3.2118 \(\int \frac {(a+b x) (a^2+2 a b x+b^2 x^2)^{5/2}}{(d+e x)^{11/2}} \, dx\)

Optimal. Leaf size=370 \[ -\frac {6 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4}{e^7 (a+b x) (d+e x)^{5/2}}+\frac {12 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5}{7 e^7 (a+b x) (d+e x)^{7/2}}-\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^6}{9 e^7 (a+b x) (d+e x)^{9/2}}+\frac {2 b^6 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2}}{3 e^7 (a+b x)}-\frac {12 b^5 \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x} (b d-a e)}{e^7 (a+b x)}-\frac {30 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}{e^7 (a+b x) \sqrt {d+e x}}+\frac {40 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3}{3 e^7 (a+b x) (d+e x)^{3/2}} \]

[Out]

-2/9*(-a*e+b*d)^6*((b*x+a)^2)^(1/2)/e^7/(b*x+a)/(e*x+d)^(9/2)+12/7*b*(-a*e+b*d)^5*((b*x+a)^2)^(1/2)/e^7/(b*x+a
)/(e*x+d)^(7/2)-6*b^2*(-a*e+b*d)^4*((b*x+a)^2)^(1/2)/e^7/(b*x+a)/(e*x+d)^(5/2)+40/3*b^3*(-a*e+b*d)^3*((b*x+a)^
2)^(1/2)/e^7/(b*x+a)/(e*x+d)^(3/2)+2/3*b^6*(e*x+d)^(3/2)*((b*x+a)^2)^(1/2)/e^7/(b*x+a)-30*b^4*(-a*e+b*d)^2*((b
*x+a)^2)^(1/2)/e^7/(b*x+a)/(e*x+d)^(1/2)-12*b^5*(-a*e+b*d)*(e*x+d)^(1/2)*((b*x+a)^2)^(1/2)/e^7/(b*x+a)

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Rubi [A]  time = 0.15, antiderivative size = 370, 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} \[ \frac {2 b^6 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2}}{3 e^7 (a+b x)}-\frac {12 b^5 \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x} (b d-a e)}{e^7 (a+b x)}-\frac {30 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}{e^7 (a+b x) \sqrt {d+e x}}+\frac {40 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3}{3 e^7 (a+b x) (d+e x)^{3/2}}-\frac {6 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4}{e^7 (a+b x) (d+e x)^{5/2}}+\frac {12 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5}{7 e^7 (a+b x) (d+e x)^{7/2}}-\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^6}{9 e^7 (a+b x) (d+e x)^{9/2}} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^(11/2),x]

[Out]

(-2*(b*d - a*e)^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*e^7*(a + b*x)*(d + e*x)^(9/2)) + (12*b*(b*d - a*e)^5*Sqrt[
a^2 + 2*a*b*x + b^2*x^2])/(7*e^7*(a + b*x)*(d + e*x)^(7/2)) - (6*b^2*(b*d - a*e)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^
2])/(e^7*(a + b*x)*(d + e*x)^(5/2)) + (40*b^3*(b*d - a*e)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^7*(a + b*x)*(d
+ e*x)^(3/2)) - (30*b^4*(b*d - a*e)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(a + b*x)*Sqrt[d + e*x]) - (12*b^5*
(b*d - a*e)*Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(a + b*x)) + (2*b^6*(d + e*x)^(3/2)*Sqrt[a^2 + 2
*a*b*x + b^2*x^2])/(3*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)^{11/2}} \, 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)^{11/2}} \, 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)^{11/2}} \, 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)^{11/2}}-\frac {6 b (b d-a e)^5}{e^6 (d+e x)^{9/2}}+\frac {15 b^2 (b d-a e)^4}{e^6 (d+e x)^{7/2}}-\frac {20 b^3 (b d-a e)^3}{e^6 (d+e x)^{5/2}}+\frac {15 b^4 (b d-a e)^2}{e^6 (d+e x)^{3/2}}-\frac {6 b^5 (b d-a e)}{e^6 \sqrt {d+e x}}+\frac {b^6 \sqrt {d+e x}}{e^6}\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}}{9 e^7 (a+b x) (d+e x)^{9/2}}+\frac {12 b (b d-a e)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^7 (a+b x) (d+e x)^{7/2}}-\frac {6 b^2 (b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x) (d+e x)^{5/2}}+\frac {40 b^3 (b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^7 (a+b x) (d+e x)^{3/2}}-\frac {30 b^4 (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x) \sqrt {d+e x}}-\frac {12 b^5 (b d-a e) \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x)}+\frac {2 b^6 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^7 (a+b x)}\\ \end {align*}

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Mathematica [A]  time = 0.15, size = 163, normalized size = 0.44 \[ \frac {2 \sqrt {(a+b x)^2} \left (-378 b^5 (d+e x)^5 (b d-a e)-945 b^4 (d+e x)^4 (b d-a e)^2+420 b^3 (d+e x)^3 (b d-a e)^3-189 b^2 (d+e x)^2 (b d-a e)^4+54 b (d+e x) (b d-a e)^5-7 (b d-a e)^6+21 b^6 (d+e x)^6\right )}{63 e^7 (a+b x) (d+e x)^{9/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^(11/2),x]

[Out]

(2*Sqrt[(a + b*x)^2]*(-7*(b*d - a*e)^6 + 54*b*(b*d - a*e)^5*(d + e*x) - 189*b^2*(b*d - a*e)^4*(d + e*x)^2 + 42
0*b^3*(b*d - a*e)^3*(d + e*x)^3 - 945*b^4*(b*d - a*e)^2*(d + e*x)^4 - 378*b^5*(b*d - a*e)*(d + e*x)^5 + 21*b^6
*(d + e*x)^6))/(63*e^7*(a + b*x)*(d + e*x)^(9/2))

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fricas [A]  time = 0.93, size = 409, normalized size = 1.11 \[ \frac {2 \, {\left (21 \, b^{6} e^{6} x^{6} - 1024 \, b^{6} d^{6} + 1536 \, a b^{5} d^{5} e - 384 \, a^{2} b^{4} d^{4} e^{2} - 64 \, a^{3} b^{3} d^{3} e^{3} - 24 \, a^{4} b^{2} d^{2} e^{4} - 12 \, a^{5} b d e^{5} - 7 \, a^{6} e^{6} - 126 \, {\left (2 \, b^{6} d e^{5} - 3 \, a b^{5} e^{6}\right )} x^{5} - 315 \, {\left (8 \, b^{6} d^{2} e^{4} - 12 \, a b^{5} d e^{5} + 3 \, a^{2} b^{4} e^{6}\right )} x^{4} - 420 \, {\left (16 \, b^{6} d^{3} e^{3} - 24 \, a b^{5} d^{2} e^{4} + 6 \, a^{2} b^{4} d e^{5} + a^{3} b^{3} e^{6}\right )} x^{3} - 63 \, {\left (128 \, b^{6} d^{4} e^{2} - 192 \, a b^{5} d^{3} e^{3} + 48 \, a^{2} b^{4} d^{2} e^{4} + 8 \, a^{3} b^{3} d e^{5} + 3 \, a^{4} b^{2} e^{6}\right )} x^{2} - 18 \, {\left (256 \, b^{6} d^{5} e - 384 \, a b^{5} d^{4} e^{2} + 96 \, a^{2} b^{4} d^{3} e^{3} + 16 \, a^{3} b^{3} d^{2} e^{4} + 6 \, a^{4} b^{2} d e^{5} + 3 \, a^{5} b e^{6}\right )} x\right )} \sqrt {e x + d}}{63 \, {\left (e^{12} x^{5} + 5 \, d e^{11} x^{4} + 10 \, d^{2} e^{10} x^{3} + 10 \, d^{3} e^{9} x^{2} + 5 \, d^{4} e^{8} x + d^{5} e^{7}\right )}} \]

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)^(11/2),x, algorithm="fricas")

[Out]

2/63*(21*b^6*e^6*x^6 - 1024*b^6*d^6 + 1536*a*b^5*d^5*e - 384*a^2*b^4*d^4*e^2 - 64*a^3*b^3*d^3*e^3 - 24*a^4*b^2
*d^2*e^4 - 12*a^5*b*d*e^5 - 7*a^6*e^6 - 126*(2*b^6*d*e^5 - 3*a*b^5*e^6)*x^5 - 315*(8*b^6*d^2*e^4 - 12*a*b^5*d*
e^5 + 3*a^2*b^4*e^6)*x^4 - 420*(16*b^6*d^3*e^3 - 24*a*b^5*d^2*e^4 + 6*a^2*b^4*d*e^5 + a^3*b^3*e^6)*x^3 - 63*(1
28*b^6*d^4*e^2 - 192*a*b^5*d^3*e^3 + 48*a^2*b^4*d^2*e^4 + 8*a^3*b^3*d*e^5 + 3*a^4*b^2*e^6)*x^2 - 18*(256*b^6*d
^5*e - 384*a*b^5*d^4*e^2 + 96*a^2*b^4*d^3*e^3 + 16*a^3*b^3*d^2*e^4 + 6*a^4*b^2*d*e^5 + 3*a^5*b*e^6)*x)*sqrt(e*
x + d)/(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)

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giac [B]  time = 0.27, size = 623, normalized size = 1.68 \[ \frac {2}{3} \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} b^{6} e^{14} \mathrm {sgn}\left (b x + a\right ) - 18 \, \sqrt {x e + d} b^{6} d e^{14} \mathrm {sgn}\left (b x + a\right ) + 18 \, \sqrt {x e + d} a b^{5} e^{15} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-21\right )} - \frac {2 \, {\left (945 \, {\left (x e + d\right )}^{4} b^{6} d^{2} \mathrm {sgn}\left (b x + a\right ) - 420 \, {\left (x e + d\right )}^{3} b^{6} d^{3} \mathrm {sgn}\left (b x + a\right ) + 189 \, {\left (x e + d\right )}^{2} b^{6} d^{4} \mathrm {sgn}\left (b x + a\right ) - 54 \, {\left (x e + d\right )} b^{6} d^{5} \mathrm {sgn}\left (b x + a\right ) + 7 \, b^{6} d^{6} \mathrm {sgn}\left (b x + a\right ) - 1890 \, {\left (x e + d\right )}^{4} a b^{5} d e \mathrm {sgn}\left (b x + a\right ) + 1260 \, {\left (x e + d\right )}^{3} a b^{5} d^{2} e \mathrm {sgn}\left (b x + a\right ) - 756 \, {\left (x e + d\right )}^{2} a b^{5} d^{3} e \mathrm {sgn}\left (b x + a\right ) + 270 \, {\left (x e + d\right )} a b^{5} d^{4} e \mathrm {sgn}\left (b x + a\right ) - 42 \, a b^{5} d^{5} e \mathrm {sgn}\left (b x + a\right ) + 945 \, {\left (x e + d\right )}^{4} a^{2} b^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) - 1260 \, {\left (x e + d\right )}^{3} a^{2} b^{4} d e^{2} \mathrm {sgn}\left (b x + a\right ) + 1134 \, {\left (x e + d\right )}^{2} a^{2} b^{4} d^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) - 540 \, {\left (x e + d\right )} a^{2} b^{4} d^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) + 105 \, a^{2} b^{4} d^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) + 420 \, {\left (x e + d\right )}^{3} a^{3} b^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) - 756 \, {\left (x e + d\right )}^{2} a^{3} b^{3} d e^{3} \mathrm {sgn}\left (b x + a\right ) + 540 \, {\left (x e + d\right )} a^{3} b^{3} d^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) - 140 \, a^{3} b^{3} d^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) + 189 \, {\left (x e + d\right )}^{2} a^{4} b^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) - 270 \, {\left (x e + d\right )} a^{4} b^{2} d e^{4} \mathrm {sgn}\left (b x + a\right ) + 105 \, a^{4} b^{2} d^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) + 54 \, {\left (x e + d\right )} a^{5} b e^{5} \mathrm {sgn}\left (b x + a\right ) - 42 \, a^{5} b d e^{5} \mathrm {sgn}\left (b x + a\right ) + 7 \, a^{6} e^{6} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-7\right )}}{63 \, {\left (x e + d\right )}^{\frac {9}{2}}} \]

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)^(11/2),x, algorithm="giac")

[Out]

2/3*((x*e + d)^(3/2)*b^6*e^14*sgn(b*x + a) - 18*sqrt(x*e + d)*b^6*d*e^14*sgn(b*x + a) + 18*sqrt(x*e + d)*a*b^5
*e^15*sgn(b*x + a))*e^(-21) - 2/63*(945*(x*e + d)^4*b^6*d^2*sgn(b*x + a) - 420*(x*e + d)^3*b^6*d^3*sgn(b*x + a
) + 189*(x*e + d)^2*b^6*d^4*sgn(b*x + a) - 54*(x*e + d)*b^6*d^5*sgn(b*x + a) + 7*b^6*d^6*sgn(b*x + a) - 1890*(
x*e + d)^4*a*b^5*d*e*sgn(b*x + a) + 1260*(x*e + d)^3*a*b^5*d^2*e*sgn(b*x + a) - 756*(x*e + d)^2*a*b^5*d^3*e*sg
n(b*x + a) + 270*(x*e + d)*a*b^5*d^4*e*sgn(b*x + a) - 42*a*b^5*d^5*e*sgn(b*x + a) + 945*(x*e + d)^4*a^2*b^4*e^
2*sgn(b*x + a) - 1260*(x*e + d)^3*a^2*b^4*d*e^2*sgn(b*x + a) + 1134*(x*e + d)^2*a^2*b^4*d^2*e^2*sgn(b*x + a) -
540*(x*e + d)*a^2*b^4*d^3*e^2*sgn(b*x + a) + 105*a^2*b^4*d^4*e^2*sgn(b*x + a) + 420*(x*e + d)^3*a^3*b^3*e^3*s
gn(b*x + a) - 756*(x*e + d)^2*a^3*b^3*d*e^3*sgn(b*x + a) + 540*(x*e + d)*a^3*b^3*d^2*e^3*sgn(b*x + a) - 140*a^
3*b^3*d^3*e^3*sgn(b*x + a) + 189*(x*e + d)^2*a^4*b^2*e^4*sgn(b*x + a) - 270*(x*e + d)*a^4*b^2*d*e^4*sgn(b*x +
a) + 105*a^4*b^2*d^2*e^4*sgn(b*x + a) + 54*(x*e + d)*a^5*b*e^5*sgn(b*x + a) - 42*a^5*b*d*e^5*sgn(b*x + a) + 7*
a^6*e^6*sgn(b*x + a))*e^(-7)/(x*e + d)^(9/2)

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maple [A]  time = 0.05, size = 393, normalized size = 1.06 \[ -\frac {2 \left (-21 b^{6} e^{6} x^{6}-378 a \,b^{5} e^{6} x^{5}+252 b^{6} d \,e^{5} x^{5}+945 a^{2} b^{4} e^{6} x^{4}-3780 a \,b^{5} d \,e^{5} x^{4}+2520 b^{6} d^{2} e^{4} x^{4}+420 a^{3} b^{3} e^{6} x^{3}+2520 a^{2} b^{4} d \,e^{5} x^{3}-10080 a \,b^{5} d^{2} e^{4} x^{3}+6720 b^{6} d^{3} e^{3} x^{3}+189 a^{4} b^{2} e^{6} x^{2}+504 a^{3} b^{3} d \,e^{5} x^{2}+3024 a^{2} b^{4} d^{2} e^{4} x^{2}-12096 a \,b^{5} d^{3} e^{3} x^{2}+8064 b^{6} d^{4} e^{2} x^{2}+54 a^{5} b \,e^{6} x +108 a^{4} b^{2} d \,e^{5} x +288 a^{3} b^{3} d^{2} e^{4} x +1728 a^{2} b^{4} d^{3} e^{3} x -6912 a \,b^{5} d^{4} e^{2} x +4608 b^{6} d^{5} e x +7 a^{6} e^{6}+12 a^{5} b d \,e^{5}+24 a^{4} b^{2} d^{2} e^{4}+64 a^{3} b^{3} d^{3} e^{3}+384 a^{2} b^{4} d^{4} e^{2}-1536 a \,b^{5} d^{5} e +1024 b^{6} d^{6}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{63 \left (e x +d \right )^{\frac {9}{2}} \left (b x +a \right )^{5} e^{7}} \]

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)^(11/2),x)

[Out]

-2/63/(e*x+d)^(9/2)*(-21*b^6*e^6*x^6-378*a*b^5*e^6*x^5+252*b^6*d*e^5*x^5+945*a^2*b^4*e^6*x^4-3780*a*b^5*d*e^5*
x^4+2520*b^6*d^2*e^4*x^4+420*a^3*b^3*e^6*x^3+2520*a^2*b^4*d*e^5*x^3-10080*a*b^5*d^2*e^4*x^3+6720*b^6*d^3*e^3*x
^3+189*a^4*b^2*e^6*x^2+504*a^3*b^3*d*e^5*x^2+3024*a^2*b^4*d^2*e^4*x^2-12096*a*b^5*d^3*e^3*x^2+8064*b^6*d^4*e^2
*x^2+54*a^5*b*e^6*x+108*a^4*b^2*d*e^5*x+288*a^3*b^3*d^2*e^4*x+1728*a^2*b^4*d^3*e^3*x-6912*a*b^5*d^4*e^2*x+4608
*b^6*d^5*e*x+7*a^6*e^6+12*a^5*b*d*e^5+24*a^4*b^2*d^2*e^4+64*a^3*b^3*d^3*e^3+384*a^2*b^4*d^4*e^2-1536*a*b^5*d^5
*e+1024*b^6*d^6)*((b*x+a)^2)^(5/2)/e^7/(b*x+a)^5

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maxima [B]  time = 0.92, size = 687, normalized size = 1.86 \[ \frac {2 \, {\left (63 \, b^{5} e^{5} x^{5} + 256 \, b^{5} d^{5} - 128 \, a b^{4} d^{4} e - 32 \, a^{2} b^{3} d^{3} e^{2} - 16 \, a^{3} b^{2} d^{2} e^{3} - 10 \, a^{4} b d e^{4} - 7 \, a^{5} e^{5} + 315 \, {\left (2 \, b^{5} d e^{4} - a b^{4} e^{5}\right )} x^{4} + 210 \, {\left (8 \, b^{5} d^{2} e^{3} - 4 \, a b^{4} d e^{4} - a^{2} b^{3} e^{5}\right )} x^{3} + 126 \, {\left (16 \, b^{5} d^{3} e^{2} - 8 \, a b^{4} d^{2} e^{3} - 2 \, a^{2} b^{3} d e^{4} - a^{3} b^{2} e^{5}\right )} x^{2} + 9 \, {\left (128 \, b^{5} d^{4} e - 64 \, a b^{4} d^{3} e^{2} - 16 \, a^{2} b^{3} d^{2} e^{3} - 8 \, a^{3} b^{2} d e^{4} - 5 \, a^{4} b e^{5}\right )} x\right )} a}{63 \, {\left (e^{10} x^{4} + 4 \, d e^{9} x^{3} + 6 \, d^{2} e^{8} x^{2} + 4 \, d^{3} e^{7} x + d^{4} e^{6}\right )} \sqrt {e x + d}} + \frac {2 \, {\left (21 \, b^{5} e^{6} x^{6} - 1024 \, b^{5} d^{6} + 1280 \, a b^{4} d^{5} e - 256 \, a^{2} b^{3} d^{4} e^{2} - 32 \, a^{3} b^{2} d^{3} e^{3} - 8 \, a^{4} b d^{2} e^{4} - 2 \, a^{5} d e^{5} - 63 \, {\left (4 \, b^{5} d e^{5} - 5 \, a b^{4} e^{6}\right )} x^{5} - 630 \, {\left (4 \, b^{5} d^{2} e^{4} - 5 \, a b^{4} d e^{5} + a^{2} b^{3} e^{6}\right )} x^{4} - 210 \, {\left (32 \, b^{5} d^{3} e^{3} - 40 \, a b^{4} d^{2} e^{4} + 8 \, a^{2} b^{3} d e^{5} + a^{3} b^{2} e^{6}\right )} x^{3} - 63 \, {\left (128 \, b^{5} d^{4} e^{2} - 160 \, a b^{4} d^{3} e^{3} + 32 \, a^{2} b^{3} d^{2} e^{4} + 4 \, a^{3} b^{2} d e^{5} + a^{4} b e^{6}\right )} x^{2} - 9 \, {\left (512 \, b^{5} d^{5} e - 640 \, a b^{4} d^{4} e^{2} + 128 \, a^{2} b^{3} d^{3} e^{3} + 16 \, a^{3} b^{2} d^{2} e^{4} + 4 \, a^{4} b d e^{5} + a^{5} e^{6}\right )} x\right )} b}{63 \, {\left (e^{11} x^{4} + 4 \, d e^{10} x^{3} + 6 \, d^{2} e^{9} x^{2} + 4 \, d^{3} e^{8} x + d^{4} e^{7}\right )} \sqrt {e x + d}} \]

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)^(11/2),x, algorithm="maxima")

[Out]

2/63*(63*b^5*e^5*x^5 + 256*b^5*d^5 - 128*a*b^4*d^4*e - 32*a^2*b^3*d^3*e^2 - 16*a^3*b^2*d^2*e^3 - 10*a^4*b*d*e^
4 - 7*a^5*e^5 + 315*(2*b^5*d*e^4 - a*b^4*e^5)*x^4 + 210*(8*b^5*d^2*e^3 - 4*a*b^4*d*e^4 - a^2*b^3*e^5)*x^3 + 12
6*(16*b^5*d^3*e^2 - 8*a*b^4*d^2*e^3 - 2*a^2*b^3*d*e^4 - a^3*b^2*e^5)*x^2 + 9*(128*b^5*d^4*e - 64*a*b^4*d^3*e^2
- 16*a^2*b^3*d^2*e^3 - 8*a^3*b^2*d*e^4 - 5*a^4*b*e^5)*x)*a/((e^10*x^4 + 4*d*e^9*x^3 + 6*d^2*e^8*x^2 + 4*d^3*e
^7*x + d^4*e^6)*sqrt(e*x + d)) + 2/63*(21*b^5*e^6*x^6 - 1024*b^5*d^6 + 1280*a*b^4*d^5*e - 256*a^2*b^3*d^4*e^2
- 32*a^3*b^2*d^3*e^3 - 8*a^4*b*d^2*e^4 - 2*a^5*d*e^5 - 63*(4*b^5*d*e^5 - 5*a*b^4*e^6)*x^5 - 630*(4*b^5*d^2*e^4
- 5*a*b^4*d*e^5 + a^2*b^3*e^6)*x^4 - 210*(32*b^5*d^3*e^3 - 40*a*b^4*d^2*e^4 + 8*a^2*b^3*d*e^5 + a^3*b^2*e^6)*
x^3 - 63*(128*b^5*d^4*e^2 - 160*a*b^4*d^3*e^3 + 32*a^2*b^3*d^2*e^4 + 4*a^3*b^2*d*e^5 + a^4*b*e^6)*x^2 - 9*(512
*b^5*d^5*e - 640*a*b^4*d^4*e^2 + 128*a^2*b^3*d^3*e^3 + 16*a^3*b^2*d^2*e^4 + 4*a^4*b*d*e^5 + a^5*e^6)*x)*b/((e^
11*x^4 + 4*d*e^10*x^3 + 6*d^2*e^9*x^2 + 4*d^3*e^8*x + d^4*e^7)*sqrt(e*x + d))

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mupad [B]  time = 3.28, size = 508, normalized size = 1.37 \[ -\frac {\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}\,\left (\frac {14\,a^6\,e^6+24\,a^5\,b\,d\,e^5+48\,a^4\,b^2\,d^2\,e^4+128\,a^3\,b^3\,d^3\,e^3+768\,a^2\,b^4\,d^4\,e^2-3072\,a\,b^5\,d^5\,e+2048\,b^6\,d^6}{63\,b\,e^{11}}-\frac {2\,b^5\,x^6}{3\,e^5}+\frac {x\,\left (108\,a^5\,b\,e^6+216\,a^4\,b^2\,d\,e^5+576\,a^3\,b^3\,d^2\,e^4+3456\,a^2\,b^4\,d^3\,e^3-13824\,a\,b^5\,d^4\,e^2+9216\,b^6\,d^5\,e\right )}{63\,b\,e^{11}}+\frac {40\,b^2\,x^3\,\left (a^3\,e^3+6\,a^2\,b\,d\,e^2-24\,a\,b^2\,d^2\,e+16\,b^3\,d^3\right )}{3\,e^8}+\frac {2\,b\,x^2\,\left (3\,a^4\,e^4+8\,a^3\,b\,d\,e^3+48\,a^2\,b^2\,d^2\,e^2-192\,a\,b^3\,d^3\,e+128\,b^4\,d^4\right )}{e^9}-\frac {4\,b^4\,x^5\,\left (3\,a\,e-2\,b\,d\right )}{e^6}+\frac {10\,b^3\,x^4\,\left (3\,a^2\,e^2-12\,a\,b\,d\,e+8\,b^2\,d^2\right )}{e^7}\right )}{x^5\,\sqrt {d+e\,x}+\frac {a\,d^4\,\sqrt {d+e\,x}}{b\,e^4}+\frac {x^4\,\left (63\,a\,e^{11}+252\,b\,d\,e^{10}\right )\,\sqrt {d+e\,x}}{63\,b\,e^{11}}+\frac {2\,d\,x^3\,\left (2\,a\,e+3\,b\,d\right )\,\sqrt {d+e\,x}}{b\,e^2}+\frac {d^3\,x\,\left (4\,a\,e+b\,d\right )\,\sqrt {d+e\,x}}{b\,e^4}+\frac {2\,d^2\,x^2\,\left (3\,a\,e+2\,b\,d\right )\,\sqrt {d+e\,x}}{b\,e^3}} \]

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)^(11/2),x)

[Out]

-((a^2 + b^2*x^2 + 2*a*b*x)^(1/2)*((14*a^6*e^6 + 2048*b^6*d^6 + 768*a^2*b^4*d^4*e^2 + 128*a^3*b^3*d^3*e^3 + 48
*a^4*b^2*d^2*e^4 - 3072*a*b^5*d^5*e + 24*a^5*b*d*e^5)/(63*b*e^11) - (2*b^5*x^6)/(3*e^5) + (x*(108*a^5*b*e^6 +
9216*b^6*d^5*e - 13824*a*b^5*d^4*e^2 + 216*a^4*b^2*d*e^5 + 3456*a^2*b^4*d^3*e^3 + 576*a^3*b^3*d^2*e^4))/(63*b*
e^11) + (40*b^2*x^3*(a^3*e^3 + 16*b^3*d^3 - 24*a*b^2*d^2*e + 6*a^2*b*d*e^2))/(3*e^8) + (2*b*x^2*(3*a^4*e^4 + 1
28*b^4*d^4 + 48*a^2*b^2*d^2*e^2 - 192*a*b^3*d^3*e + 8*a^3*b*d*e^3))/e^9 - (4*b^4*x^5*(3*a*e - 2*b*d))/e^6 + (1
0*b^3*x^4*(3*a^2*e^2 + 8*b^2*d^2 - 12*a*b*d*e))/e^7))/(x^5*(d + e*x)^(1/2) + (a*d^4*(d + e*x)^(1/2))/(b*e^4) +
(x^4*(63*a*e^11 + 252*b*d*e^10)*(d + e*x)^(1/2))/(63*b*e^11) + (2*d*x^3*(2*a*e + 3*b*d)*(d + e*x)^(1/2))/(b*e
^2) + (d^3*x*(4*a*e + b*d)*(d + e*x)^(1/2))/(b*e^4) + (2*d^2*x^2*(3*a*e + 2*b*d)*(d + e*x)^(1/2))/(b*e^3))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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)**(11/2),x)

[Out]

Timed out

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