3.17.41 \(\int \frac {(a^2+2 a b x+b^2 x^2)^3}{\sqrt {d+e x}} \, dx\) [1641]

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

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.04, 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, 45} \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 45

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*}

________________________________________________________________________________________

Mathematica [A]
time = 0.13, size = 290, normalized size = 1.60 \begin {gather*} \frac {2 \sqrt {d+e x} \left (3003 a^6 e^6+6006 a^5 b e^5 (-2 d+e x)+3003 a^4 b^2 e^4 \left (8 d^2-4 d e x+3 e^2 x^2\right )+1716 a^3 b^3 e^3 \left (-16 d^3+8 d^2 e x-6 d e^2 x^2+5 e^3 x^3\right )+143 a^2 b^4 e^2 \left (128 d^4-64 d^3 e x+48 d^2 e^2 x^2-40 d e^3 x^3+35 e^4 x^4\right )+26 a b^5 e \left (-256 d^5+128 d^4 e x-96 d^3 e^2 x^2+80 d^2 e^3 x^3-70 d e^4 x^4+63 e^5 x^5\right )+b^6 \left (1024 d^6-512 d^5 e x+384 d^4 e^2 x^2-320 d^3 e^3 x^3+280 d^2 e^4 x^4-252 d e^5 x^5+231 e^6 x^6\right )\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*a^6*e^6 + 6006*a^5*b*e^5*(-2*d + e*x) + 3003*a^4*b^2*e^4*(8*d^2 - 4*d*e*x + 3*e^2*x^2)
+ 1716*a^3*b^3*e^3*(-16*d^3 + 8*d^2*e*x - 6*d*e^2*x^2 + 5*e^3*x^3) + 143*a^2*b^4*e^2*(128*d^4 - 64*d^3*e*x + 4
8*d^2*e^2*x^2 - 40*d*e^3*x^3 + 35*e^4*x^4) + 26*a*b^5*e*(-256*d^5 + 128*d^4*e*x - 96*d^3*e^2*x^2 + 80*d^2*e^3*
x^3 - 70*d*e^4*x^4 + 63*e^5*x^5) + b^6*(1024*d^6 - 512*d^5*e*x + 384*d^4*e^2*x^2 - 320*d^3*e^3*x^3 + 280*d^2*e
^4*x^4 - 252*d*e^5*x^5 + 231*e^6*x^6)))/(3003*e^7)

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(454\) vs. \(2(159)=318\).
time = 0.63, size = 455, normalized size = 2.51 Too large to display

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,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 570 vs. \(2 (165) = 330\).
time = 0.32, size = 570, normalized size = 3.15 \begin {gather*} \frac {2}{15015} \, {\left (3432 \, {\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 )} + 572 \, {\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 )} + 130 \, {\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 )} + 5 \, {\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 )} + 15015 \, \sqrt {x e + d} a^{6} + 3003 \, {\left (10 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} a b e^{\left (-1\right )} + {\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 )} b^{2} e^{\left (-2\right )}\right )} a^{4} + 143 \, {\left (84 \, {\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} e^{\left (-2\right )} + 36 \, {\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} e^{\left (-3\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^{4} e^{\left (-4\right )}\right )} a^{2}\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="maxima")

[Out]

2/15015*(3432*(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) + 572*(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) + 130*(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) + 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) + 15015*sqrt(x*e + d)*a^6 + 3003
*(10*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*a*b*e^(-1) + (3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*
e + d)*d^2)*b^2*e^(-2))*a^4 + 143*(84*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*a^2*b^
2*e^(-2) + 36*(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
*e^(-3) + (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 + 31
5*sqrt(x*e + d)*d^4)*b^4*e^(-4))*a^2)*e^(-1)

________________________________________________________________________________________

Fricas [A]
time = 2.38, size = 327, normalized size = 1.81 \begin {gather*} \frac {2}{3003} \, {\left (1024 \, b^{6} d^{6} + {\left (231 \, b^{6} x^{6} + 1638 \, a b^{5} x^{5} + 5005 \, a^{2} b^{4} x^{4} + 8580 \, a^{3} b^{3} x^{3} + 9009 \, a^{4} b^{2} x^{2} + 6006 \, a^{5} b x + 3003 \, a^{6}\right )} e^{6} - 4 \, {\left (63 \, b^{6} d x^{5} + 455 \, a b^{5} d x^{4} + 1430 \, a^{2} b^{4} d x^{3} + 2574 \, a^{3} b^{3} d x^{2} + 3003 \, a^{4} b^{2} d x + 3003 \, a^{5} b d\right )} e^{5} + 8 \, {\left (35 \, b^{6} d^{2} x^{4} + 260 \, a b^{5} d^{2} x^{3} + 858 \, a^{2} b^{4} d^{2} x^{2} + 1716 \, a^{3} b^{3} d^{2} x + 3003 \, a^{4} b^{2} d^{2}\right )} e^{4} - 64 \, {\left (5 \, b^{6} d^{3} x^{3} + 39 \, a b^{5} d^{3} x^{2} + 143 \, a^{2} b^{4} d^{3} x + 429 \, a^{3} b^{3} d^{3}\right )} e^{3} + 128 \, {\left (3 \, b^{6} d^{4} x^{2} + 26 \, a b^{5} d^{4} x + 143 \, a^{2} b^{4} d^{4}\right )} e^{2} - 512 \, {\left (b^{6} d^{5} x + 13 \, a b^{5} d^{5}\right )} e\right )} \sqrt {x e + d} e^{\left (-7\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="fricas")

[Out]

2/3003*(1024*b^6*d^6 + (231*b^6*x^6 + 1638*a*b^5*x^5 + 5005*a^2*b^4*x^4 + 8580*a^3*b^3*x^3 + 9009*a^4*b^2*x^2
+ 6006*a^5*b*x + 3003*a^6)*e^6 - 4*(63*b^6*d*x^5 + 455*a*b^5*d*x^4 + 1430*a^2*b^4*d*x^3 + 2574*a^3*b^3*d*x^2 +
3003*a^4*b^2*d*x + 3003*a^5*b*d)*e^5 + 8*(35*b^6*d^2*x^4 + 260*a*b^5*d^2*x^3 + 858*a^2*b^4*d^2*x^2 + 1716*a^3
*b^3*d^2*x + 3003*a^4*b^2*d^2)*e^4 - 64*(5*b^6*d^3*x^3 + 39*a*b^5*d^3*x^2 + 143*a^2*b^4*d^3*x + 429*a^3*b^3*d^
3)*e^3 + 128*(3*b^6*d^4*x^2 + 26*a*b^5*d^4*x + 143*a^2*b^4*d^4)*e^2 - 512*(b^6*d^5*x + 13*a*b^5*d^5)*e)*sqrt(x
*e + d)*e^(-7)

________________________________________________________________________________________

Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1003 vs. \(2 (168) = 336\).
time = 52.49, size = 1003, normalized size = 5.54 \begin {gather*} \begin {cases} \frac {- \frac {2 a^{6} d}{\sqrt {d + e x}} - 2 a^{6} \left (- \frac {d}{\sqrt {d + e x}} - \sqrt {d + e x}\right ) - \frac {12 a^{5} b d \left (- \frac {d}{\sqrt {d + e x}} - \sqrt {d + e x}\right )}{e} - \frac {12 a^{5} b \left (\frac {d^{2}}{\sqrt {d + e x}} + 2 d \sqrt {d + e x} - \frac {\left (d + e x\right )^{\frac {3}{2}}}{3}\right )}{e} - \frac {30 a^{4} b^{2} d \left (\frac {d^{2}}{\sqrt {d + e x}} + 2 d \sqrt {d + e x} - \frac {\left (d + e x\right )^{\frac {3}{2}}}{3}\right )}{e^{2}} - \frac {30 a^{4} b^{2} \left (- \frac {d^{3}}{\sqrt {d + e x}} - 3 d^{2} \sqrt {d + e x} + d \left (d + e x\right )^{\frac {3}{2}} - \frac {\left (d + e x\right )^{\frac {5}{2}}}{5}\right )}{e^{2}} - \frac {40 a^{3} b^{3} d \left (- \frac {d^{3}}{\sqrt {d + e x}} - 3 d^{2} \sqrt {d + e x} + d \left (d + e x\right )^{\frac {3}{2}} - \frac {\left (d + e x\right )^{\frac {5}{2}}}{5}\right )}{e^{3}} - \frac {40 a^{3} b^{3} \left (\frac {d^{4}}{\sqrt {d + e x}} + 4 d^{3} \sqrt {d + e x} - 2 d^{2} \left (d + e x\right )^{\frac {3}{2}} + \frac {4 d \left (d + e x\right )^{\frac {5}{2}}}{5} - \frac {\left (d + e x\right )^{\frac {7}{2}}}{7}\right )}{e^{3}} - \frac {30 a^{2} b^{4} d \left (\frac {d^{4}}{\sqrt {d + e x}} + 4 d^{3} \sqrt {d + e x} - 2 d^{2} \left (d + e x\right )^{\frac {3}{2}} + \frac {4 d \left (d + e x\right )^{\frac {5}{2}}}{5} - \frac {\left (d + e x\right )^{\frac {7}{2}}}{7}\right )}{e^{4}} - \frac {30 a^{2} b^{4} \left (- \frac {d^{5}}{\sqrt {d + e x}} - 5 d^{4} \sqrt {d + e x} + \frac {10 d^{3} \left (d + e x\right )^{\frac {3}{2}}}{3} - 2 d^{2} \left (d + e x\right )^{\frac {5}{2}} + \frac {5 d \left (d + e x\right )^{\frac {7}{2}}}{7} - \frac {\left (d + e x\right )^{\frac {9}{2}}}{9}\right )}{e^{4}} - \frac {12 a b^{5} d \left (- \frac {d^{5}}{\sqrt {d + e x}} - 5 d^{4} \sqrt {d + e x} + \frac {10 d^{3} \left (d + e x\right )^{\frac {3}{2}}}{3} - 2 d^{2} \left (d + e x\right )^{\frac {5}{2}} + \frac {5 d \left (d + e x\right )^{\frac {7}{2}}}{7} - \frac {\left (d + e x\right )^{\frac {9}{2}}}{9}\right )}{e^{5}} - \frac {12 a b^{5} \left (\frac {d^{6}}{\sqrt {d + e x}} + 6 d^{5} \sqrt {d + e x} - 5 d^{4} \left (d + e x\right )^{\frac {3}{2}} + 4 d^{3} \left (d + e x\right )^{\frac {5}{2}} - \frac {15 d^{2} \left (d + e x\right )^{\frac {7}{2}}}{7} + \frac {2 d \left (d + e x\right )^{\frac {9}{2}}}{3} - \frac {\left (d + e x\right )^{\frac {11}{2}}}{11}\right )}{e^{5}} - \frac {2 b^{6} d \left (\frac {d^{6}}{\sqrt {d + e x}} + 6 d^{5} \sqrt {d + e x} - 5 d^{4} \left (d + e x\right )^{\frac {3}{2}} + 4 d^{3} \left (d + e x\right )^{\frac {5}{2}} - \frac {15 d^{2} \left (d + e x\right )^{\frac {7}{2}}}{7} + \frac {2 d \left (d + e x\right )^{\frac {9}{2}}}{3} - \frac {\left (d + e x\right )^{\frac {11}{2}}}{11}\right )}{e^{6}} - \frac {2 b^{6} \left (- \frac {d^{7}}{\sqrt {d + e x}} - 7 d^{6} \sqrt {d + e x} + 7 d^{5} \left (d + e x\right )^{\frac {3}{2}} - 7 d^{4} \left (d + e x\right )^{\frac {5}{2}} + 5 d^{3} \left (d + e x\right )^{\frac {7}{2}} - \frac {7 d^{2} \left (d + e x\right )^{\frac {9}{2}}}{3} + \frac {7 d \left (d + e x\right )^{\frac {11}{2}}}{11} - \frac {\left (d + e x\right )^{\frac {13}{2}}}{13}\right )}{e^{6}}}{e} & \text {for}\: e \neq 0 \\\frac {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} + \frac {b^{6} x^{7}}{7}}{\sqrt {d}} & \text {otherwise} \end {cases} \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)

[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))

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 395 vs. \(2 (165) = 330\).
time = 0.94, 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)

________________________________________________________________________________________

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

________________________________________________________________________________________