3.19.94 \(\int \frac {(a+b x) (a^2+2 a b x+b^2 x^2)^{5/2}}{(d+e x)^{13/2}} \, dx\)

Optimal. Leaf size=368 \[ -\frac {30 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4}{7 e^7 (a+b x) (d+e x)^{7/2}}+\frac {4 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5}{3 e^7 (a+b x) (d+e x)^{9/2}}-\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^6}{11 e^7 (a+b x) (d+e x)^{11/2}}+\frac {2 b^6 \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x}}{e^7 (a+b x)}+\frac {12 b^5 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}{e^7 (a+b x) \sqrt {d+e x}}-\frac {10 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}{e^7 (a+b x) (d+e x)^{3/2}}+\frac {8 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3}{e^7 (a+b x) (d+e x)^{5/2}} \]

________________________________________________________________________________________

Rubi [A]  time = 0.14, antiderivative size = 368, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.086, Rules used = {770, 21, 43} \begin {gather*} \frac {2 b^6 \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x}}{e^7 (a+b x)}+\frac {12 b^5 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}{e^7 (a+b x) \sqrt {d+e x}}-\frac {10 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}{e^7 (a+b x) (d+e x)^{3/2}}+\frac {8 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3}{e^7 (a+b x) (d+e x)^{5/2}}-\frac {30 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4}{7 e^7 (a+b x) (d+e x)^{7/2}}+\frac {4 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5}{3 e^7 (a+b x) (d+e x)^{9/2}}-\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^6}{11 e^7 (a+b x) (d+e x)^{11/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^(13/2),x]

[Out]

(-2*(b*d - a*e)^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*e^7*(a + b*x)*(d + e*x)^(11/2)) + (4*b*(b*d - a*e)^5*Sqrt
[a^2 + 2*a*b*x + b^2*x^2])/(3*e^7*(a + b*x)*(d + e*x)^(9/2)) - (30*b^2*(b*d - a*e)^4*Sqrt[a^2 + 2*a*b*x + b^2*
x^2])/(7*e^7*(a + b*x)*(d + e*x)^(7/2)) + (8*b^3*(b*d - a*e)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(a + b*x)*(
d + e*x)^(5/2)) - (10*b^4*(b*d - a*e)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(a + b*x)*(d + e*x)^(3/2)) + (12*b
^5*(b*d - a*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(a + b*x)*Sqrt[d + e*x]) + (2*b^6*Sqrt[d + e*x]*Sqrt[a^2 +
2*a*b*x + b^2*x^2])/(e^7*(a + b*x))

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

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])

Rule 770

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dis
t[(a + b*x + c*x^2)^FracPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*FracPart[p])), Int[(d + e*x)^m*(f + g*x)*(b/2 + c
*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && EqQ[b^2 - 4*a*c, 0]

Rubi steps

\begin {align*} \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{13/2}} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {(a+b x) \left (a b+b^2 x\right )^5}{(d+e x)^{13/2}} \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac {\left (b \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int \frac {(a+b x)^6}{(d+e x)^{13/2}} \, dx}{a b+b^2 x}\\ &=\frac {\left (b \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int \left (\frac {(-b d+a e)^6}{e^6 (d+e x)^{13/2}}-\frac {6 b (b d-a e)^5}{e^6 (d+e x)^{11/2}}+\frac {15 b^2 (b d-a e)^4}{e^6 (d+e x)^{9/2}}-\frac {20 b^3 (b d-a e)^3}{e^6 (d+e x)^{7/2}}+\frac {15 b^4 (b d-a e)^2}{e^6 (d+e x)^{5/2}}-\frac {6 b^5 (b d-a e)}{e^6 (d+e x)^{3/2}}+\frac {b^6}{e^6 \sqrt {d+e x}}\right ) \, dx}{a b+b^2 x}\\ &=-\frac {2 (b d-a e)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{11 e^7 (a+b x) (d+e x)^{11/2}}+\frac {4 b (b d-a e)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^7 (a+b x) (d+e x)^{9/2}}-\frac {30 b^2 (b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^7 (a+b x) (d+e x)^{7/2}}+\frac {8 b^3 (b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x) (d+e x)^{5/2}}-\frac {10 b^4 (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x) (d+e x)^{3/2}}+\frac {12 b^5 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x) \sqrt {d+e x}}+\frac {2 b^6 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x)}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.15, size = 163, normalized size = 0.44 \begin {gather*} \frac {2 \sqrt {(a+b x)^2} \left (1386 b^5 (d+e x)^5 (b d-a e)-1155 b^4 (d+e x)^4 (b d-a e)^2+924 b^3 (d+e x)^3 (b d-a e)^3-495 b^2 (d+e x)^2 (b d-a e)^4+154 b (d+e x) (b d-a e)^5-21 (b d-a e)^6+231 b^6 (d+e x)^6\right )}{231 e^7 (a+b x) (d+e x)^{11/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^(13/2),x]

[Out]

(2*Sqrt[(a + b*x)^2]*(-21*(b*d - a*e)^6 + 154*b*(b*d - a*e)^5*(d + e*x) - 495*b^2*(b*d - a*e)^4*(d + e*x)^2 +
924*b^3*(b*d - a*e)^3*(d + e*x)^3 - 1155*b^4*(b*d - a*e)^2*(d + e*x)^4 + 1386*b^5*(b*d - a*e)*(d + e*x)^5 + 23
1*b^6*(d + e*x)^6))/(231*e^7*(a + b*x)*(d + e*x)^(11/2))

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 36.78, size = 466, normalized size = 1.27 \begin {gather*} \frac {2 \sqrt {\frac {(a e+b e x)^2}{e^2}} \left (-21 a^6 e^6-154 a^5 b e^5 (d+e x)+126 a^5 b d e^5-315 a^4 b^2 d^2 e^4-495 a^4 b^2 e^4 (d+e x)^2+770 a^4 b^2 d e^4 (d+e x)+420 a^3 b^3 d^3 e^3-1540 a^3 b^3 d^2 e^3 (d+e x)-924 a^3 b^3 e^3 (d+e x)^3+1980 a^3 b^3 d e^3 (d+e x)^2-315 a^2 b^4 d^4 e^2+1540 a^2 b^4 d^3 e^2 (d+e x)-2970 a^2 b^4 d^2 e^2 (d+e x)^2-1155 a^2 b^4 e^2 (d+e x)^4+2772 a^2 b^4 d e^2 (d+e x)^3+126 a b^5 d^5 e-770 a b^5 d^4 e (d+e x)+1980 a b^5 d^3 e (d+e x)^2-2772 a b^5 d^2 e (d+e x)^3-1386 a b^5 e (d+e x)^5+2310 a b^5 d e (d+e x)^4-21 b^6 d^6+154 b^6 d^5 (d+e x)-495 b^6 d^4 (d+e x)^2+924 b^6 d^3 (d+e x)^3-1155 b^6 d^2 (d+e x)^4+231 b^6 (d+e x)^6+1386 b^6 d (d+e x)^5\right )}{231 e^6 (d+e x)^{11/2} (a e+b e x)} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^(13/2),x]

[Out]

(2*Sqrt[(a*e + b*e*x)^2/e^2]*(-21*b^6*d^6 + 126*a*b^5*d^5*e - 315*a^2*b^4*d^4*e^2 + 420*a^3*b^3*d^3*e^3 - 315*
a^4*b^2*d^2*e^4 + 126*a^5*b*d*e^5 - 21*a^6*e^6 + 154*b^6*d^5*(d + e*x) - 770*a*b^5*d^4*e*(d + e*x) + 1540*a^2*
b^4*d^3*e^2*(d + e*x) - 1540*a^3*b^3*d^2*e^3*(d + e*x) + 770*a^4*b^2*d*e^4*(d + e*x) - 154*a^5*b*e^5*(d + e*x)
- 495*b^6*d^4*(d + e*x)^2 + 1980*a*b^5*d^3*e*(d + e*x)^2 - 2970*a^2*b^4*d^2*e^2*(d + e*x)^2 + 1980*a^3*b^3*d*
e^3*(d + e*x)^2 - 495*a^4*b^2*e^4*(d + e*x)^2 + 924*b^6*d^3*(d + e*x)^3 - 2772*a*b^5*d^2*e*(d + e*x)^3 + 2772*
a^2*b^4*d*e^2*(d + e*x)^3 - 924*a^3*b^3*e^3*(d + e*x)^3 - 1155*b^6*d^2*(d + e*x)^4 + 2310*a*b^5*d*e*(d + e*x)^
4 - 1155*a^2*b^4*e^2*(d + e*x)^4 + 1386*b^6*d*(d + e*x)^5 - 1386*a*b^5*e*(d + e*x)^5 + 231*b^6*(d + e*x)^6))/(
231*e^6*(d + e*x)^(11/2)*(a*e + b*e*x))

________________________________________________________________________________________

fricas [A]  time = 0.44, size = 421, normalized size = 1.14 \begin {gather*} \frac {2 \, {\left (231 \, b^{6} e^{6} x^{6} + 1024 \, b^{6} d^{6} - 512 \, a b^{5} d^{5} e - 128 \, a^{2} b^{4} d^{4} e^{2} - 64 \, a^{3} b^{3} d^{3} e^{3} - 40 \, a^{4} b^{2} d^{2} e^{4} - 28 \, a^{5} b d e^{5} - 21 \, a^{6} e^{6} + 1386 \, {\left (2 \, b^{6} d e^{5} - a b^{5} e^{6}\right )} x^{5} + 1155 \, {\left (8 \, b^{6} d^{2} e^{4} - 4 \, a b^{5} d e^{5} - a^{2} b^{4} e^{6}\right )} x^{4} + 924 \, {\left (16 \, b^{6} d^{3} e^{3} - 8 \, a b^{5} d^{2} e^{4} - 2 \, a^{2} b^{4} d e^{5} - a^{3} b^{3} e^{6}\right )} x^{3} + 99 \, {\left (128 \, b^{6} d^{4} e^{2} - 64 \, a b^{5} d^{3} e^{3} - 16 \, a^{2} b^{4} d^{2} e^{4} - 8 \, a^{3} b^{3} d e^{5} - 5 \, a^{4} b^{2} e^{6}\right )} x^{2} + 22 \, {\left (256 \, b^{6} d^{5} e - 128 \, a b^{5} d^{4} e^{2} - 32 \, a^{2} b^{4} d^{3} e^{3} - 16 \, a^{3} b^{3} d^{2} e^{4} - 10 \, a^{4} b^{2} d e^{5} - 7 \, a^{5} b e^{6}\right )} x\right )} \sqrt {e x + d}}{231 \, {\left (e^{13} x^{6} + 6 \, d e^{12} x^{5} + 15 \, d^{2} e^{11} x^{4} + 20 \, d^{3} e^{10} x^{3} + 15 \, d^{4} e^{9} x^{2} + 6 \, d^{5} e^{8} x + d^{6} e^{7}\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)^(5/2)/(e*x+d)^(13/2),x, algorithm="fricas")

[Out]

2/231*(231*b^6*e^6*x^6 + 1024*b^6*d^6 - 512*a*b^5*d^5*e - 128*a^2*b^4*d^4*e^2 - 64*a^3*b^3*d^3*e^3 - 40*a^4*b^
2*d^2*e^4 - 28*a^5*b*d*e^5 - 21*a^6*e^6 + 1386*(2*b^6*d*e^5 - a*b^5*e^6)*x^5 + 1155*(8*b^6*d^2*e^4 - 4*a*b^5*d
*e^5 - a^2*b^4*e^6)*x^4 + 924*(16*b^6*d^3*e^3 - 8*a*b^5*d^2*e^4 - 2*a^2*b^4*d*e^5 - a^3*b^3*e^6)*x^3 + 99*(128
*b^6*d^4*e^2 - 64*a*b^5*d^3*e^3 - 16*a^2*b^4*d^2*e^4 - 8*a^3*b^3*d*e^5 - 5*a^4*b^2*e^6)*x^2 + 22*(256*b^6*d^5*
e - 128*a*b^5*d^4*e^2 - 32*a^2*b^4*d^3*e^3 - 16*a^3*b^3*d^2*e^4 - 10*a^4*b^2*d*e^5 - 7*a^5*b*e^6)*x)*sqrt(e*x
+ d)/(e^13*x^6 + 6*d*e^12*x^5 + 15*d^2*e^11*x^4 + 20*d^3*e^10*x^3 + 15*d^4*e^9*x^2 + 6*d^5*e^8*x + d^6*e^7)

________________________________________________________________________________________

giac [B]  time = 0.28, size = 617, normalized size = 1.68 \begin {gather*} 2 \, \sqrt {x e + d} b^{6} e^{\left (-7\right )} \mathrm {sgn}\left (b x + a\right ) + \frac {2 \, {\left (1386 \, {\left (x e + d\right )}^{5} b^{6} d \mathrm {sgn}\left (b x + a\right ) - 1155 \, {\left (x e + d\right )}^{4} b^{6} d^{2} \mathrm {sgn}\left (b x + a\right ) + 924 \, {\left (x e + d\right )}^{3} b^{6} d^{3} \mathrm {sgn}\left (b x + a\right ) - 495 \, {\left (x e + d\right )}^{2} b^{6} d^{4} \mathrm {sgn}\left (b x + a\right ) + 154 \, {\left (x e + d\right )} b^{6} d^{5} \mathrm {sgn}\left (b x + a\right ) - 21 \, b^{6} d^{6} \mathrm {sgn}\left (b x + a\right ) - 1386 \, {\left (x e + d\right )}^{5} a b^{5} e \mathrm {sgn}\left (b x + a\right ) + 2310 \, {\left (x e + d\right )}^{4} a b^{5} d e \mathrm {sgn}\left (b x + a\right ) - 2772 \, {\left (x e + d\right )}^{3} a b^{5} d^{2} e \mathrm {sgn}\left (b x + a\right ) + 1980 \, {\left (x e + d\right )}^{2} a b^{5} d^{3} e \mathrm {sgn}\left (b x + a\right ) - 770 \, {\left (x e + d\right )} a b^{5} d^{4} e \mathrm {sgn}\left (b x + a\right ) + 126 \, a b^{5} d^{5} e \mathrm {sgn}\left (b x + a\right ) - 1155 \, {\left (x e + d\right )}^{4} a^{2} b^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) + 2772 \, {\left (x e + d\right )}^{3} a^{2} b^{4} d e^{2} \mathrm {sgn}\left (b x + a\right ) - 2970 \, {\left (x e + d\right )}^{2} a^{2} b^{4} d^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) + 1540 \, {\left (x e + d\right )} a^{2} b^{4} d^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) - 315 \, a^{2} b^{4} d^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) - 924 \, {\left (x e + d\right )}^{3} a^{3} b^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) + 1980 \, {\left (x e + d\right )}^{2} a^{3} b^{3} d e^{3} \mathrm {sgn}\left (b x + a\right ) - 1540 \, {\left (x e + d\right )} a^{3} b^{3} d^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) + 420 \, a^{3} b^{3} d^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) - 495 \, {\left (x e + d\right )}^{2} a^{4} b^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) + 770 \, {\left (x e + d\right )} a^{4} b^{2} d e^{4} \mathrm {sgn}\left (b x + a\right ) - 315 \, a^{4} b^{2} d^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) - 154 \, {\left (x e + d\right )} a^{5} b e^{5} \mathrm {sgn}\left (b x + a\right ) + 126 \, a^{5} b d e^{5} \mathrm {sgn}\left (b x + a\right ) - 21 \, a^{6} e^{6} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-7\right )}}{231 \, {\left (x e + d\right )}^{\frac {11}{2}}} \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)^(5/2)/(e*x+d)^(13/2),x, algorithm="giac")

[Out]

2*sqrt(x*e + d)*b^6*e^(-7)*sgn(b*x + a) + 2/231*(1386*(x*e + d)^5*b^6*d*sgn(b*x + a) - 1155*(x*e + d)^4*b^6*d^
2*sgn(b*x + a) + 924*(x*e + d)^3*b^6*d^3*sgn(b*x + a) - 495*(x*e + d)^2*b^6*d^4*sgn(b*x + a) + 154*(x*e + d)*b
^6*d^5*sgn(b*x + a) - 21*b^6*d^6*sgn(b*x + a) - 1386*(x*e + d)^5*a*b^5*e*sgn(b*x + a) + 2310*(x*e + d)^4*a*b^5
*d*e*sgn(b*x + a) - 2772*(x*e + d)^3*a*b^5*d^2*e*sgn(b*x + a) + 1980*(x*e + d)^2*a*b^5*d^3*e*sgn(b*x + a) - 77
0*(x*e + d)*a*b^5*d^4*e*sgn(b*x + a) + 126*a*b^5*d^5*e*sgn(b*x + a) - 1155*(x*e + d)^4*a^2*b^4*e^2*sgn(b*x + a
) + 2772*(x*e + d)^3*a^2*b^4*d*e^2*sgn(b*x + a) - 2970*(x*e + d)^2*a^2*b^4*d^2*e^2*sgn(b*x + a) + 1540*(x*e +
d)*a^2*b^4*d^3*e^2*sgn(b*x + a) - 315*a^2*b^4*d^4*e^2*sgn(b*x + a) - 924*(x*e + d)^3*a^3*b^3*e^3*sgn(b*x + a)
+ 1980*(x*e + d)^2*a^3*b^3*d*e^3*sgn(b*x + a) - 1540*(x*e + d)*a^3*b^3*d^2*e^3*sgn(b*x + a) + 420*a^3*b^3*d^3*
e^3*sgn(b*x + a) - 495*(x*e + d)^2*a^4*b^2*e^4*sgn(b*x + a) + 770*(x*e + d)*a^4*b^2*d*e^4*sgn(b*x + a) - 315*a
^4*b^2*d^2*e^4*sgn(b*x + a) - 154*(x*e + d)*a^5*b*e^5*sgn(b*x + a) + 126*a^5*b*d*e^5*sgn(b*x + a) - 21*a^6*e^6
*sgn(b*x + a))*e^(-7)/(x*e + d)^(11/2)

________________________________________________________________________________________

maple [A]  time = 0.06, size = 393, normalized size = 1.07 \begin {gather*} -\frac {2 \left (-231 b^{6} e^{6} x^{6}+1386 a \,b^{5} e^{6} x^{5}-2772 b^{6} d \,e^{5} x^{5}+1155 a^{2} b^{4} e^{6} x^{4}+4620 a \,b^{5} d \,e^{5} x^{4}-9240 b^{6} d^{2} e^{4} x^{4}+924 a^{3} b^{3} e^{6} x^{3}+1848 a^{2} b^{4} d \,e^{5} x^{3}+7392 a \,b^{5} d^{2} e^{4} x^{3}-14784 b^{6} d^{3} e^{3} x^{3}+495 a^{4} b^{2} e^{6} x^{2}+792 a^{3} b^{3} d \,e^{5} x^{2}+1584 a^{2} b^{4} d^{2} e^{4} x^{2}+6336 a \,b^{5} d^{3} e^{3} x^{2}-12672 b^{6} d^{4} e^{2} x^{2}+154 a^{5} b \,e^{6} x +220 a^{4} b^{2} d \,e^{5} x +352 a^{3} b^{3} d^{2} e^{4} x +704 a^{2} b^{4} d^{3} e^{3} x +2816 a \,b^{5} d^{4} e^{2} x -5632 b^{6} d^{5} e x +21 a^{6} e^{6}+28 a^{5} b d \,e^{5}+40 a^{4} b^{2} d^{2} e^{4}+64 a^{3} b^{3} d^{3} e^{3}+128 a^{2} b^{4} d^{4} e^{2}+512 a \,b^{5} d^{5} e -1024 b^{6} d^{6}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{231 \left (e x +d \right )^{\frac {11}{2}} \left (b x +a \right )^{5} e^{7}} \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)^(5/2)/(e*x+d)^(13/2),x)

[Out]

-2/231/(e*x+d)^(11/2)*(-231*b^6*e^6*x^6+1386*a*b^5*e^6*x^5-2772*b^6*d*e^5*x^5+1155*a^2*b^4*e^6*x^4+4620*a*b^5*
d*e^5*x^4-9240*b^6*d^2*e^4*x^4+924*a^3*b^3*e^6*x^3+1848*a^2*b^4*d*e^5*x^3+7392*a*b^5*d^2*e^4*x^3-14784*b^6*d^3
*e^3*x^3+495*a^4*b^2*e^6*x^2+792*a^3*b^3*d*e^5*x^2+1584*a^2*b^4*d^2*e^4*x^2+6336*a*b^5*d^3*e^3*x^2-12672*b^6*d
^4*e^2*x^2+154*a^5*b*e^6*x+220*a^4*b^2*d*e^5*x+352*a^3*b^3*d^2*e^4*x+704*a^2*b^4*d^3*e^3*x+2816*a*b^5*d^4*e^2*
x-5632*b^6*d^5*e*x+21*a^6*e^6+28*a^5*b*d*e^5+40*a^4*b^2*d^2*e^4+64*a^3*b^3*d^3*e^3+128*a^2*b^4*d^4*e^2+512*a*b
^5*d^5*e-1024*b^6*d^6)*((b*x+a)^2)^(5/2)/e^7/(b*x+a)^5

________________________________________________________________________________________

maxima [B]  time = 0.98, size = 712, normalized size = 1.93 \begin {gather*} -\frac {2 \, {\left (693 \, b^{5} e^{5} x^{5} + 256 \, b^{5} d^{5} + 128 \, a b^{4} d^{4} e + 96 \, a^{2} b^{3} d^{3} e^{2} + 80 \, a^{3} b^{2} d^{2} e^{3} + 70 \, a^{4} b d e^{4} + 63 \, a^{5} e^{5} + 1155 \, {\left (2 \, b^{5} d e^{4} + a b^{4} e^{5}\right )} x^{4} + 462 \, {\left (8 \, b^{5} d^{2} e^{3} + 4 \, a b^{4} d e^{4} + 3 \, a^{2} b^{3} e^{5}\right )} x^{3} + 198 \, {\left (16 \, b^{5} d^{3} e^{2} + 8 \, a b^{4} d^{2} e^{3} + 6 \, a^{2} b^{3} d e^{4} + 5 \, a^{3} b^{2} e^{5}\right )} x^{2} + 11 \, {\left (128 \, b^{5} d^{4} e + 64 \, a b^{4} d^{3} e^{2} + 48 \, a^{2} b^{3} d^{2} e^{3} + 40 \, a^{3} b^{2} d e^{4} + 35 \, a^{4} b e^{5}\right )} x\right )} a}{693 \, {\left (e^{11} x^{5} + 5 \, d e^{10} x^{4} + 10 \, d^{2} e^{9} x^{3} + 10 \, d^{3} e^{8} x^{2} + 5 \, d^{4} e^{7} x + d^{5} e^{6}\right )} \sqrt {e x + d}} + \frac {2 \, {\left (693 \, b^{5} e^{6} x^{6} + 3072 \, b^{5} d^{6} - 1280 \, a b^{4} d^{5} e - 256 \, a^{2} b^{3} d^{4} e^{2} - 96 \, a^{3} b^{2} d^{3} e^{3} - 40 \, a^{4} b d^{2} e^{4} - 14 \, a^{5} d e^{5} + 693 \, {\left (12 \, b^{5} d e^{5} - 5 \, a b^{4} e^{6}\right )} x^{5} + 2310 \, {\left (12 \, b^{5} d^{2} e^{4} - 5 \, a b^{4} d e^{5} - a^{2} b^{3} e^{6}\right )} x^{4} + 462 \, {\left (96 \, b^{5} d^{3} e^{3} - 40 \, a b^{4} d^{2} e^{4} - 8 \, a^{2} b^{3} d e^{5} - 3 \, a^{3} b^{2} e^{6}\right )} x^{3} + 99 \, {\left (384 \, b^{5} d^{4} e^{2} - 160 \, a b^{4} d^{3} e^{3} - 32 \, a^{2} b^{3} d^{2} e^{4} - 12 \, a^{3} b^{2} d e^{5} - 5 \, a^{4} b e^{6}\right )} x^{2} + 11 \, {\left (1536 \, b^{5} d^{5} e - 640 \, a b^{4} d^{4} e^{2} - 128 \, a^{2} b^{3} d^{3} e^{3} - 48 \, a^{3} b^{2} d^{2} e^{4} - 20 \, a^{4} b d e^{5} - 7 \, a^{5} e^{6}\right )} x\right )} b}{693 \, {\left (e^{12} x^{5} + 5 \, d e^{11} x^{4} + 10 \, d^{2} e^{10} x^{3} + 10 \, d^{3} e^{9} x^{2} + 5 \, d^{4} e^{8} x + d^{5} e^{7}\right )} \sqrt {e x + d}} \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)^(5/2)/(e*x+d)^(13/2),x, algorithm="maxima")

[Out]

-2/693*(693*b^5*e^5*x^5 + 256*b^5*d^5 + 128*a*b^4*d^4*e + 96*a^2*b^3*d^3*e^2 + 80*a^3*b^2*d^2*e^3 + 70*a^4*b*d
*e^4 + 63*a^5*e^5 + 1155*(2*b^5*d*e^4 + a*b^4*e^5)*x^4 + 462*(8*b^5*d^2*e^3 + 4*a*b^4*d*e^4 + 3*a^2*b^3*e^5)*x
^3 + 198*(16*b^5*d^3*e^2 + 8*a*b^4*d^2*e^3 + 6*a^2*b^3*d*e^4 + 5*a^3*b^2*e^5)*x^2 + 11*(128*b^5*d^4*e + 64*a*b
^4*d^3*e^2 + 48*a^2*b^3*d^2*e^3 + 40*a^3*b^2*d*e^4 + 35*a^4*b*e^5)*x)*a/((e^11*x^5 + 5*d*e^10*x^4 + 10*d^2*e^9
*x^3 + 10*d^3*e^8*x^2 + 5*d^4*e^7*x + d^5*e^6)*sqrt(e*x + d)) + 2/693*(693*b^5*e^6*x^6 + 3072*b^5*d^6 - 1280*a
*b^4*d^5*e - 256*a^2*b^3*d^4*e^2 - 96*a^3*b^2*d^3*e^3 - 40*a^4*b*d^2*e^4 - 14*a^5*d*e^5 + 693*(12*b^5*d*e^5 -
5*a*b^4*e^6)*x^5 + 2310*(12*b^5*d^2*e^4 - 5*a*b^4*d*e^5 - a^2*b^3*e^6)*x^4 + 462*(96*b^5*d^3*e^3 - 40*a*b^4*d^
2*e^4 - 8*a^2*b^3*d*e^5 - 3*a^3*b^2*e^6)*x^3 + 99*(384*b^5*d^4*e^2 - 160*a*b^4*d^3*e^3 - 32*a^2*b^3*d^2*e^4 -
12*a^3*b^2*d*e^5 - 5*a^4*b*e^6)*x^2 + 11*(1536*b^5*d^5*e - 640*a*b^4*d^4*e^2 - 128*a^2*b^3*d^3*e^3 - 48*a^3*b^
2*d^2*e^4 - 20*a^4*b*d*e^5 - 7*a^5*e^6)*x)*b/((e^12*x^5 + 5*d*e^11*x^4 + 10*d^2*e^10*x^3 + 10*d^3*e^9*x^2 + 5*
d^4*e^8*x + d^5*e^7)*sqrt(e*x + d))

________________________________________________________________________________________

mupad [B]  time = 3.37, size = 532, normalized size = 1.45 \begin {gather*} -\frac {\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}\,\left (\frac {42\,a^6\,e^6+56\,a^5\,b\,d\,e^5+80\,a^4\,b^2\,d^2\,e^4+128\,a^3\,b^3\,d^3\,e^3+256\,a^2\,b^4\,d^4\,e^2+1024\,a\,b^5\,d^5\,e-2048\,b^6\,d^6}{231\,b\,e^{12}}-\frac {2\,b^5\,x^6}{e^6}+\frac {x\,\left (308\,a^5\,b\,e^6+440\,a^4\,b^2\,d\,e^5+704\,a^3\,b^3\,d^2\,e^4+1408\,a^2\,b^4\,d^3\,e^3+5632\,a\,b^5\,d^4\,e^2-11264\,b^6\,d^5\,e\right )}{231\,b\,e^{12}}+\frac {8\,b^2\,x^3\,\left (a^3\,e^3+2\,a^2\,b\,d\,e^2+8\,a\,b^2\,d^2\,e-16\,b^3\,d^3\right )}{e^9}+\frac {6\,b\,x^2\,\left (5\,a^4\,e^4+8\,a^3\,b\,d\,e^3+16\,a^2\,b^2\,d^2\,e^2+64\,a\,b^3\,d^3\,e-128\,b^4\,d^4\right )}{7\,e^{10}}+\frac {12\,b^4\,x^5\,\left (a\,e-2\,b\,d\right )}{e^7}+\frac {10\,b^3\,x^4\,\left (a^2\,e^2+4\,a\,b\,d\,e-8\,b^2\,d^2\right )}{e^8}\right )}{x^6\,\sqrt {d+e\,x}+\frac {a\,d^5\,\sqrt {d+e\,x}}{b\,e^5}+\frac {x^5\,\left (231\,a\,e^{12}+1155\,b\,d\,e^{11}\right )\,\sqrt {d+e\,x}}{231\,b\,e^{12}}+\frac {5\,d\,x^4\,\left (a\,e+2\,b\,d\right )\,\sqrt {d+e\,x}}{b\,e^2}+\frac {d^4\,x\,\left (5\,a\,e+b\,d\right )\,\sqrt {d+e\,x}}{b\,e^5}+\frac {10\,d^2\,x^3\,\left (a\,e+b\,d\right )\,\sqrt {d+e\,x}}{b\,e^3}+\frac {5\,d^3\,x^2\,\left (2\,a\,e+b\,d\right )\,\sqrt {d+e\,x}}{b\,e^4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + b*x)*(a^2 + b^2*x^2 + 2*a*b*x)^(5/2))/(d + e*x)^(13/2),x)

[Out]

-((a^2 + b^2*x^2 + 2*a*b*x)^(1/2)*((42*a^6*e^6 - 2048*b^6*d^6 + 256*a^2*b^4*d^4*e^2 + 128*a^3*b^3*d^3*e^3 + 80
*a^4*b^2*d^2*e^4 + 1024*a*b^5*d^5*e + 56*a^5*b*d*e^5)/(231*b*e^12) - (2*b^5*x^6)/e^6 + (x*(308*a^5*b*e^6 - 112
64*b^6*d^5*e + 5632*a*b^5*d^4*e^2 + 440*a^4*b^2*d*e^5 + 1408*a^2*b^4*d^3*e^3 + 704*a^3*b^3*d^2*e^4))/(231*b*e^
12) + (8*b^2*x^3*(a^3*e^3 - 16*b^3*d^3 + 8*a*b^2*d^2*e + 2*a^2*b*d*e^2))/e^9 + (6*b*x^2*(5*a^4*e^4 - 128*b^4*d
^4 + 16*a^2*b^2*d^2*e^2 + 64*a*b^3*d^3*e + 8*a^3*b*d*e^3))/(7*e^10) + (12*b^4*x^5*(a*e - 2*b*d))/e^7 + (10*b^3
*x^4*(a^2*e^2 - 8*b^2*d^2 + 4*a*b*d*e))/e^8))/(x^6*(d + e*x)^(1/2) + (a*d^5*(d + e*x)^(1/2))/(b*e^5) + (x^5*(2
31*a*e^12 + 1155*b*d*e^11)*(d + e*x)^(1/2))/(231*b*e^12) + (5*d*x^4*(a*e + 2*b*d)*(d + e*x)^(1/2))/(b*e^2) + (
d^4*x*(5*a*e + b*d)*(d + e*x)^(1/2))/(b*e^5) + (10*d^2*x^3*(a*e + b*d)*(d + e*x)^(1/2))/(b*e^3) + (5*d^3*x^2*(
2*a*e + b*d)*(d + e*x)^(1/2))/(b*e^4))

________________________________________________________________________________________

sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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)**(5/2)/(e*x+d)**(13/2),x)

[Out]

Timed out

________________________________________________________________________________________