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

Optimal. Leaf size=129 \[ -\frac {8 b^3 (d+e x)^{9/2} (b d-a e)}{9 e^5}+\frac {12 b^2 (d+e x)^{7/2} (b d-a e)^2}{7 e^5}-\frac {8 b (d+e x)^{5/2} (b d-a e)^3}{5 e^5}+\frac {2 (d+e x)^{3/2} (b d-a e)^4}{3 e^5}+\frac {2 b^4 (d+e x)^{11/2}}{11 e^5} \]

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Rubi [A]  time = 0.05, antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {27, 43} \begin {gather*} -\frac {8 b^3 (d+e x)^{9/2} (b d-a e)}{9 e^5}+\frac {12 b^2 (d+e x)^{7/2} (b d-a e)^2}{7 e^5}-\frac {8 b (d+e x)^{5/2} (b d-a e)^3}{5 e^5}+\frac {2 (d+e x)^{3/2} (b d-a e)^4}{3 e^5}+\frac {2 b^4 (d+e x)^{11/2}}{11 e^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(b*d - a*e)^4*(d + e*x)^(3/2))/(3*e^5) - (8*b*(b*d - a*e)^3*(d + e*x)^(5/2))/(5*e^5) + (12*b^2*(b*d - a*e)^
2*(d + e*x)^(7/2))/(7*e^5) - (8*b^3*(b*d - a*e)*(d + e*x)^(9/2))/(9*e^5) + (2*b^4*(d + e*x)^(11/2))/(11*e^5)

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 \sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx &=\int (a+b x)^4 \sqrt {d+e x} \, dx\\ &=\int \left (\frac {(-b d+a e)^4 \sqrt {d+e x}}{e^4}-\frac {4 b (b d-a e)^3 (d+e x)^{3/2}}{e^4}+\frac {6 b^2 (b d-a e)^2 (d+e x)^{5/2}}{e^4}-\frac {4 b^3 (b d-a e) (d+e x)^{7/2}}{e^4}+\frac {b^4 (d+e x)^{9/2}}{e^4}\right ) \, dx\\ &=\frac {2 (b d-a e)^4 (d+e x)^{3/2}}{3 e^5}-\frac {8 b (b d-a e)^3 (d+e x)^{5/2}}{5 e^5}+\frac {12 b^2 (b d-a e)^2 (d+e x)^{7/2}}{7 e^5}-\frac {8 b^3 (b d-a e) (d+e x)^{9/2}}{9 e^5}+\frac {2 b^4 (d+e x)^{11/2}}{11 e^5}\\ \end {align*}

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Mathematica [A]  time = 0.08, size = 101, normalized size = 0.78 \begin {gather*} \frac {2 (d+e x)^{3/2} \left (-1540 b^3 (d+e x)^3 (b d-a e)+2970 b^2 (d+e x)^2 (b d-a e)^2-2772 b (d+e x) (b d-a e)^3+1155 (b d-a e)^4+315 b^4 (d+e x)^4\right )}{3465 e^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(d + e*x)^(3/2)*(1155*(b*d - a*e)^4 - 2772*b*(b*d - a*e)^3*(d + e*x) + 2970*b^2*(b*d - a*e)^2*(d + e*x)^2 -
1540*b^3*(b*d - a*e)*(d + e*x)^3 + 315*b^4*(d + e*x)^4))/(3465*e^5)

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IntegrateAlgebraic [A]  time = 0.08, size = 213, normalized size = 1.65 \begin {gather*} \frac {2 (d+e x)^{3/2} \left (1155 a^4 e^4+2772 a^3 b e^3 (d+e x)-4620 a^3 b d e^3+6930 a^2 b^2 d^2 e^2+2970 a^2 b^2 e^2 (d+e x)^2-8316 a^2 b^2 d e^2 (d+e x)-4620 a b^3 d^3 e+8316 a b^3 d^2 e (d+e x)+1540 a b^3 e (d+e x)^3-5940 a b^3 d e (d+e x)^2+1155 b^4 d^4-2772 b^4 d^3 (d+e x)+2970 b^4 d^2 (d+e x)^2+315 b^4 (d+e x)^4-1540 b^4 d (d+e x)^3\right )}{3465 e^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(d + e*x)^(3/2)*(1155*b^4*d^4 - 4620*a*b^3*d^3*e + 6930*a^2*b^2*d^2*e^2 - 4620*a^3*b*d*e^3 + 1155*a^4*e^4 -
2772*b^4*d^3*(d + e*x) + 8316*a*b^3*d^2*e*(d + e*x) - 8316*a^2*b^2*d*e^2*(d + e*x) + 2772*a^3*b*e^3*(d + e*x)
+ 2970*b^4*d^2*(d + e*x)^2 - 5940*a*b^3*d*e*(d + e*x)^2 + 2970*a^2*b^2*e^2*(d + e*x)^2 - 1540*b^4*d*(d + e*x)
^3 + 1540*a*b^3*e*(d + e*x)^3 + 315*b^4*(d + e*x)^4))/(3465*e^5)

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fricas [B]  time = 0.39, size = 245, normalized size = 1.90 \begin {gather*} \frac {2 \, {\left (315 \, b^{4} e^{5} x^{5} + 128 \, b^{4} d^{5} - 704 \, a b^{3} d^{4} e + 1584 \, a^{2} b^{2} d^{3} e^{2} - 1848 \, a^{3} b d^{2} e^{3} + 1155 \, a^{4} d e^{4} + 35 \, {\left (b^{4} d e^{4} + 44 \, a b^{3} e^{5}\right )} x^{4} - 10 \, {\left (4 \, b^{4} d^{2} e^{3} - 22 \, a b^{3} d e^{4} - 297 \, a^{2} b^{2} e^{5}\right )} x^{3} + 6 \, {\left (8 \, b^{4} d^{3} e^{2} - 44 \, a b^{3} d^{2} e^{3} + 99 \, a^{2} b^{2} d e^{4} + 462 \, a^{3} b e^{5}\right )} x^{2} - {\left (64 \, b^{4} d^{4} e - 352 \, a b^{3} d^{3} e^{2} + 792 \, a^{2} b^{2} d^{2} e^{3} - 924 \, a^{3} b d e^{4} - 1155 \, a^{4} e^{5}\right )} x\right )} \sqrt {e x + d}}{3465 \, e^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3465*(315*b^4*e^5*x^5 + 128*b^4*d^5 - 704*a*b^3*d^4*e + 1584*a^2*b^2*d^3*e^2 - 1848*a^3*b*d^2*e^3 + 1155*a^4
*d*e^4 + 35*(b^4*d*e^4 + 44*a*b^3*e^5)*x^4 - 10*(4*b^4*d^2*e^3 - 22*a*b^3*d*e^4 - 297*a^2*b^2*e^5)*x^3 + 6*(8*
b^4*d^3*e^2 - 44*a*b^3*d^2*e^3 + 99*a^2*b^2*d*e^4 + 462*a^3*b*e^5)*x^2 - (64*b^4*d^4*e - 352*a*b^3*d^3*e^2 + 7
92*a^2*b^2*d^2*e^3 - 924*a^3*b*d*e^4 - 1155*a^4*e^5)*x)*sqrt(e*x + d)/e^5

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giac [B]  time = 0.20, size = 496, normalized size = 3.84 \begin {gather*} \frac {2}{3465} \, {\left (4620 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} a^{3} b d e^{\left (-1\right )} + 1386 \, {\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^{2} d e^{\left (-2\right )} + 396 \, {\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^{3} d e^{\left (-3\right )} + 11 \, {\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^{4} d e^{\left (-4\right )} + 924 \, {\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^{3} b e^{\left (-1\right )} + 594 \, {\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^{2} b^{2} e^{\left (-2\right )} + 44 \, {\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 )} a b^{3} e^{\left (-3\right )} + 5 \, {\left (63 \, {\left (x e + d\right )}^{\frac {11}{2}} - 385 \, {\left (x e + d\right )}^{\frac {9}{2}} d + 990 \, {\left (x e + d\right )}^{\frac {7}{2}} d^{2} - 1386 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{3} + 1155 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{4} - 693 \, \sqrt {x e + d} d^{5}\right )} b^{4} e^{\left (-4\right )} + 3465 \, \sqrt {x e + d} a^{4} d + 1155 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} a^{4}\right )} e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3465*(4620*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*a^3*b*d*e^(-1) + 1386*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/
2)*d + 15*sqrt(x*e + d)*d^2)*a^2*b^2*d*e^(-2) + 396*(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^3*d*e^(-3) + 11*(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^4*d*e^(-4) + 924*(3*(x*e + d)^(5/2) - 10*(x*
e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*a^3*b*e^(-1) + 594*(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^2*b^2*e^(-2) + 44*(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)*a*b^3*e^(-3) + 5*(63*(x*e + d)^(11/2) -
385*(x*e + d)^(9/2)*d + 990*(x*e + d)^(7/2)*d^2 - 1386*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4 - 693*sq
rt(x*e + d)*d^5)*b^4*e^(-4) + 3465*sqrt(x*e + d)*a^4*d + 1155*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*a^4)*e^(-1
)

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maple [A]  time = 0.05, size = 186, normalized size = 1.44 \begin {gather*} \frac {2 \left (e x +d \right )^{\frac {3}{2}} \left (315 b^{4} e^{4} x^{4}+1540 a \,b^{3} e^{4} x^{3}-280 b^{4} d \,e^{3} x^{3}+2970 a^{2} b^{2} e^{4} x^{2}-1320 a \,b^{3} d \,e^{3} x^{2}+240 b^{4} d^{2} e^{2} x^{2}+2772 a^{3} b \,e^{4} x -2376 a^{2} b^{2} d \,e^{3} x +1056 a \,b^{3} d^{2} e^{2} x -192 b^{4} d^{3} e x +1155 a^{4} e^{4}-1848 a^{3} b d \,e^{3}+1584 a^{2} b^{2} d^{2} e^{2}-704 a \,b^{3} d^{3} e +128 b^{4} d^{4}\right )}{3465 e^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3465*(e*x+d)^(3/2)*(315*b^4*e^4*x^4+1540*a*b^3*e^4*x^3-280*b^4*d*e^3*x^3+2970*a^2*b^2*e^4*x^2-1320*a*b^3*d*e
^3*x^2+240*b^4*d^2*e^2*x^2+2772*a^3*b*e^4*x-2376*a^2*b^2*d*e^3*x+1056*a*b^3*d^2*e^2*x-192*b^4*d^3*e*x+1155*a^4
*e^4-1848*a^3*b*d*e^3+1584*a^2*b^2*d^2*e^2-704*a*b^3*d^3*e+128*b^4*d^4)/e^5

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maxima [A]  time = 1.07, size = 181, normalized size = 1.40 \begin {gather*} \frac {2 \, {\left (315 \, {\left (e x + d\right )}^{\frac {11}{2}} b^{4} - 1540 \, {\left (b^{4} d - a b^{3} e\right )} {\left (e x + d\right )}^{\frac {9}{2}} + 2970 \, {\left (b^{4} d^{2} - 2 \, a b^{3} d e + a^{2} b^{2} e^{2}\right )} {\left (e x + d\right )}^{\frac {7}{2}} - 2772 \, {\left (b^{4} d^{3} - 3 \, a b^{3} d^{2} e + 3 \, a^{2} b^{2} d e^{2} - a^{3} b e^{3}\right )} {\left (e x + d\right )}^{\frac {5}{2}} + 1155 \, {\left (b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}\right )} {\left (e x + d\right )}^{\frac {3}{2}}\right )}}{3465 \, e^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3465*(315*(e*x + d)^(11/2)*b^4 - 1540*(b^4*d - a*b^3*e)*(e*x + d)^(9/2) + 2970*(b^4*d^2 - 2*a*b^3*d*e + a^2*
b^2*e^2)*(e*x + d)^(7/2) - 2772*(b^4*d^3 - 3*a*b^3*d^2*e + 3*a^2*b^2*d*e^2 - a^3*b*e^3)*(e*x + d)^(5/2) + 1155
*(b^4*d^4 - 4*a*b^3*d^3*e + 6*a^2*b^2*d^2*e^2 - 4*a^3*b*d*e^3 + a^4*e^4)*(e*x + d)^(3/2))/e^5

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mupad [B]  time = 0.04, size = 112, normalized size = 0.87 \begin {gather*} \frac {2\,b^4\,{\left (d+e\,x\right )}^{11/2}}{11\,e^5}-\frac {\left (8\,b^4\,d-8\,a\,b^3\,e\right )\,{\left (d+e\,x\right )}^{9/2}}{9\,e^5}+\frac {2\,{\left (a\,e-b\,d\right )}^4\,{\left (d+e\,x\right )}^{3/2}}{3\,e^5}+\frac {12\,b^2\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{7/2}}{7\,e^5}+\frac {8\,b\,{\left (a\,e-b\,d\right )}^3\,{\left (d+e\,x\right )}^{5/2}}{5\,e^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*b^4*(d + e*x)^(11/2))/(11*e^5) - ((8*b^4*d - 8*a*b^3*e)*(d + e*x)^(9/2))/(9*e^5) + (2*(a*e - b*d)^4*(d + e*
x)^(3/2))/(3*e^5) + (12*b^2*(a*e - b*d)^2*(d + e*x)^(7/2))/(7*e^5) + (8*b*(a*e - b*d)^3*(d + e*x)^(5/2))/(5*e^
5)

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sympy [A]  time = 5.28, size = 223, normalized size = 1.73 \begin {gather*} \frac {2 \left (\frac {b^{4} \left (d + e x\right )^{\frac {11}{2}}}{11 e^{4}} + \frac {\left (d + e x\right )^{\frac {9}{2}} \left (4 a b^{3} e - 4 b^{4} d\right )}{9 e^{4}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \left (6 a^{2} b^{2} e^{2} - 12 a b^{3} d e + 6 b^{4} d^{2}\right )}{7 e^{4}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \left (4 a^{3} b e^{3} - 12 a^{2} b^{2} d e^{2} + 12 a b^{3} d^{2} e - 4 b^{4} d^{3}\right )}{5 e^{4}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \left (a^{4} e^{4} - 4 a^{3} b d e^{3} + 6 a^{2} b^{2} d^{2} e^{2} - 4 a b^{3} d^{3} e + b^{4} d^{4}\right )}{3 e^{4}}\right )}{e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(b**4*(d + e*x)**(11/2)/(11*e**4) + (d + e*x)**(9/2)*(4*a*b**3*e - 4*b**4*d)/(9*e**4) + (d + e*x)**(7/2)*(6*
a**2*b**2*e**2 - 12*a*b**3*d*e + 6*b**4*d**2)/(7*e**4) + (d + e*x)**(5/2)*(4*a**3*b*e**3 - 12*a**2*b**2*d*e**2
+ 12*a*b**3*d**2*e - 4*b**4*d**3)/(5*e**4) + (d + e*x)**(3/2)*(a**4*e**4 - 4*a**3*b*d*e**3 + 6*a**2*b**2*d**2
*e**2 - 4*a*b**3*d**3*e + b**4*d**4)/(3*e**4))/e

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