3.423 \(\int \frac {x^4 (A+B x)}{\sqrt {a+b x}} \, dx\)

Optimal. Leaf size=149 \[ \frac {2 a^4 \sqrt {a+b x} (A b-a B)}{b^6}-\frac {2 a^3 (a+b x)^{3/2} (4 A b-5 a B)}{3 b^6}+\frac {4 a^2 (a+b x)^{5/2} (3 A b-5 a B)}{5 b^6}+\frac {2 (a+b x)^{9/2} (A b-5 a B)}{9 b^6}-\frac {4 a (a+b x)^{7/2} (2 A b-5 a B)}{7 b^6}+\frac {2 B (a+b x)^{11/2}}{11 b^6} \]

[Out]

-2/3*a^3*(4*A*b-5*B*a)*(b*x+a)^(3/2)/b^6+4/5*a^2*(3*A*b-5*B*a)*(b*x+a)^(5/2)/b^6-4/7*a*(2*A*b-5*B*a)*(b*x+a)^(
7/2)/b^6+2/9*(A*b-5*B*a)*(b*x+a)^(9/2)/b^6+2/11*B*(b*x+a)^(11/2)/b^6+2*a^4*(A*b-B*a)*(b*x+a)^(1/2)/b^6

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Rubi [A]  time = 0.06, antiderivative size = 149, 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} \[ \frac {4 a^2 (a+b x)^{5/2} (3 A b-5 a B)}{5 b^6}-\frac {2 a^3 (a+b x)^{3/2} (4 A b-5 a B)}{3 b^6}+\frac {2 a^4 \sqrt {a+b x} (A b-a B)}{b^6}+\frac {2 (a+b x)^{9/2} (A b-5 a B)}{9 b^6}-\frac {4 a (a+b x)^{7/2} (2 A b-5 a B)}{7 b^6}+\frac {2 B (a+b x)^{11/2}}{11 b^6} \]

Antiderivative was successfully verified.

[In]

Int[(x^4*(A + B*x))/Sqrt[a + b*x],x]

[Out]

(2*a^4*(A*b - a*B)*Sqrt[a + b*x])/b^6 - (2*a^3*(4*A*b - 5*a*B)*(a + b*x)^(3/2))/(3*b^6) + (4*a^2*(3*A*b - 5*a*
B)*(a + b*x)^(5/2))/(5*b^6) - (4*a*(2*A*b - 5*a*B)*(a + b*x)^(7/2))/(7*b^6) + (2*(A*b - 5*a*B)*(a + b*x)^(9/2)
)/(9*b^6) + (2*B*(a + b*x)^(11/2))/(11*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)}{\sqrt {a+b x}} \, dx &=\int \left (-\frac {a^4 (-A b+a B)}{b^5 \sqrt {a+b x}}+\frac {a^3 (-4 A b+5 a B) \sqrt {a+b x}}{b^5}-\frac {2 a^2 (-3 A b+5 a B) (a+b x)^{3/2}}{b^5}+\frac {2 a (-2 A b+5 a B) (a+b x)^{5/2}}{b^5}+\frac {(A b-5 a B) (a+b x)^{7/2}}{b^5}+\frac {B (a+b x)^{9/2}}{b^5}\right ) \, dx\\ &=\frac {2 a^4 (A b-a B) \sqrt {a+b x}}{b^6}-\frac {2 a^3 (4 A b-5 a B) (a+b x)^{3/2}}{3 b^6}+\frac {4 a^2 (3 A b-5 a B) (a+b x)^{5/2}}{5 b^6}-\frac {4 a (2 A b-5 a B) (a+b x)^{7/2}}{7 b^6}+\frac {2 (A b-5 a B) (a+b x)^{9/2}}{9 b^6}+\frac {2 B (a+b x)^{11/2}}{11 b^6}\\ \end {align*}

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Mathematica [A]  time = 0.11, size = 106, normalized size = 0.71 \[ \frac {2 \sqrt {a+b x} \left (-1280 a^5 B+128 a^4 b (11 A+5 B x)-32 a^3 b^2 x (22 A+15 B x)+16 a^2 b^3 x^2 (33 A+25 B x)-10 a b^4 x^3 (44 A+35 B x)+35 b^5 x^4 (11 A+9 B x)\right )}{3465 b^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[a + b*x]*(-1280*a^5*B + 128*a^4*b*(11*A + 5*B*x) + 35*b^5*x^4*(11*A + 9*B*x) - 32*a^3*b^2*x*(22*A + 15
*B*x) + 16*a^2*b^3*x^2*(33*A + 25*B*x) - 10*a*b^4*x^3*(44*A + 35*B*x)))/(3465*b^6)

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fricas [A]  time = 0.67, size = 120, normalized size = 0.81 \[ \frac {2 \, {\left (315 \, B b^{5} x^{5} - 1280 \, B a^{5} + 1408 \, A a^{4} b - 35 \, {\left (10 \, B a b^{4} - 11 \, A b^{5}\right )} x^{4} + 40 \, {\left (10 \, B a^{2} b^{3} - 11 \, A a b^{4}\right )} x^{3} - 48 \, {\left (10 \, B a^{3} b^{2} - 11 \, A a^{2} b^{3}\right )} x^{2} + 64 \, {\left (10 \, B a^{4} b - 11 \, A a^{3} b^{2}\right )} x\right )} \sqrt {b x + a}}{3465 \, b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(B*x+A)/(b*x+a)^(1/2),x, algorithm="fricas")

[Out]

2/3465*(315*B*b^5*x^5 - 1280*B*a^5 + 1408*A*a^4*b - 35*(10*B*a*b^4 - 11*A*b^5)*x^4 + 40*(10*B*a^2*b^3 - 11*A*a
*b^4)*x^3 - 48*(10*B*a^3*b^2 - 11*A*a^2*b^3)*x^2 + 64*(10*B*a^4*b - 11*A*a^3*b^2)*x)*sqrt(b*x + a)/b^6

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giac [A]  time = 1.24, size = 142, normalized size = 0.95 \[ \frac {2 \, {\left (\frac {11 \, {\left (35 \, {\left (b x + a\right )}^{\frac {9}{2}} - 180 \, {\left (b x + a\right )}^{\frac {7}{2}} a + 378 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{2} - 420 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{3} + 315 \, \sqrt {b x + a} a^{4}\right )} A}{b^{4}} + \frac {5 \, {\left (63 \, {\left (b x + a\right )}^{\frac {11}{2}} - 385 \, {\left (b x + a\right )}^{\frac {9}{2}} a + 990 \, {\left (b x + a\right )}^{\frac {7}{2}} a^{2} - 1386 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{3} + 1155 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{4} - 693 \, \sqrt {b x + a} a^{5}\right )} B}{b^{5}}\right )}}{3465 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(B*x+A)/(b*x+a)^(1/2),x, algorithm="giac")

[Out]

2/3465*(11*(35*(b*x + a)^(9/2) - 180*(b*x + a)^(7/2)*a + 378*(b*x + a)^(5/2)*a^2 - 420*(b*x + a)^(3/2)*a^3 + 3
15*sqrt(b*x + a)*a^4)*A/b^4 + 5*(63*(b*x + a)^(11/2) - 385*(b*x + a)^(9/2)*a + 990*(b*x + a)^(7/2)*a^2 - 1386*
(b*x + a)^(5/2)*a^3 + 1155*(b*x + a)^(3/2)*a^4 - 693*sqrt(b*x + a)*a^5)*B/b^5)/b

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maple [A]  time = 0.01, size = 119, normalized size = 0.80 \[ \frac {2 \sqrt {b x +a}\, \left (315 B \,b^{5} x^{5}+385 A \,b^{5} x^{4}-350 B a \,b^{4} x^{4}-440 A a \,b^{4} x^{3}+400 B \,a^{2} b^{3} x^{3}+528 A \,a^{2} b^{3} x^{2}-480 B \,a^{3} b^{2} x^{2}-704 A \,a^{3} b^{2} x +640 B \,a^{4} b x +1408 A \,a^{4} b -1280 B \,a^{5}\right )}{3465 b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(B*x+A)/(b*x+a)^(1/2),x)

[Out]

2/3465*(b*x+a)^(1/2)*(315*B*b^5*x^5+385*A*b^5*x^4-350*B*a*b^4*x^4-440*A*a*b^4*x^3+400*B*a^2*b^3*x^3+528*A*a^2*
b^3*x^2-480*B*a^3*b^2*x^2-704*A*a^3*b^2*x+640*B*a^4*b*x+1408*A*a^4*b-1280*B*a^5)/b^6

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maxima [A]  time = 0.86, size = 123, normalized size = 0.83 \[ \frac {2 \, {\left (315 \, {\left (b x + a\right )}^{\frac {11}{2}} B - 385 \, {\left (5 \, B a - A b\right )} {\left (b x + a\right )}^{\frac {9}{2}} + 990 \, {\left (5 \, B a^{2} - 2 \, A a b\right )} {\left (b x + a\right )}^{\frac {7}{2}} - 1386 \, {\left (5 \, B a^{3} - 3 \, A a^{2} b\right )} {\left (b x + a\right )}^{\frac {5}{2}} + 1155 \, {\left (5 \, B a^{4} - 4 \, A a^{3} b\right )} {\left (b x + a\right )}^{\frac {3}{2}} - 3465 \, {\left (B a^{5} - A a^{4} b\right )} \sqrt {b x + a}\right )}}{3465 \, b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(B*x+A)/(b*x+a)^(1/2),x, algorithm="maxima")

[Out]

2/3465*(315*(b*x + a)^(11/2)*B - 385*(5*B*a - A*b)*(b*x + a)^(9/2) + 990*(5*B*a^2 - 2*A*a*b)*(b*x + a)^(7/2) -
1386*(5*B*a^3 - 3*A*a^2*b)*(b*x + a)^(5/2) + 1155*(5*B*a^4 - 4*A*a^3*b)*(b*x + a)^(3/2) - 3465*(B*a^5 - A*a^4
*b)*sqrt(b*x + a))/b^6

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mupad [B]  time = 0.06, size = 137, normalized size = 0.92 \[ \frac {\left (20\,B\,a^2-8\,A\,a\,b\right )\,{\left (a+b\,x\right )}^{7/2}}{7\,b^6}+\frac {2\,B\,{\left (a+b\,x\right )}^{11/2}}{11\,b^6}+\frac {\left (2\,A\,b-10\,B\,a\right )\,{\left (a+b\,x\right )}^{9/2}}{9\,b^6}-\frac {\left (2\,B\,a^5-2\,A\,a^4\,b\right )\,\sqrt {a+b\,x}}{b^6}+\frac {\left (10\,B\,a^4-8\,A\,a^3\,b\right )\,{\left (a+b\,x\right )}^{3/2}}{3\,b^6}-\frac {\left (20\,B\,a^3-12\,A\,a^2\,b\right )\,{\left (a+b\,x\right )}^{5/2}}{5\,b^6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4*(A + B*x))/(a + b*x)^(1/2),x)

[Out]

((20*B*a^2 - 8*A*a*b)*(a + b*x)^(7/2))/(7*b^6) + (2*B*(a + b*x)^(11/2))/(11*b^6) + ((2*A*b - 10*B*a)*(a + b*x)
^(9/2))/(9*b^6) - ((2*B*a^5 - 2*A*a^4*b)*(a + b*x)^(1/2))/b^6 + ((10*B*a^4 - 8*A*a^3*b)*(a + b*x)^(3/2))/(3*b^
6) - ((20*B*a^3 - 12*A*a^2*b)*(a + b*x)^(5/2))/(5*b^6)

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sympy [A]  time = 43.16, size = 362, normalized size = 2.43 \[ \begin {cases} \frac {- \frac {2 A a \left (\frac {a^{4}}{\sqrt {a + b x}} + 4 a^{3} \sqrt {a + b x} - 2 a^{2} \left (a + b x\right )^{\frac {3}{2}} + \frac {4 a \left (a + b x\right )^{\frac {5}{2}}}{5} - \frac {\left (a + b x\right )^{\frac {7}{2}}}{7}\right )}{b^{4}} - \frac {2 A \left (- \frac {a^{5}}{\sqrt {a + b x}} - 5 a^{4} \sqrt {a + b x} + \frac {10 a^{3} \left (a + b x\right )^{\frac {3}{2}}}{3} - 2 a^{2} \left (a + b x\right )^{\frac {5}{2}} + \frac {5 a \left (a + b x\right )^{\frac {7}{2}}}{7} - \frac {\left (a + b x\right )^{\frac {9}{2}}}{9}\right )}{b^{4}} - \frac {2 B a \left (- \frac {a^{5}}{\sqrt {a + b x}} - 5 a^{4} \sqrt {a + b x} + \frac {10 a^{3} \left (a + b x\right )^{\frac {3}{2}}}{3} - 2 a^{2} \left (a + b x\right )^{\frac {5}{2}} + \frac {5 a \left (a + b x\right )^{\frac {7}{2}}}{7} - \frac {\left (a + b x\right )^{\frac {9}{2}}}{9}\right )}{b^{5}} - \frac {2 B \left (\frac {a^{6}}{\sqrt {a + b x}} + 6 a^{5} \sqrt {a + b x} - 5 a^{4} \left (a + b x\right )^{\frac {3}{2}} + 4 a^{3} \left (a + b x\right )^{\frac {5}{2}} - \frac {15 a^{2} \left (a + b x\right )^{\frac {7}{2}}}{7} + \frac {2 a \left (a + b x\right )^{\frac {9}{2}}}{3} - \frac {\left (a + b x\right )^{\frac {11}{2}}}{11}\right )}{b^{5}}}{b} & \text {for}\: b \neq 0 \\\frac {\frac {A x^{5}}{5} + \frac {B x^{6}}{6}}{\sqrt {a}} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(B*x+A)/(b*x+a)**(1/2),x)

[Out]

Piecewise(((-2*A*a*(a**4/sqrt(a + b*x) + 4*a**3*sqrt(a + b*x) - 2*a**2*(a + b*x)**(3/2) + 4*a*(a + b*x)**(5/2)
/5 - (a + b*x)**(7/2)/7)/b**4 - 2*A*(-a**5/sqrt(a + b*x) - 5*a**4*sqrt(a + b*x) + 10*a**3*(a + b*x)**(3/2)/3 -
2*a**2*(a + b*x)**(5/2) + 5*a*(a + b*x)**(7/2)/7 - (a + b*x)**(9/2)/9)/b**4 - 2*B*a*(-a**5/sqrt(a + b*x) - 5*
a**4*sqrt(a + b*x) + 10*a**3*(a + b*x)**(3/2)/3 - 2*a**2*(a + b*x)**(5/2) + 5*a*(a + b*x)**(7/2)/7 - (a + b*x)
**(9/2)/9)/b**5 - 2*B*(a**6/sqrt(a + b*x) + 6*a**5*sqrt(a + b*x) - 5*a**4*(a + b*x)**(3/2) + 4*a**3*(a + b*x)*
*(5/2) - 15*a**2*(a + b*x)**(7/2)/7 + 2*a*(a + b*x)**(9/2)/3 - (a + b*x)**(11/2)/11)/b**5)/b, Ne(b, 0)), ((A*x
**5/5 + B*x**6/6)/sqrt(a), True))

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