3.1737 \(\int (a+b x)^3 (A+B x) \sqrt{d+e x} \, dx\)

Optimal. Leaf size=173 \[ -\frac{2 b^2 (d+e x)^{9/2} (-3 a B e-A b e+4 b B d)}{9 e^5}+\frac{6 b (d+e x)^{7/2} (b d-a e) (-a B e-A b e+2 b B d)}{7 e^5}-\frac{2 (d+e x)^{5/2} (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{5 e^5}+\frac{2 (d+e x)^{3/2} (b d-a e)^3 (B d-A e)}{3 e^5}+\frac{2 b^3 B (d+e x)^{11/2}}{11 e^5} \]

[Out]

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

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Rubi [A]  time = 0.075903, antiderivative size = 173, normalized size of antiderivative = 1., 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} \[ -\frac{2 b^2 (d+e x)^{9/2} (-3 a B e-A b e+4 b B d)}{9 e^5}+\frac{6 b (d+e x)^{7/2} (b d-a e) (-a B e-A b e+2 b B d)}{7 e^5}-\frac{2 (d+e x)^{5/2} (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{5 e^5}+\frac{2 (d+e x)^{3/2} (b d-a e)^3 (B d-A e)}{3 e^5}+\frac{2 b^3 B (d+e x)^{11/2}}{11 e^5} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^3*(A + B*x)*Sqrt[d + e*x],x]

[Out]

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

Mathematica [A]  time = 0.156408, size = 145, normalized size = 0.84 \[ \frac{2 (d+e x)^{3/2} \left (-385 b^2 (d+e x)^3 (-3 a B e-A b e+4 b B d)+1485 b (d+e x)^2 (b d-a e) (-a B e-A b e+2 b B d)-693 (d+e x) (b d-a e)^2 (-a B e-3 A b e+4 b B d)+1155 (b d-a e)^3 (B d-A e)+315 b^3 B (d+e x)^4\right )}{3465 e^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^3*(A + B*x)*Sqrt[d + e*x],x]

[Out]

(2*(d + e*x)^(3/2)*(1155*(b*d - a*e)^3*(B*d - A*e) - 693*(b*d - a*e)^2*(4*b*B*d - 3*A*b*e - a*B*e)*(d + e*x) +
1485*b*(b*d - a*e)*(2*b*B*d - A*b*e - a*B*e)*(d + e*x)^2 - 385*b^2*(4*b*B*d - A*b*e - 3*a*B*e)*(d + e*x)^3 +
315*b^3*B*(d + e*x)^4))/(3465*e^5)

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Maple [A]  time = 0.005, size = 301, normalized size = 1.7 \begin{align*}{\frac{630\,{b}^{3}B{x}^{4}{e}^{4}+770\,A{b}^{3}{e}^{4}{x}^{3}+2310\,Ba{b}^{2}{e}^{4}{x}^{3}-560\,B{b}^{3}d{e}^{3}{x}^{3}+2970\,Aa{b}^{2}{e}^{4}{x}^{2}-660\,A{b}^{3}d{e}^{3}{x}^{2}+2970\,B{a}^{2}b{e}^{4}{x}^{2}-1980\,Ba{b}^{2}d{e}^{3}{x}^{2}+480\,B{b}^{3}{d}^{2}{e}^{2}{x}^{2}+4158\,A{a}^{2}b{e}^{4}x-2376\,Aa{b}^{2}d{e}^{3}x+528\,A{b}^{3}{d}^{2}{e}^{2}x+1386\,B{a}^{3}{e}^{4}x-2376\,B{a}^{2}bd{e}^{3}x+1584\,Ba{b}^{2}{d}^{2}{e}^{2}x-384\,B{b}^{3}{d}^{3}ex+2310\,{a}^{3}A{e}^{4}-2772\,A{a}^{2}bd{e}^{3}+1584\,Aa{b}^{2}{d}^{2}{e}^{2}-352\,A{b}^{3}{d}^{3}e-924\,B{a}^{3}d{e}^{3}+1584\,B{a}^{2}b{d}^{2}{e}^{2}-1056\,Ba{b}^{2}{d}^{3}e+256\,B{b}^{3}{d}^{4}}{3465\,{e}^{5}} \left ( ex+d \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^3*(B*x+A)*(e*x+d)^(1/2),x)

[Out]

2/3465*(e*x+d)^(3/2)*(315*B*b^3*e^4*x^4+385*A*b^3*e^4*x^3+1155*B*a*b^2*e^4*x^3-280*B*b^3*d*e^3*x^3+1485*A*a*b^
2*e^4*x^2-330*A*b^3*d*e^3*x^2+1485*B*a^2*b*e^4*x^2-990*B*a*b^2*d*e^3*x^2+240*B*b^3*d^2*e^2*x^2+2079*A*a^2*b*e^
4*x-1188*A*a*b^2*d*e^3*x+264*A*b^3*d^2*e^2*x+693*B*a^3*e^4*x-1188*B*a^2*b*d*e^3*x+792*B*a*b^2*d^2*e^2*x-192*B*
b^3*d^3*e*x+1155*A*a^3*e^4-1386*A*a^2*b*d*e^3+792*A*a*b^2*d^2*e^2-176*A*b^3*d^3*e-462*B*a^3*d*e^3+792*B*a^2*b*
d^2*e^2-528*B*a*b^2*d^3*e+128*B*b^3*d^4)/e^5

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Maxima [A]  time = 1.34275, size = 358, normalized size = 2.07 \begin{align*} \frac{2 \,{\left (315 \,{\left (e x + d\right )}^{\frac{11}{2}} B b^{3} - 385 \,{\left (4 \, B b^{3} d -{\left (3 \, B a b^{2} + A b^{3}\right )} e\right )}{\left (e x + d\right )}^{\frac{9}{2}} + 1485 \,{\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{7}{2}} - 693 \,{\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 )}{\left (e x + d\right )}^{\frac{5}{2}} + 1155 \,{\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 )}{\left (e x + d\right )}^{\frac{3}{2}}\right )}}{3465 \, e^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3*(B*x+A)*(e*x+d)^(1/2),x, algorithm="maxima")

[Out]

2/3465*(315*(e*x + d)^(11/2)*B*b^3 - 385*(4*B*b^3*d - (3*B*a*b^2 + A*b^3)*e)*(e*x + d)^(9/2) + 1485*(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)^(7/2) - 693*(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)*(e*x + d)^(5/2) + 1155*(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)*(e*x + d)^(3/2))/
e^5

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Fricas [B]  time = 1.37261, size = 780, normalized size = 4.51 \begin{align*} \frac{2 \,{\left (315 \, B b^{3} e^{5} x^{5} + 128 \, B b^{3} d^{5} + 1155 \, A a^{3} d e^{4} - 176 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{4} e + 792 \,{\left (B a^{2} b + A a b^{2}\right )} d^{3} e^{2} - 462 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} d^{2} e^{3} + 35 \,{\left (B b^{3} d e^{4} + 11 \,{\left (3 \, B a b^{2} + A b^{3}\right )} e^{5}\right )} x^{4} - 5 \,{\left (8 \, B b^{3} d^{2} e^{3} - 11 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d e^{4} - 297 \,{\left (B a^{2} b + A a b^{2}\right )} e^{5}\right )} x^{3} + 3 \,{\left (16 \, B b^{3} d^{3} e^{2} - 22 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{3} + 99 \,{\left (B a^{2} b + A a b^{2}\right )} d e^{4} + 231 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} e^{5}\right )} x^{2} -{\left (64 \, B b^{3} d^{4} e - 1155 \, A a^{3} e^{5} - 88 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e^{2} + 396 \,{\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{3} - 231 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{4}\right )} x\right )} \sqrt{e x + d}}{3465 \, e^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3*(B*x+A)*(e*x+d)^(1/2),x, algorithm="fricas")

[Out]

2/3465*(315*B*b^3*e^5*x^5 + 128*B*b^3*d^5 + 1155*A*a^3*d*e^4 - 176*(3*B*a*b^2 + A*b^3)*d^4*e + 792*(B*a^2*b +
A*a*b^2)*d^3*e^2 - 462*(B*a^3 + 3*A*a^2*b)*d^2*e^3 + 35*(B*b^3*d*e^4 + 11*(3*B*a*b^2 + A*b^3)*e^5)*x^4 - 5*(8*
B*b^3*d^2*e^3 - 11*(3*B*a*b^2 + A*b^3)*d*e^4 - 297*(B*a^2*b + A*a*b^2)*e^5)*x^3 + 3*(16*B*b^3*d^3*e^2 - 22*(3*
B*a*b^2 + A*b^3)*d^2*e^3 + 99*(B*a^2*b + A*a*b^2)*d*e^4 + 231*(B*a^3 + 3*A*a^2*b)*e^5)*x^2 - (64*B*b^3*d^4*e -
1155*A*a^3*e^5 - 88*(3*B*a*b^2 + A*b^3)*d^3*e^2 + 396*(B*a^2*b + A*a*b^2)*d^2*e^3 - 231*(B*a^3 + 3*A*a^2*b)*d
*e^4)*x)*sqrt(e*x + d)/e^5

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Sympy [B]  time = 4.60009, size = 342, normalized size = 1.98 \begin{align*} \frac{2 \left (\frac{B b^{3} \left (d + e x\right )^{\frac{11}{2}}}{11 e^{4}} + \frac{\left (d + e x\right )^{\frac{9}{2}} \left (A b^{3} e + 3 B a b^{2} e - 4 B b^{3} d\right )}{9 e^{4}} + \frac{\left (d + e x\right )^{\frac{7}{2}} \left (3 A a b^{2} e^{2} - 3 A b^{3} d e + 3 B a^{2} b e^{2} - 9 B a b^{2} d e + 6 B b^{3} d^{2}\right )}{7 e^{4}} + \frac{\left (d + e x\right )^{\frac{5}{2}} \left (3 A a^{2} b e^{3} - 6 A a b^{2} d e^{2} + 3 A b^{3} d^{2} e + B a^{3} e^{3} - 6 B a^{2} b d e^{2} + 9 B a b^{2} d^{2} e - 4 B b^{3} d^{3}\right )}{5 e^{4}} + \frac{\left (d + e x\right )^{\frac{3}{2}} \left (A a^{3} e^{4} - 3 A a^{2} b d e^{3} + 3 A a b^{2} d^{2} e^{2} - A b^{3} d^{3} e - B a^{3} d e^{3} + 3 B a^{2} b d^{2} e^{2} - 3 B a b^{2} d^{3} e + B b^{3} d^{4}\right )}{3 e^{4}}\right )}{e} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**3*(B*x+A)*(e*x+d)**(1/2),x)

[Out]

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

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Giac [B]  time = 2.03297, size = 471, normalized size = 2.72 \begin{align*} \frac{2}{3465} \,{\left (231 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 5 \,{\left (x e + d\right )}^{\frac{3}{2}} d\right )} B a^{3} e^{\left (-1\right )} + 693 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 5 \,{\left (x e + d\right )}^{\frac{3}{2}} d\right )} A a^{2} b e^{\left (-1\right )} + 99 \,{\left (15 \,{\left (x e + d\right )}^{\frac{7}{2}} - 42 \,{\left (x e + d\right )}^{\frac{5}{2}} d + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2}\right )} B a^{2} b e^{\left (-2\right )} + 99 \,{\left (15 \,{\left (x e + d\right )}^{\frac{7}{2}} - 42 \,{\left (x e + d\right )}^{\frac{5}{2}} d + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2}\right )} A a b^{2} e^{\left (-2\right )} + 33 \,{\left (35 \,{\left (x e + d\right )}^{\frac{9}{2}} - 135 \,{\left (x e + d\right )}^{\frac{7}{2}} d + 189 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{2} - 105 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{3}\right )} B a b^{2} e^{\left (-3\right )} + 11 \,{\left (35 \,{\left (x e + d\right )}^{\frac{9}{2}} - 135 \,{\left (x e + d\right )}^{\frac{7}{2}} d + 189 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{2} - 105 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{3}\right )} A b^{3} e^{\left (-3\right )} +{\left (315 \,{\left (x e + d\right )}^{\frac{11}{2}} - 1540 \,{\left (x e + d\right )}^{\frac{9}{2}} d + 2970 \,{\left (x e + d\right )}^{\frac{7}{2}} d^{2} - 2772 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{3} + 1155 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{4}\right )} B b^{3} e^{\left (-4\right )} + 1155 \,{\left (x e + d\right )}^{\frac{3}{2}} A a^{3}\right )} e^{\left (-1\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3*(B*x+A)*(e*x+d)^(1/2),x, algorithm="giac")

[Out]

2/3465*(231*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*B*a^3*e^(-1) + 693*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2
)*d)*A*a^2*b*e^(-1) + 99*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*B*a^2*b*e^(-2) +
99*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*A*a*b^2*e^(-2) + 33*(35*(x*e + d)^(9/
2) - 135*(x*e + d)^(7/2)*d + 189*(x*e + d)^(5/2)*d^2 - 105*(x*e + d)^(3/2)*d^3)*B*a*b^2*e^(-3) + 11*(35*(x*e +
d)^(9/2) - 135*(x*e + d)^(7/2)*d + 189*(x*e + d)^(5/2)*d^2 - 105*(x*e + d)^(3/2)*d^3)*A*b^3*e^(-3) + (315*(x*
e + d)^(11/2) - 1540*(x*e + d)^(9/2)*d + 2970*(x*e + d)^(7/2)*d^2 - 2772*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^
(3/2)*d^4)*B*b^3*e^(-4) + 1155*(x*e + d)^(3/2)*A*a^3)*e^(-1)