3.14.43 \(\int \frac {(a^2+2 a b x+b^2 x^2)^3}{\sqrt {d+e x}} \, dx\)

Optimal. Leaf size=181 \[ -\frac {12 b^5 (d+e x)^{11/2} (b d-a e)}{11 e^7}+\frac {10 b^4 (d+e x)^{9/2} (b d-a e)^2}{3 e^7}-\frac {40 b^3 (d+e x)^{7/2} (b d-a e)^3}{7 e^7}+\frac {6 b^2 (d+e x)^{5/2} (b d-a e)^4}{e^7}-\frac {4 b (d+e x)^{3/2} (b d-a e)^5}{e^7}+\frac {2 \sqrt {d+e x} (b d-a e)^6}{e^7}+\frac {2 b^6 (d+e x)^{13/2}}{13 e^7} \]

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Rubi [A]  time = 0.06, antiderivative size = 181, 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 {12 b^5 (d+e x)^{11/2} (b d-a e)}{11 e^7}+\frac {10 b^4 (d+e x)^{9/2} (b d-a e)^2}{3 e^7}-\frac {40 b^3 (d+e x)^{7/2} (b d-a e)^3}{7 e^7}+\frac {6 b^2 (d+e x)^{5/2} (b d-a e)^4}{e^7}-\frac {4 b (d+e x)^{3/2} (b d-a e)^5}{e^7}+\frac {2 \sqrt {d+e x} (b d-a e)^6}{e^7}+\frac {2 b^6 (d+e x)^{13/2}}{13 e^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a^2 + 2*a*b*x + b^2*x^2)^3/Sqrt[d + e*x],x]

[Out]

(2*(b*d - a*e)^6*Sqrt[d + e*x])/e^7 - (4*b*(b*d - a*e)^5*(d + e*x)^(3/2))/e^7 + (6*b^2*(b*d - a*e)^4*(d + e*x)
^(5/2))/e^7 - (40*b^3*(b*d - a*e)^3*(d + e*x)^(7/2))/(7*e^7) + (10*b^4*(b*d - a*e)^2*(d + e*x)^(9/2))/(3*e^7)
- (12*b^5*(b*d - a*e)*(d + e*x)^(11/2))/(11*e^7) + (2*b^6*(d + e*x)^(13/2))/(13*e^7)

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 \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{\sqrt {d+e x}} \, dx &=\int \frac {(a+b x)^6}{\sqrt {d+e x}} \, dx\\ &=\int \left (\frac {(-b d+a e)^6}{e^6 \sqrt {d+e x}}-\frac {6 b (b d-a e)^5 \sqrt {d+e x}}{e^6}+\frac {15 b^2 (b d-a e)^4 (d+e x)^{3/2}}{e^6}-\frac {20 b^3 (b d-a e)^3 (d+e x)^{5/2}}{e^6}+\frac {15 b^4 (b d-a e)^2 (d+e x)^{7/2}}{e^6}-\frac {6 b^5 (b d-a e) (d+e x)^{9/2}}{e^6}+\frac {b^6 (d+e x)^{11/2}}{e^6}\right ) \, dx\\ &=\frac {2 (b d-a e)^6 \sqrt {d+e x}}{e^7}-\frac {4 b (b d-a e)^5 (d+e x)^{3/2}}{e^7}+\frac {6 b^2 (b d-a e)^4 (d+e x)^{5/2}}{e^7}-\frac {40 b^3 (b d-a e)^3 (d+e x)^{7/2}}{7 e^7}+\frac {10 b^4 (b d-a e)^2 (d+e x)^{9/2}}{3 e^7}-\frac {12 b^5 (b d-a e) (d+e x)^{11/2}}{11 e^7}+\frac {2 b^6 (d+e x)^{13/2}}{13 e^7}\\ \end {align*}

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Mathematica [A]  time = 0.08, size = 145, normalized size = 0.80 \begin {gather*} \frac {2 \sqrt {d+e x} \left (-1638 b^5 (d+e x)^5 (b d-a e)+5005 b^4 (d+e x)^4 (b d-a e)^2-8580 b^3 (d+e x)^3 (b d-a e)^3+9009 b^2 (d+e x)^2 (b d-a e)^4-6006 b (d+e x) (b d-a e)^5+3003 (b d-a e)^6+231 b^6 (d+e x)^6\right )}{3003 e^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a^2 + 2*a*b*x + b^2*x^2)^3/Sqrt[d + e*x],x]

[Out]

(2*Sqrt[d + e*x]*(3003*(b*d - a*e)^6 - 6006*b*(b*d - a*e)^5*(d + e*x) + 9009*b^2*(b*d - a*e)^4*(d + e*x)^2 - 8
580*b^3*(b*d - a*e)^3*(d + e*x)^3 + 5005*b^4*(b*d - a*e)^2*(d + e*x)^4 - 1638*b^5*(b*d - a*e)*(d + e*x)^5 + 23
1*b^6*(d + e*x)^6))/(3003*e^7)

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IntegrateAlgebraic [B]  time = 0.13, size = 438, normalized size = 2.42 \begin {gather*} \frac {2 \sqrt {d+e x} \left (3003 a^6 e^6+6006 a^5 b e^5 (d+e x)-18018 a^5 b d e^5+45045 a^4 b^2 d^2 e^4+9009 a^4 b^2 e^4 (d+e x)^2-30030 a^4 b^2 d e^4 (d+e x)-60060 a^3 b^3 d^3 e^3+60060 a^3 b^3 d^2 e^3 (d+e x)+8580 a^3 b^3 e^3 (d+e x)^3-36036 a^3 b^3 d e^3 (d+e x)^2+45045 a^2 b^4 d^4 e^2-60060 a^2 b^4 d^3 e^2 (d+e x)+54054 a^2 b^4 d^2 e^2 (d+e x)^2+5005 a^2 b^4 e^2 (d+e x)^4-25740 a^2 b^4 d e^2 (d+e x)^3-18018 a b^5 d^5 e+30030 a b^5 d^4 e (d+e x)-36036 a b^5 d^3 e (d+e x)^2+25740 a b^5 d^2 e (d+e x)^3+1638 a b^5 e (d+e x)^5-10010 a b^5 d e (d+e x)^4+3003 b^6 d^6-6006 b^6 d^5 (d+e x)+9009 b^6 d^4 (d+e x)^2-8580 b^6 d^3 (d+e x)^3+5005 b^6 d^2 (d+e x)^4+231 b^6 (d+e x)^6-1638 b^6 d (d+e x)^5\right )}{3003 e^7} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(a^2 + 2*a*b*x + b^2*x^2)^3/Sqrt[d + e*x],x]

[Out]

(2*Sqrt[d + e*x]*(3003*b^6*d^6 - 18018*a*b^5*d^5*e + 45045*a^2*b^4*d^4*e^2 - 60060*a^3*b^3*d^3*e^3 + 45045*a^4
*b^2*d^2*e^4 - 18018*a^5*b*d*e^5 + 3003*a^6*e^6 - 6006*b^6*d^5*(d + e*x) + 30030*a*b^5*d^4*e*(d + e*x) - 60060
*a^2*b^4*d^3*e^2*(d + e*x) + 60060*a^3*b^3*d^2*e^3*(d + e*x) - 30030*a^4*b^2*d*e^4*(d + e*x) + 6006*a^5*b*e^5*
(d + e*x) + 9009*b^6*d^4*(d + e*x)^2 - 36036*a*b^5*d^3*e*(d + e*x)^2 + 54054*a^2*b^4*d^2*e^2*(d + e*x)^2 - 360
36*a^3*b^3*d*e^3*(d + e*x)^2 + 9009*a^4*b^2*e^4*(d + e*x)^2 - 8580*b^6*d^3*(d + e*x)^3 + 25740*a*b^5*d^2*e*(d
+ e*x)^3 - 25740*a^2*b^4*d*e^2*(d + e*x)^3 + 8580*a^3*b^3*e^3*(d + e*x)^3 + 5005*b^6*d^2*(d + e*x)^4 - 10010*a
*b^5*d*e*(d + e*x)^4 + 5005*a^2*b^4*e^2*(d + e*x)^4 - 1638*b^6*d*(d + e*x)^5 + 1638*a*b^5*e*(d + e*x)^5 + 231*
b^6*(d + e*x)^6))/(3003*e^7)

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fricas [B]  time = 0.39, size = 356, normalized size = 1.97 \begin {gather*} \frac {2 \, {\left (231 \, b^{6} e^{6} x^{6} + 1024 \, b^{6} d^{6} - 6656 \, a b^{5} d^{5} e + 18304 \, a^{2} b^{4} d^{4} e^{2} - 27456 \, a^{3} b^{3} d^{3} e^{3} + 24024 \, a^{4} b^{2} d^{2} e^{4} - 12012 \, a^{5} b d e^{5} + 3003 \, a^{6} e^{6} - 126 \, {\left (2 \, b^{6} d e^{5} - 13 \, a b^{5} e^{6}\right )} x^{5} + 35 \, {\left (8 \, b^{6} d^{2} e^{4} - 52 \, a b^{5} d e^{5} + 143 \, a^{2} b^{4} e^{6}\right )} x^{4} - 20 \, {\left (16 \, b^{6} d^{3} e^{3} - 104 \, a b^{5} d^{2} e^{4} + 286 \, a^{2} b^{4} d e^{5} - 429 \, a^{3} b^{3} e^{6}\right )} x^{3} + 3 \, {\left (128 \, b^{6} d^{4} e^{2} - 832 \, a b^{5} d^{3} e^{3} + 2288 \, a^{2} b^{4} d^{2} e^{4} - 3432 \, a^{3} b^{3} d e^{5} + 3003 \, a^{4} b^{2} e^{6}\right )} x^{2} - 2 \, {\left (256 \, b^{6} d^{5} e - 1664 \, a b^{5} d^{4} e^{2} + 4576 \, a^{2} b^{4} d^{3} e^{3} - 6864 \, a^{3} b^{3} d^{2} e^{4} + 6006 \, a^{4} b^{2} d e^{5} - 3003 \, a^{5} b e^{6}\right )} x\right )} \sqrt {e x + d}}{3003 \, e^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3003*(231*b^6*e^6*x^6 + 1024*b^6*d^6 - 6656*a*b^5*d^5*e + 18304*a^2*b^4*d^4*e^2 - 27456*a^3*b^3*d^3*e^3 + 24
024*a^4*b^2*d^2*e^4 - 12012*a^5*b*d*e^5 + 3003*a^6*e^6 - 126*(2*b^6*d*e^5 - 13*a*b^5*e^6)*x^5 + 35*(8*b^6*d^2*
e^4 - 52*a*b^5*d*e^5 + 143*a^2*b^4*e^6)*x^4 - 20*(16*b^6*d^3*e^3 - 104*a*b^5*d^2*e^4 + 286*a^2*b^4*d*e^5 - 429
*a^3*b^3*e^6)*x^3 + 3*(128*b^6*d^4*e^2 - 832*a*b^5*d^3*e^3 + 2288*a^2*b^4*d^2*e^4 - 3432*a^3*b^3*d*e^5 + 3003*
a^4*b^2*e^6)*x^2 - 2*(256*b^6*d^5*e - 1664*a*b^5*d^4*e^2 + 4576*a^2*b^4*d^3*e^3 - 6864*a^3*b^3*d^2*e^4 + 6006*
a^4*b^2*d*e^5 - 3003*a^5*b*e^6)*x)*sqrt(e*x + d)/e^7

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giac [B]  time = 0.22, size = 395, normalized size = 2.18 \begin {gather*} \frac {2}{3003} \, {\left (6006 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} a^{5} b e^{\left (-1\right )} + 3003 \, {\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^{4} b^{2} e^{\left (-2\right )} + 1716 \, {\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^{3} b^{3} e^{\left (-3\right )} + 143 \, {\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^{2} b^{4} e^{\left (-4\right )} + 26 \, {\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 )} a b^{5} e^{\left (-5\right )} + {\left (231 \, {\left (x e + d\right )}^{\frac {13}{2}} - 1638 \, {\left (x e + d\right )}^{\frac {11}{2}} d + 5005 \, {\left (x e + d\right )}^{\frac {9}{2}} d^{2} - 8580 \, {\left (x e + d\right )}^{\frac {7}{2}} d^{3} + 9009 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{4} - 6006 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{5} + 3003 \, \sqrt {x e + d} d^{6}\right )} b^{6} e^{\left (-6\right )} + 3003 \, \sqrt {x e + d} a^{6}\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)^3/(e*x+d)^(1/2),x, algorithm="giac")

[Out]

2/3003*(6006*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*a^5*b*e^(-1) + 3003*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)
*d + 15*sqrt(x*e + d)*d^2)*a^4*b^2*e^(-2) + 1716*(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^3*b^3*e^(-3) + 143*(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^2*b^4*e^(-4) + 26*(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*sqrt(x*
e + d)*d^5)*a*b^5*e^(-5) + (231*(x*e + d)^(13/2) - 1638*(x*e + d)^(11/2)*d + 5005*(x*e + d)^(9/2)*d^2 - 8580*(
x*e + d)^(7/2)*d^3 + 9009*(x*e + d)^(5/2)*d^4 - 6006*(x*e + d)^(3/2)*d^5 + 3003*sqrt(x*e + d)*d^6)*b^6*e^(-6)
+ 3003*sqrt(x*e + d)*a^6)*e^(-1)

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maple [B]  time = 0.04, size = 377, normalized size = 2.08 \begin {gather*} \frac {2 \left (231 b^{6} e^{6} x^{6}+1638 a \,b^{5} e^{6} x^{5}-252 b^{6} d \,e^{5} x^{5}+5005 a^{2} b^{4} e^{6} x^{4}-1820 a \,b^{5} d \,e^{5} x^{4}+280 b^{6} d^{2} e^{4} x^{4}+8580 a^{3} b^{3} e^{6} x^{3}-5720 a^{2} b^{4} d \,e^{5} x^{3}+2080 a \,b^{5} d^{2} e^{4} x^{3}-320 b^{6} d^{3} e^{3} x^{3}+9009 a^{4} b^{2} e^{6} x^{2}-10296 a^{3} b^{3} d \,e^{5} x^{2}+6864 a^{2} b^{4} d^{2} e^{4} x^{2}-2496 a \,b^{5} d^{3} e^{3} x^{2}+384 b^{6} d^{4} e^{2} x^{2}+6006 a^{5} b \,e^{6} x -12012 a^{4} b^{2} d \,e^{5} x +13728 a^{3} b^{3} d^{2} e^{4} x -9152 a^{2} b^{4} d^{3} e^{3} x +3328 a \,b^{5} d^{4} e^{2} x -512 b^{6} d^{5} e x +3003 a^{6} e^{6}-12012 a^{5} b d \,e^{5}+24024 a^{4} b^{2} d^{2} e^{4}-27456 a^{3} b^{3} d^{3} e^{3}+18304 a^{2} b^{4} d^{4} e^{2}-6656 a \,b^{5} d^{5} e +1024 b^{6} d^{6}\right ) \sqrt {e x +d}}{3003 e^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3003*(231*b^6*e^6*x^6+1638*a*b^5*e^6*x^5-252*b^6*d*e^5*x^5+5005*a^2*b^4*e^6*x^4-1820*a*b^5*d*e^5*x^4+280*b^6
*d^2*e^4*x^4+8580*a^3*b^3*e^6*x^3-5720*a^2*b^4*d*e^5*x^3+2080*a*b^5*d^2*e^4*x^3-320*b^6*d^3*e^3*x^3+9009*a^4*b
^2*e^6*x^2-10296*a^3*b^3*d*e^5*x^2+6864*a^2*b^4*d^2*e^4*x^2-2496*a*b^5*d^3*e^3*x^2+384*b^6*d^4*e^2*x^2+6006*a^
5*b*e^6*x-12012*a^4*b^2*d*e^5*x+13728*a^3*b^3*d^2*e^4*x-9152*a^2*b^4*d^3*e^3*x+3328*a*b^5*d^4*e^2*x-512*b^6*d^
5*e*x+3003*a^6*e^6-12012*a^5*b*d*e^5+24024*a^4*b^2*d^2*e^4-27456*a^3*b^3*d^3*e^3+18304*a^2*b^4*d^4*e^2-6656*a*
b^5*d^5*e+1024*b^6*d^6)*(e*x+d)^(1/2)/e^7

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maxima [B]  time = 1.17, size = 540, normalized size = 2.98 \begin {gather*} \frac {2 \, {\left (15015 \, \sqrt {e x + d} a^{6} + 3003 \, {\left (\frac {10 \, {\left ({\left (e x + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {e x + d} d\right )} a b}{e} + \frac {{\left (3 \, {\left (e x + d\right )}^{\frac {5}{2}} - 10 \, {\left (e x + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {e x + d} d^{2}\right )} b^{2}}{e^{2}}\right )} a^{4} + \frac {3432 \, {\left (5 \, {\left (e x + d\right )}^{\frac {7}{2}} - 21 \, {\left (e x + d\right )}^{\frac {5}{2}} d + 35 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {e x + d} d^{3}\right )} a^{3} b^{3}}{e^{3}} + 143 \, {\left (\frac {84 \, {\left (3 \, {\left (e x + d\right )}^{\frac {5}{2}} - 10 \, {\left (e x + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {e x + d} d^{2}\right )} a^{2} b^{2}}{e^{2}} + \frac {36 \, {\left (5 \, {\left (e x + d\right )}^{\frac {7}{2}} - 21 \, {\left (e x + d\right )}^{\frac {5}{2}} d + 35 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {e x + d} d^{3}\right )} a b^{3}}{e^{3}} + \frac {{\left (35 \, {\left (e x + d\right )}^{\frac {9}{2}} - 180 \, {\left (e x + d\right )}^{\frac {7}{2}} d + 378 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {e x + d} d^{4}\right )} b^{4}}{e^{4}}\right )} a^{2} + \frac {572 \, {\left (35 \, {\left (e x + d\right )}^{\frac {9}{2}} - 180 \, {\left (e x + d\right )}^{\frac {7}{2}} d + 378 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {e x + d} d^{4}\right )} a^{2} b^{4}}{e^{4}} + \frac {130 \, {\left (63 \, {\left (e x + d\right )}^{\frac {11}{2}} - 385 \, {\left (e x + d\right )}^{\frac {9}{2}} d + 990 \, {\left (e x + d\right )}^{\frac {7}{2}} d^{2} - 1386 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{3} + 1155 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{4} - 693 \, \sqrt {e x + d} d^{5}\right )} a b^{5}}{e^{5}} + \frac {5 \, {\left (231 \, {\left (e x + d\right )}^{\frac {13}{2}} - 1638 \, {\left (e x + d\right )}^{\frac {11}{2}} d + 5005 \, {\left (e x + d\right )}^{\frac {9}{2}} d^{2} - 8580 \, {\left (e x + d\right )}^{\frac {7}{2}} d^{3} + 9009 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{4} - 6006 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{5} + 3003 \, \sqrt {e x + d} d^{6}\right )} b^{6}}{e^{6}}\right )}}{15015 \, 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)^3/(e*x+d)^(1/2),x, algorithm="maxima")

[Out]

2/15015*(15015*sqrt(e*x + d)*a^6 + 3003*(10*((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d)*a*b/e + (3*(e*x + d)^(5/2) -
10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*b^2/e^2)*a^4 + 3432*(5*(e*x + d)^(7/2) - 21*(e*x + d)^(5/2)*d +
35*(e*x + d)^(3/2)*d^2 - 35*sqrt(e*x + d)*d^3)*a^3*b^3/e^3 + 143*(84*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d
+ 15*sqrt(e*x + d)*d^2)*a^2*b^2/e^2 + 36*(5*(e*x + d)^(7/2) - 21*(e*x + d)^(5/2)*d + 35*(e*x + d)^(3/2)*d^2 -
35*sqrt(e*x + d)*d^3)*a*b^3/e^3 + (35*(e*x + d)^(9/2) - 180*(e*x + d)^(7/2)*d + 378*(e*x + d)^(5/2)*d^2 - 420
*(e*x + d)^(3/2)*d^3 + 315*sqrt(e*x + d)*d^4)*b^4/e^4)*a^2 + 572*(35*(e*x + d)^(9/2) - 180*(e*x + d)^(7/2)*d +
378*(e*x + d)^(5/2)*d^2 - 420*(e*x + d)^(3/2)*d^3 + 315*sqrt(e*x + d)*d^4)*a^2*b^4/e^4 + 130*(63*(e*x + d)^(1
1/2) - 385*(e*x + d)^(9/2)*d + 990*(e*x + d)^(7/2)*d^2 - 1386*(e*x + d)^(5/2)*d^3 + 1155*(e*x + d)^(3/2)*d^4 -
693*sqrt(e*x + d)*d^5)*a*b^5/e^5 + 5*(231*(e*x + d)^(13/2) - 1638*(e*x + d)^(11/2)*d + 5005*(e*x + d)^(9/2)*d
^2 - 8580*(e*x + d)^(7/2)*d^3 + 9009*(e*x + d)^(5/2)*d^4 - 6006*(e*x + d)^(3/2)*d^5 + 3003*sqrt(e*x + d)*d^6)*
b^6/e^6)/e

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mupad [B]  time = 0.55, size = 162, normalized size = 0.90 \begin {gather*} \frac {2\,b^6\,{\left (d+e\,x\right )}^{13/2}}{13\,e^7}-\frac {\left (12\,b^6\,d-12\,a\,b^5\,e\right )\,{\left (d+e\,x\right )}^{11/2}}{11\,e^7}+\frac {2\,{\left (a\,e-b\,d\right )}^6\,\sqrt {d+e\,x}}{e^7}+\frac {6\,b^2\,{\left (a\,e-b\,d\right )}^4\,{\left (d+e\,x\right )}^{5/2}}{e^7}+\frac {40\,b^3\,{\left (a\,e-b\,d\right )}^3\,{\left (d+e\,x\right )}^{7/2}}{7\,e^7}+\frac {10\,b^4\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{9/2}}{3\,e^7}+\frac {4\,b\,{\left (a\,e-b\,d\right )}^5\,{\left (d+e\,x\right )}^{3/2}}{e^7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*b^6*(d + e*x)^(13/2))/(13*e^7) - ((12*b^6*d - 12*a*b^5*e)*(d + e*x)^(11/2))/(11*e^7) + (2*(a*e - b*d)^6*(d
+ e*x)^(1/2))/e^7 + (6*b^2*(a*e - b*d)^4*(d + e*x)^(5/2))/e^7 + (40*b^3*(a*e - b*d)^3*(d + e*x)^(7/2))/(7*e^7)
+ (10*b^4*(a*e - b*d)^2*(d + e*x)^(9/2))/(3*e^7) + (4*b*(a*e - b*d)^5*(d + e*x)^(3/2))/e^7

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sympy [A]  time = 107.36, size = 1003, normalized size = 5.54

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise(((-2*a**6*d/sqrt(d + e*x) - 2*a**6*(-d/sqrt(d + e*x) - sqrt(d + e*x)) - 12*a**5*b*d*(-d/sqrt(d + e*x
) - sqrt(d + e*x))/e - 12*a**5*b*(d**2/sqrt(d + e*x) + 2*d*sqrt(d + e*x) - (d + e*x)**(3/2)/3)/e - 30*a**4*b**
2*d*(d**2/sqrt(d + e*x) + 2*d*sqrt(d + e*x) - (d + e*x)**(3/2)/3)/e**2 - 30*a**4*b**2*(-d**3/sqrt(d + e*x) - 3
*d**2*sqrt(d + e*x) + d*(d + e*x)**(3/2) - (d + e*x)**(5/2)/5)/e**2 - 40*a**3*b**3*d*(-d**3/sqrt(d + e*x) - 3*
d**2*sqrt(d + e*x) + d*(d + e*x)**(3/2) - (d + e*x)**(5/2)/5)/e**3 - 40*a**3*b**3*(d**4/sqrt(d + e*x) + 4*d**3
*sqrt(d + e*x) - 2*d**2*(d + e*x)**(3/2) + 4*d*(d + e*x)**(5/2)/5 - (d + e*x)**(7/2)/7)/e**3 - 30*a**2*b**4*d*
(d**4/sqrt(d + e*x) + 4*d**3*sqrt(d + e*x) - 2*d**2*(d + e*x)**(3/2) + 4*d*(d + e*x)**(5/2)/5 - (d + e*x)**(7/
2)/7)/e**4 - 30*a**2*b**4*(-d**5/sqrt(d + e*x) - 5*d**4*sqrt(d + e*x) + 10*d**3*(d + e*x)**(3/2)/3 - 2*d**2*(d
+ e*x)**(5/2) + 5*d*(d + e*x)**(7/2)/7 - (d + e*x)**(9/2)/9)/e**4 - 12*a*b**5*d*(-d**5/sqrt(d + e*x) - 5*d**4
*sqrt(d + e*x) + 10*d**3*(d + e*x)**(3/2)/3 - 2*d**2*(d + e*x)**(5/2) + 5*d*(d + e*x)**(7/2)/7 - (d + e*x)**(9
/2)/9)/e**5 - 12*a*b**5*(d**6/sqrt(d + e*x) + 6*d**5*sqrt(d + e*x) - 5*d**4*(d + e*x)**(3/2) + 4*d**3*(d + e*x
)**(5/2) - 15*d**2*(d + e*x)**(7/2)/7 + 2*d*(d + e*x)**(9/2)/3 - (d + e*x)**(11/2)/11)/e**5 - 2*b**6*d*(d**6/s
qrt(d + e*x) + 6*d**5*sqrt(d + e*x) - 5*d**4*(d + e*x)**(3/2) + 4*d**3*(d + e*x)**(5/2) - 15*d**2*(d + e*x)**(
7/2)/7 + 2*d*(d + e*x)**(9/2)/3 - (d + e*x)**(11/2)/11)/e**6 - 2*b**6*(-d**7/sqrt(d + e*x) - 7*d**6*sqrt(d + e
*x) + 7*d**5*(d + e*x)**(3/2) - 7*d**4*(d + e*x)**(5/2) + 5*d**3*(d + e*x)**(7/2) - 7*d**2*(d + e*x)**(9/2)/3
+ 7*d*(d + e*x)**(11/2)/11 - (d + e*x)**(13/2)/13)/e**6)/e, Ne(e, 0)), ((a**6*x + 3*a**5*b*x**2 + 5*a**4*b**2*
x**3 + 5*a**3*b**3*x**4 + 3*a**2*b**4*x**5 + a*b**5*x**6 + b**6*x**7/7)/sqrt(d), True))

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