B b^{4} x^{4} - 128 \, B a^{4} + 112 \, A a^{3} b - {\left (8 \, B a b^{3} - 7 \, A b^{4}\right )} x^{3} + 2 \, {\left (8 \, B a^{2} b^{2} - 7 \, A a b^{3}\right )} x^{2} - 8 \, {\left (8 \, B a^{3} b - 7 \, A a^{2} b^{2}\right )} x\right )} \sqrt {b x + a}}{35 \, {\left (b^{6} x + a b^{5}\right )}} \]

[In]

integrate(x^3*(B*x+A)/(b*x+a)^(3/2),x, algorithm="fricas")

[Out]

2/35*(5*B*b^4*x^4 - 128*B*a^4 + 112*A*a^3*b - (8*B*a*b^3 - 7*A*b^4)*x^3 + 2*(8*B*a^2*b^2 - 7*A*a*b^3)*x^2 - 8*
(8*B*a^3*b - 7*A*a^2*b^2)*x)*sqrt(b*x + a)/(b^6*x + a*b^5)

Sympy [A] (verification not implemented)

Time = 1.44 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.16 \[ \int \frac {x^3 (A+B x)}{(a+b x)^{3/2}} \, dx=\begin {cases} \frac {2 \left (\frac {B \left (a + b x\right )^{\frac {7}{2}}}{7 b} - \frac {a^{3} \left (- A b + B a\right )}{b \sqrt {a + b x}} + \frac {\left (a + b x\right )^{\frac {5}{2}} \left (A b - 4 B a\right )}{5 b} + \frac {\left (a + b x\right )^{\frac {3}{2}} \left (- 3 A a b + 6 B a^{2}\right )}{3 b} + \frac {\sqrt {a + b x} \left (3 A a^{2} b - 4 B a^{3}\right )}{b}\right )}{b^{4}} & \text {for}\: b \neq 0 \\\frac {\frac {A x^{4}}{4} + \frac {B x^{5}}{5}}{a^{\frac {3}{2}}} & \text {otherwise} \end {cases} \]

[In]

integrate(x**3*(B*x+A)/(b*x+a)**(3/2),x)

[Out]

Piecewise((2*(B*(a + b*x)**(7/2)/(7*b) - a**3*(-A*b + B*a)/(b*sqrt(a + b*x)) + (a + b*x)**(5/2)*(A*b - 4*B*a)/
(5*b) + (a + b*x)**(3/2)*(-3*A*a*b + 6*B*a**2)/(3*b) + sqrt(a + b*x)*(3*A*a**2*b - 4*B*a**3)/b)/b**4, Ne(b, 0)
), ((A*x**4/4 + B*x**5/5)/a**(3/2), True))

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.93 \[ \int \frac {x^3 (A+B x)}{(a+b x)^{3/2}} \, dx=\frac {2 \, {\left (\frac {5 \, {\left (b x + a\right )}^{\frac {7}{2}} B - 7 \, {\left (4 \, B a - A b\right )} {\left (b x + a\right )}^{\frac {5}{2}} + 35 \, {\left (2 \, B a^{2} - A a b\right )} {\left (b x + a\right )}^{\frac {3}{2}} - 35 \, {\left (4 \, B a^{3} - 3 \, A a^{2} b\right )} \sqrt {b x + a}}{b} - \frac {35 \, {\left (B a^{4} - A a^{3} b\right )}}{\sqrt {b x + a} b}\right )}}{35 \, b^{4}} \]

[In]

integrate(x^3*(B*x+A)/(b*x+a)^(3/2),x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.16 \[ \int \frac {x^3 (A+B x)}{(a+b x)^{3/2}} \, dx=-\frac {2 \, {\left (B a^{4} - A a^{3} b\right )}}{\sqrt {b x + a} b^{5}} + \frac {2 \, {\left (5 \, {\left (b x + a\right )}^{\frac {7}{2}} B b^{30} - 28 \, {\left (b x + a\right )}^{\frac {5}{2}} B a b^{30} + 70 \, {\left (b x + a\right )}^{\frac {3}{2}} B a^{2} b^{30} - 140 \, \sqrt {b x + a} B a^{3} b^{30} + 7 \, {\left (b x + a\right )}^{\frac {5}{2}} A b^{31} - 35 \, {\left (b x + a\right )}^{\frac {3}{2}} A a b^{31} + 105 \, \sqrt {b x + a} A a^{2} b^{31}\right )}}{35 \, b^{35}} \]

[In]

integrate(x^3*(B*x+A)/(b*x+a)^(3/2),x, algorithm="giac")

[Out]

-2*(B*a^4 - A*a^3*b)/(sqrt(b*x + a)*b^5) + 2/35*(5*(b*x + a)^(7/2)*B*b^30 - 28*(b*x + a)^(5/2)*B*a*b^30 + 70*(
b*x + a)^(3/2)*B*a^2*b^30 - 140*sqrt(b*x + a)*B*a^3*b^30 + 7*(b*x + a)^(5/2)*A*b^31 - 35*(b*x + a)^(3/2)*A*a*b
^31 + 105*sqrt(b*x + a)*A*a^2*b^31)/b^35

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.96 \[ \int \frac {x^3 (A+B x)}{(a+b x)^{3/2}} \, dx=\frac {\left (12\,B\,a^2-6\,A\,a\,b\right )\,{\left (a+b\,x\right )}^{3/2}}{3\,b^5}+\frac {2\,B\,{\left (a+b\,x\right )}^{7/2}}{7\,b^5}+\frac {\left (2\,A\,b-8\,B\,a\right )\,{\left (a+b\,x\right )}^{5/2}}{5\,b^5}-\frac {2\,B\,a^4-2\,A\,a^3\,b}{b^5\,\sqrt {a+b\,x}}-\frac {\left (8\,B\,a^3-6\,A\,a^2\,b\right )\,\sqrt {a+b\,x}}{b^5} \]

[In]

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

[Out]

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