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

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

[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)

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Rubi [A]  time = 0.0571883, antiderivative size = 149, normalized size of antiderivative = 1., 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*}

Mathematica [A]  time = 0.0781851, size = 106, normalized size = 0.71 \[ \frac{2 \sqrt{a+b x} \left (16 a^2 b^3 x^2 (33 A+25 B x)-32 a^3 b^2 x (22 A+15 B x)+128 a^4 b (11 A+5 B x)-1280 a^5 B-10 a b^4 x^3 (44 A+35 B x)+35 b^5 x^4 (11 A+9 B x)\right )}{3465 b^6} \]

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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Maple [A]  time = 0.005, size = 119, normalized size = 0.8 \begin{align*}{\frac{630\,{b}^{5}B{x}^{5}+770\,A{x}^{4}{b}^{5}-700\,B{x}^{4}a{b}^{4}-880\,A{x}^{3}a{b}^{4}+800\,B{x}^{3}{a}^{2}{b}^{3}+1056\,A{x}^{2}{a}^{2}{b}^{3}-960\,B{x}^{2}{a}^{3}{b}^{2}-1408\,{a}^{3}{b}^{2}Ax+1280\,{a}^{4}bBx+2816\,A{a}^{4}b-2560\,B{a}^{5}}{3465\,{b}^{6}}\sqrt{bx+a}} \end{align*}

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 = 1.12021, size = 166, normalized size = 1.11 \begin{align*} \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}} \end{align*}

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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Fricas [A]  time = 2.41637, size = 289, normalized size = 1.94 \begin{align*} \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}} \end{align*}

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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Sympy [A]  time = 47.2333, size = 362, normalized size = 2.43 \begin{align*} \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} \end{align*}

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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Giac [A]  time = 1.15269, size = 192, normalized size = 1.29 \begin{align*} \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} \end{align*}

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