3.5.9 \(\int \frac {x^4 (A+B x)}{(a+b x)^{3/2}} \, dx\)

Optimal. Leaf size=147 \[ -\frac {2 a^4 (A b-a B)}{b^6 \sqrt {a+b x}}-\frac {2 a^3 \sqrt {a+b x} (4 A b-5 a B)}{b^6}+\frac {4 a^2 (a+b x)^{3/2} (3 A b-5 a B)}{3 b^6}+\frac {2 (a+b x)^{7/2} (A b-5 a B)}{7 b^6}-\frac {4 a (a+b x)^{5/2} (2 A b-5 a B)}{5 b^6}+\frac {2 B (a+b x)^{9/2}}{9 b^6} \]

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Rubi [A]  time = 0.06, antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {77} \begin {gather*} \frac {4 a^2 (a+b x)^{3/2} (3 A b-5 a B)}{3 b^6}-\frac {2 a^3 \sqrt {a+b x} (4 A b-5 a B)}{b^6}-\frac {2 a^4 (A b-a B)}{b^6 \sqrt {a+b x}}+\frac {2 (a+b x)^{7/2} (A b-5 a B)}{7 b^6}-\frac {4 a (a+b x)^{5/2} (2 A b-5 a B)}{5 b^6}+\frac {2 B (a+b x)^{9/2}}{9 b^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^4*(A + B*x))/(a + b*x)^(3/2),x]

[Out]

(-2*a^4*(A*b - a*B))/(b^6*Sqrt[a + b*x]) - (2*a^3*(4*A*b - 5*a*B)*Sqrt[a + b*x])/b^6 + (4*a^2*(3*A*b - 5*a*B)*
(a + b*x)^(3/2))/(3*b^6) - (4*a*(2*A*b - 5*a*B)*(a + b*x)^(5/2))/(5*b^6) + (2*(A*b - 5*a*B)*(a + b*x)^(7/2))/(
7*b^6) + (2*B*(a + b*x)^(9/2))/(9*b^6)

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 {x^4 (A+B x)}{(a+b x)^{3/2}} \, dx &=\int \left (-\frac {a^4 (-A b+a B)}{b^5 (a+b x)^{3/2}}+\frac {a^3 (-4 A b+5 a B)}{b^5 \sqrt {a+b x}}-\frac {2 a^2 (-3 A b+5 a B) \sqrt {a+b x}}{b^5}+\frac {2 a (-2 A b+5 a B) (a+b x)^{3/2}}{b^5}+\frac {(A b-5 a B) (a+b x)^{5/2}}{b^5}+\frac {B (a+b x)^{7/2}}{b^5}\right ) \, dx\\ &=-\frac {2 a^4 (A b-a B)}{b^6 \sqrt {a+b x}}-\frac {2 a^3 (4 A b-5 a B) \sqrt {a+b x}}{b^6}+\frac {4 a^2 (3 A b-5 a B) (a+b x)^{3/2}}{3 b^6}-\frac {4 a (2 A b-5 a B) (a+b x)^{5/2}}{5 b^6}+\frac {2 (A b-5 a B) (a+b x)^{7/2}}{7 b^6}+\frac {2 B (a+b x)^{9/2}}{9 b^6}\\ \end {align*}

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Mathematica [A]  time = 0.10, size = 106, normalized size = 0.72 \begin {gather*} \frac {2560 a^5 B-256 a^4 b (9 A-5 B x)-64 a^3 b^2 x (18 A+5 B x)+32 a^2 b^3 x^2 (9 A+5 B x)-4 a b^4 x^3 (36 A+25 B x)+10 b^5 x^4 (9 A+7 B x)}{315 b^6 \sqrt {a+b x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x^4*(A + B*x))/(a + b*x)^(3/2),x]

[Out]

(2560*a^5*B - 256*a^4*b*(9*A - 5*B*x) + 32*a^2*b^3*x^2*(9*A + 5*B*x) - 64*a^3*b^2*x*(18*A + 5*B*x) + 10*b^5*x^
4*(9*A + 7*B*x) - 4*a*b^4*x^3*(36*A + 25*B*x))/(315*b^6*Sqrt[a + b*x])

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IntegrateAlgebraic [A]  time = 0.06, size = 137, normalized size = 0.93 \begin {gather*} \frac {2 \left (315 a^5 B-315 a^4 A b+1575 a^4 B (a+b x)-1260 a^3 A b (a+b x)-1050 a^3 B (a+b x)^2+630 a^2 A b (a+b x)^2+630 a^2 B (a+b x)^3-252 a A b (a+b x)^3+45 A b (a+b x)^4-225 a B (a+b x)^4+35 B (a+b x)^5\right )}{315 b^6 \sqrt {a+b x}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(x^4*(A + B*x))/(a + b*x)^(3/2),x]

[Out]

(2*(-315*a^4*A*b + 315*a^5*B - 1260*a^3*A*b*(a + b*x) + 1575*a^4*B*(a + b*x) + 630*a^2*A*b*(a + b*x)^2 - 1050*
a^3*B*(a + b*x)^2 - 252*a*A*b*(a + b*x)^3 + 630*a^2*B*(a + b*x)^3 + 45*A*b*(a + b*x)^4 - 225*a*B*(a + b*x)^4 +
35*B*(a + b*x)^5))/(315*b^6*Sqrt[a + b*x])

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fricas [A]  time = 1.00, size = 130, normalized size = 0.88 \begin {gather*} \frac {2 \, {\left (35 \, B b^{5} x^{5} + 1280 \, B a^{5} - 1152 \, A a^{4} b - 5 \, {\left (10 \, B a b^{4} - 9 \, A b^{5}\right )} x^{4} + 8 \, {\left (10 \, B a^{2} b^{3} - 9 \, A a b^{4}\right )} x^{3} - 16 \, {\left (10 \, B a^{3} b^{2} - 9 \, A a^{2} b^{3}\right )} x^{2} + 64 \, {\left (10 \, B a^{4} b - 9 \, A a^{3} b^{2}\right )} x\right )} \sqrt {b x + a}}{315 \, {\left (b^{7} x + a b^{6}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(B*x+A)/(b*x+a)^(3/2),x, algorithm="fricas")

[Out]

2/315*(35*B*b^5*x^5 + 1280*B*a^5 - 1152*A*a^4*b - 5*(10*B*a*b^4 - 9*A*b^5)*x^4 + 8*(10*B*a^2*b^3 - 9*A*a*b^4)*
x^3 - 16*(10*B*a^3*b^2 - 9*A*a^2*b^3)*x^2 + 64*(10*B*a^4*b - 9*A*a^3*b^2)*x)*sqrt(b*x + a)/(b^7*x + a*b^6)

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giac [A]  time = 1.26, size = 166, normalized size = 1.13 \begin {gather*} \frac {2 \, {\left (B a^{5} - A a^{4} b\right )}}{\sqrt {b x + a} b^{6}} + \frac {2 \, {\left (35 \, {\left (b x + a\right )}^{\frac {9}{2}} B b^{48} - 225 \, {\left (b x + a\right )}^{\frac {7}{2}} B a b^{48} + 630 \, {\left (b x + a\right )}^{\frac {5}{2}} B a^{2} b^{48} - 1050 \, {\left (b x + a\right )}^{\frac {3}{2}} B a^{3} b^{48} + 1575 \, \sqrt {b x + a} B a^{4} b^{48} + 45 \, {\left (b x + a\right )}^{\frac {7}{2}} A b^{49} - 252 \, {\left (b x + a\right )}^{\frac {5}{2}} A a b^{49} + 630 \, {\left (b x + a\right )}^{\frac {3}{2}} A a^{2} b^{49} - 1260 \, \sqrt {b x + a} A a^{3} b^{49}\right )}}{315 \, b^{54}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(B*x+A)/(b*x+a)^(3/2),x, algorithm="giac")

[Out]

2*(B*a^5 - A*a^4*b)/(sqrt(b*x + a)*b^6) + 2/315*(35*(b*x + a)^(9/2)*B*b^48 - 225*(b*x + a)^(7/2)*B*a*b^48 + 63
0*(b*x + a)^(5/2)*B*a^2*b^48 - 1050*(b*x + a)^(3/2)*B*a^3*b^48 + 1575*sqrt(b*x + a)*B*a^4*b^48 + 45*(b*x + a)^
(7/2)*A*b^49 - 252*(b*x + a)^(5/2)*A*a*b^49 + 630*(b*x + a)^(3/2)*A*a^2*b^49 - 1260*sqrt(b*x + a)*A*a^3*b^49)/
b^54

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maple [A]  time = 0.01, size = 119, normalized size = 0.81 \begin {gather*} -\frac {2 \left (-35 B \,b^{5} x^{5}-45 A \,b^{5} x^{4}+50 B a \,b^{4} x^{4}+72 A a \,b^{4} x^{3}-80 B \,a^{2} b^{3} x^{3}-144 A \,a^{2} b^{3} x^{2}+160 B \,a^{3} b^{2} x^{2}+576 A \,a^{3} b^{2} x -640 B \,a^{4} b x +1152 A \,a^{4} b -1280 B \,a^{5}\right )}{315 \sqrt {b x +a}\, b^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(B*x+A)/(b*x+a)^(3/2),x)

[Out]

-2/315/(b*x+a)^(1/2)*(-35*B*b^5*x^5-45*A*b^5*x^4+50*B*a*b^4*x^4+72*A*a*b^4*x^3-80*B*a^2*b^3*x^3-144*A*a^2*b^3*
x^2+160*B*a^3*b^2*x^2+576*A*a^3*b^2*x-640*B*a^4*b*x+1152*A*a^4*b-1280*B*a^5)/b^6

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maxima [A]  time = 0.89, size = 131, normalized size = 0.89 \begin {gather*} \frac {2 \, {\left (\frac {35 \, {\left (b x + a\right )}^{\frac {9}{2}} B - 45 \, {\left (5 \, B a - A b\right )} {\left (b x + a\right )}^{\frac {7}{2}} + 126 \, {\left (5 \, B a^{2} - 2 \, A a b\right )} {\left (b x + a\right )}^{\frac {5}{2}} - 210 \, {\left (5 \, B a^{3} - 3 \, A a^{2} b\right )} {\left (b x + a\right )}^{\frac {3}{2}} + 315 \, {\left (5 \, B a^{4} - 4 \, A a^{3} b\right )} \sqrt {b x + a}}{b} + \frac {315 \, {\left (B a^{5} - A a^{4} b\right )}}{\sqrt {b x + a} b}\right )}}{315 \, b^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(B*x+A)/(b*x+a)^(3/2),x, algorithm="maxima")

[Out]

2/315*((35*(b*x + a)^(9/2)*B - 45*(5*B*a - A*b)*(b*x + a)^(7/2) + 126*(5*B*a^2 - 2*A*a*b)*(b*x + a)^(5/2) - 21
0*(5*B*a^3 - 3*A*a^2*b)*(b*x + a)^(3/2) + 315*(5*B*a^4 - 4*A*a^3*b)*sqrt(b*x + a))/b + 315*(B*a^5 - A*a^4*b)/(
sqrt(b*x + a)*b))/b^5

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mupad [B]  time = 0.38, size = 135, normalized size = 0.92 \begin {gather*} \frac {\left (20\,B\,a^2-8\,A\,a\,b\right )\,{\left (a+b\,x\right )}^{5/2}}{5\,b^6}+\frac {2\,B\,{\left (a+b\,x\right )}^{9/2}}{9\,b^6}+\frac {\left (2\,A\,b-10\,B\,a\right )\,{\left (a+b\,x\right )}^{7/2}}{7\,b^6}+\frac {2\,B\,a^5-2\,A\,a^4\,b}{b^6\,\sqrt {a+b\,x}}+\frac {\left (10\,B\,a^4-8\,A\,a^3\,b\right )\,\sqrt {a+b\,x}}{b^6}-\frac {\left (20\,B\,a^3-12\,A\,a^2\,b\right )\,{\left (a+b\,x\right )}^{3/2}}{3\,b^6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4*(A + B*x))/(a + b*x)^(3/2),x)

[Out]

((20*B*a^2 - 8*A*a*b)*(a + b*x)^(5/2))/(5*b^6) + (2*B*(a + b*x)^(9/2))/(9*b^6) + ((2*A*b - 10*B*a)*(a + b*x)^(
7/2))/(7*b^6) + (2*B*a^5 - 2*A*a^4*b)/(b^6*(a + b*x)^(1/2)) + ((10*B*a^4 - 8*A*a^3*b)*(a + b*x)^(1/2))/b^6 - (
(20*B*a^3 - 12*A*a^2*b)*(a + b*x)^(3/2))/(3*b^6)

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sympy [A]  time = 18.90, size = 146, normalized size = 0.99 \begin {gather*} \frac {2 B \left (a + b x\right )^{\frac {9}{2}}}{9 b^{6}} + \frac {2 a^{4} \left (- A b + B a\right )}{b^{6} \sqrt {a + b x}} + \frac {\left (a + b x\right )^{\frac {7}{2}} \left (2 A b - 10 B a\right )}{7 b^{6}} + \frac {\left (a + b x\right )^{\frac {5}{2}} \left (- 8 A a b + 20 B a^{2}\right )}{5 b^{6}} + \frac {\left (a + b x\right )^{\frac {3}{2}} \left (12 A a^{2} b - 20 B a^{3}\right )}{3 b^{6}} + \frac {\sqrt {a + b x} \left (- 8 A a^{3} b + 10 B a^{4}\right )}{b^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(B*x+A)/(b*x+a)**(3/2),x)

[Out]

2*B*(a + b*x)**(9/2)/(9*b**6) + 2*a**4*(-A*b + B*a)/(b**6*sqrt(a + b*x)) + (a + b*x)**(7/2)*(2*A*b - 10*B*a)/(
7*b**6) + (a + b*x)**(5/2)*(-8*A*a*b + 20*B*a**2)/(5*b**6) + (a + b*x)**(3/2)*(12*A*a**2*b - 20*B*a**3)/(3*b**
6) + sqrt(a + b*x)*(-8*A*a**3*b + 10*B*a**4)/b**6

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