3.19.20 \(\int (a+b x) \sqrt {d+e x} (a^2+2 a b x+b^2 x^2) \, dx\)

Optimal. Leaf size=100 \[ -\frac {6 b^2 (d+e x)^{7/2} (b d-a e)}{7 e^4}+\frac {6 b (d+e x)^{5/2} (b d-a e)^2}{5 e^4}-\frac {2 (d+e x)^{3/2} (b d-a e)^3}{3 e^4}+\frac {2 b^3 (d+e x)^{9/2}}{9 e^4} \]

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Rubi [A]  time = 0.03, antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {27, 43} \begin {gather*} -\frac {6 b^2 (d+e x)^{7/2} (b d-a e)}{7 e^4}+\frac {6 b (d+e x)^{5/2} (b d-a e)^2}{5 e^4}-\frac {2 (d+e x)^{3/2} (b d-a e)^3}{3 e^4}+\frac {2 b^3 (d+e x)^{9/2}}{9 e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)*Sqrt[d + e*x]*(a^2 + 2*a*b*x + b^2*x^2),x]

[Out]

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

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

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Mathematica [A]  time = 0.05, size = 79, normalized size = 0.79 \begin {gather*} \frac {2 (d+e x)^{3/2} \left (-135 b^2 (d+e x)^2 (b d-a e)+189 b (d+e x) (b d-a e)^2-105 (b d-a e)^3+35 b^3 (d+e x)^3\right )}{315 e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)*Sqrt[d + e*x]*(a^2 + 2*a*b*x + b^2*x^2),x]

[Out]

(2*(d + e*x)^(3/2)*(-105*(b*d - a*e)^3 + 189*b*(b*d - a*e)^2*(d + e*x) - 135*b^2*(b*d - a*e)*(d + e*x)^2 + 35*
b^3*(d + e*x)^3))/(315*e^4)

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IntegrateAlgebraic [A]  time = 0.06, size = 132, normalized size = 1.32 \begin {gather*} \frac {2 (d+e x)^{3/2} \left (105 a^3 e^3+189 a^2 b e^2 (d+e x)-315 a^2 b d e^2+315 a b^2 d^2 e+135 a b^2 e (d+e x)^2-378 a b^2 d e (d+e x)-105 b^3 d^3+189 b^3 d^2 (d+e x)+35 b^3 (d+e x)^3-135 b^3 d (d+e x)^2\right )}{315 e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(a + b*x)*Sqrt[d + e*x]*(a^2 + 2*a*b*x + b^2*x^2),x]

[Out]

(2*(d + e*x)^(3/2)*(-105*b^3*d^3 + 315*a*b^2*d^2*e - 315*a^2*b*d*e^2 + 105*a^3*e^3 + 189*b^3*d^2*(d + e*x) - 3
78*a*b^2*d*e*(d + e*x) + 189*a^2*b*e^2*(d + e*x) - 135*b^3*d*(d + e*x)^2 + 135*a*b^2*e*(d + e*x)^2 + 35*b^3*(d
+ e*x)^3))/(315*e^4)

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fricas [A]  time = 0.41, size = 164, normalized size = 1.64 \begin {gather*} \frac {2 \, {\left (35 \, b^{3} e^{4} x^{4} - 16 \, b^{3} d^{4} + 72 \, a b^{2} d^{3} e - 126 \, a^{2} b d^{2} e^{2} + 105 \, a^{3} d e^{3} + 5 \, {\left (b^{3} d e^{3} + 27 \, a b^{2} e^{4}\right )} x^{3} - 3 \, {\left (2 \, b^{3} d^{2} e^{2} - 9 \, a b^{2} d e^{3} - 63 \, a^{2} b e^{4}\right )} x^{2} + {\left (8 \, b^{3} d^{3} e - 36 \, a b^{2} d^{2} e^{2} + 63 \, a^{2} b d e^{3} + 105 \, a^{3} e^{4}\right )} x\right )} \sqrt {e x + d}}{315 \, e^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/315*(35*b^3*e^4*x^4 - 16*b^3*d^4 + 72*a*b^2*d^3*e - 126*a^2*b*d^2*e^2 + 105*a^3*d*e^3 + 5*(b^3*d*e^3 + 27*a*
b^2*e^4)*x^3 - 3*(2*b^3*d^2*e^2 - 9*a*b^2*d*e^3 - 63*a^2*b*e^4)*x^2 + (8*b^3*d^3*e - 36*a*b^2*d^2*e^2 + 63*a^2
*b*d*e^3 + 105*a^3*e^4)*x)*sqrt(e*x + d)/e^4

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giac [B]  time = 0.24, size = 339, normalized size = 3.39 \begin {gather*} \frac {2}{315} \, {\left (315 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} a^{2} b d e^{\left (-1\right )} + 63 \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} - 10 \, {\left (x e + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {x e + d} d^{2}\right )} a b^{2} d e^{\left (-2\right )} + 9 \, {\left (5 \, {\left (x e + d\right )}^{\frac {7}{2}} - 21 \, {\left (x e + d\right )}^{\frac {5}{2}} d + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {x e + d} d^{3}\right )} b^{3} d e^{\left (-3\right )} + 63 \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} - 10 \, {\left (x e + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {x e + d} d^{2}\right )} a^{2} b e^{\left (-1\right )} + 27 \, {\left (5 \, {\left (x e + d\right )}^{\frac {7}{2}} - 21 \, {\left (x e + d\right )}^{\frac {5}{2}} d + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {x e + d} d^{3}\right )} a b^{2} e^{\left (-2\right )} + {\left (35 \, {\left (x e + d\right )}^{\frac {9}{2}} - 180 \, {\left (x e + d\right )}^{\frac {7}{2}} d + 378 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {x e + d} d^{4}\right )} b^{3} e^{\left (-3\right )} + 315 \, \sqrt {x e + d} a^{3} d + 105 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} a^{3}\right )} e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/315*(315*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*a^2*b*d*e^(-1) + 63*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d
+ 15*sqrt(x*e + d)*d^2)*a*b^2*d*e^(-2) + 9*(5*(x*e + d)^(7/2) - 21*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2
- 35*sqrt(x*e + d)*d^3)*b^3*d*e^(-3) + 63*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*a
^2*b*e^(-1) + 27*(5*(x*e + d)^(7/2) - 21*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2 - 35*sqrt(x*e + d)*d^3)*a*
b^2*e^(-2) + (35*(x*e + d)^(9/2) - 180*(x*e + d)^(7/2)*d + 378*(x*e + d)^(5/2)*d^2 - 420*(x*e + d)^(3/2)*d^3 +
315*sqrt(x*e + d)*d^4)*b^3*e^(-3) + 315*sqrt(x*e + d)*a^3*d + 105*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*a^3)*
e^(-1)

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maple [A]  time = 0.05, size = 116, normalized size = 1.16 \begin {gather*} \frac {2 \left (e x +d \right )^{\frac {3}{2}} \left (35 b^{3} e^{3} x^{3}+135 a \,b^{2} e^{3} x^{2}-30 b^{3} d \,e^{2} x^{2}+189 a^{2} b \,e^{3} x -108 a \,b^{2} d \,e^{2} x +24 b^{3} d^{2} e x +105 a^{3} e^{3}-126 a^{2} b d \,e^{2}+72 a \,b^{2} d^{2} e -16 b^{3} d^{3}\right )}{315 e^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/315*(e*x+d)^(3/2)*(35*b^3*e^3*x^3+135*a*b^2*e^3*x^2-30*b^3*d*e^2*x^2+189*a^2*b*e^3*x-108*a*b^2*d*e^2*x+24*b^
3*d^2*e*x+105*a^3*e^3-126*a^2*b*d*e^2+72*a*b^2*d^2*e-16*b^3*d^3)/e^4

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maxima [A]  time = 0.62, size = 118, normalized size = 1.18 \begin {gather*} \frac {2 \, {\left (35 \, {\left (e x + d\right )}^{\frac {9}{2}} b^{3} - 135 \, {\left (b^{3} d - a b^{2} e\right )} {\left (e x + d\right )}^{\frac {7}{2}} + 189 \, {\left (b^{3} d^{2} - 2 \, a b^{2} d e + a^{2} b e^{2}\right )} {\left (e x + d\right )}^{\frac {5}{2}} - 105 \, {\left (b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}\right )} {\left (e x + d\right )}^{\frac {3}{2}}\right )}}{315 \, e^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/315*(35*(e*x + d)^(9/2)*b^3 - 135*(b^3*d - a*b^2*e)*(e*x + d)^(7/2) + 189*(b^3*d^2 - 2*a*b^2*d*e + a^2*b*e^2
)*(e*x + d)^(5/2) - 105*(b^3*d^3 - 3*a*b^2*d^2*e + 3*a^2*b*d*e^2 - a^3*e^3)*(e*x + d)^(3/2))/e^4

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mupad [B]  time = 0.06, size = 87, normalized size = 0.87 \begin {gather*} \frac {2\,b^3\,{\left (d+e\,x\right )}^{9/2}}{9\,e^4}-\frac {\left (6\,b^3\,d-6\,a\,b^2\,e\right )\,{\left (d+e\,x\right )}^{7/2}}{7\,e^4}+\frac {2\,{\left (a\,e-b\,d\right )}^3\,{\left (d+e\,x\right )}^{3/2}}{3\,e^4}+\frac {6\,b\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{5/2}}{5\,e^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 4.58, size = 146, normalized size = 1.46 \begin {gather*} \frac {2 \left (\frac {b^{3} \left (d + e x\right )^{\frac {9}{2}}}{9 e^{3}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \left (3 a b^{2} e - 3 b^{3} d\right )}{7 e^{3}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \left (3 a^{2} b e^{2} - 6 a b^{2} d e + 3 b^{3} d^{2}\right )}{5 e^{3}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \left (a^{3} e^{3} - 3 a^{2} b d e^{2} + 3 a b^{2} d^{2} e - b^{3} d^{3}\right )}{3 e^{3}}\right )}{e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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