3.2117 \(\int \frac {(a+b x) (a^2+2 a b x+b^2 x^2)^{5/2}}{(d+e x)^{9/2}} \, dx\)

Optimal. Leaf size=368 \[ -\frac {10 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4}{e^7 (a+b x) (d+e x)^{3/2}}+\frac {12 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5}{5 e^7 (a+b x) (d+e x)^{5/2}}-\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^6}{7 e^7 (a+b x) (d+e x)^{7/2}}+\frac {2 b^6 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2}}{5 e^7 (a+b x)}-\frac {4 b^5 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)}{e^7 (a+b x)}+\frac {30 b^4 \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x} (b d-a e)^2}{e^7 (a+b x)}+\frac {40 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3}{e^7 (a+b x) \sqrt {d+e x}} \]

[Out]

-2/7*(-a*e+b*d)^6*((b*x+a)^2)^(1/2)/e^7/(b*x+a)/(e*x+d)^(7/2)+12/5*b*(-a*e+b*d)^5*((b*x+a)^2)^(1/2)/e^7/(b*x+a
)/(e*x+d)^(5/2)-10*b^2*(-a*e+b*d)^4*((b*x+a)^2)^(1/2)/e^7/(b*x+a)/(e*x+d)^(3/2)-4*b^5*(-a*e+b*d)*(e*x+d)^(3/2)
*((b*x+a)^2)^(1/2)/e^7/(b*x+a)+2/5*b^6*(e*x+d)^(5/2)*((b*x+a)^2)^(1/2)/e^7/(b*x+a)+40*b^3*(-a*e+b*d)^3*((b*x+a
)^2)^(1/2)/e^7/(b*x+a)/(e*x+d)^(1/2)+30*b^4*(-a*e+b*d)^2*(e*x+d)^(1/2)*((b*x+a)^2)^(1/2)/e^7/(b*x+a)

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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} \[ \frac {2 b^6 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2}}{5 e^7 (a+b x)}-\frac {4 b^5 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)}{e^7 (a+b x)}+\frac {30 b^4 \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x} (b d-a e)^2}{e^7 (a+b x)}+\frac {40 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3}{e^7 (a+b x) \sqrt {d+e x}}-\frac {10 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4}{e^7 (a+b x) (d+e x)^{3/2}}+\frac {12 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5}{5 e^7 (a+b x) (d+e x)^{5/2}}-\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^6}{7 e^7 (a+b x) (d+e x)^{7/2}} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^(9/2),x]

[Out]

(-2*(b*d - a*e)^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*e^7*(a + b*x)*(d + e*x)^(7/2)) + (12*b*(b*d - a*e)^5*Sqrt[
a^2 + 2*a*b*x + b^2*x^2])/(5*e^7*(a + b*x)*(d + e*x)^(5/2)) - (10*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)^(3/2)) + (40*b^3*(b*d - a*e)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(a + b*x)*Sqr
t[d + e*x]) + (30*b^4*(b*d - a*e)^2*Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(a + b*x)) - (4*b^5*(b*d
- a*e)*(d + e*x)^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(a + b*x)) + (2*b^6*(d + e*x)^(5/2)*Sqrt[a^2 + 2*a
*b*x + b^2*x^2])/(5*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)^{9/2}} \, 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)^{9/2}} \, 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)^{9/2}} \, 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)^{9/2}}-\frac {6 b (b d-a e)^5}{e^6 (d+e x)^{7/2}}+\frac {15 b^2 (b d-a e)^4}{e^6 (d+e x)^{5/2}}-\frac {20 b^3 (b d-a e)^3}{e^6 (d+e x)^{3/2}}+\frac {15 b^4 (b d-a e)^2}{e^6 \sqrt {d+e x}}-\frac {6 b^5 (b d-a e) \sqrt {d+e x}}{e^6}+\frac {b^6 (d+e x)^{3/2}}{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}}{7 e^7 (a+b x) (d+e x)^{7/2}}+\frac {12 b (b d-a e)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^7 (a+b x) (d+e x)^{5/2}}-\frac {10 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)^{3/2}}+\frac {40 b^3 (b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x) \sqrt {d+e x}}+\frac {30 b^4 (b d-a e)^2 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x)}-\frac {4 b^5 (b d-a e) (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x)}+\frac {2 b^6 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^7 (a+b x)}\\ \end {align*}

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Mathematica [A]  time = 0.16, size = 163, normalized size = 0.44 \[ \frac {2 \sqrt {(a+b x)^2} \left (-70 b^5 (d+e x)^5 (b d-a e)+525 b^4 (d+e x)^4 (b d-a e)^2+700 b^3 (d+e x)^3 (b d-a e)^3-175 b^2 (d+e x)^2 (b d-a e)^4+42 b (d+e x) (b d-a e)^5-5 (b d-a e)^6+7 b^6 (d+e x)^6\right )}{35 e^7 (a+b x) (d+e x)^{7/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^(9/2),x]

[Out]

(2*Sqrt[(a + b*x)^2]*(-5*(b*d - a*e)^6 + 42*b*(b*d - a*e)^5*(d + e*x) - 175*b^2*(b*d - a*e)^4*(d + e*x)^2 + 70
0*b^3*(b*d - a*e)^3*(d + e*x)^3 + 525*b^4*(b*d - a*e)^2*(d + e*x)^4 - 70*b^5*(b*d - a*e)*(d + e*x)^5 + 7*b^6*(
d + e*x)^6))/(35*e^7*(a + b*x)*(d + e*x)^(7/2))

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fricas [A]  time = 1.00, size = 399, normalized size = 1.08 \[ \frac {2 \, {\left (7 \, b^{6} e^{6} x^{6} + 1024 \, b^{6} d^{6} - 2560 \, a b^{5} d^{5} e + 1920 \, a^{2} b^{4} d^{4} e^{2} - 320 \, a^{3} b^{3} d^{3} e^{3} - 40 \, a^{4} b^{2} d^{2} e^{4} - 12 \, a^{5} b d e^{5} - 5 \, a^{6} e^{6} - 14 \, {\left (2 \, b^{6} d e^{5} - 5 \, a b^{5} e^{6}\right )} x^{5} + 35 \, {\left (8 \, b^{6} d^{2} e^{4} - 20 \, a b^{5} d e^{5} + 15 \, a^{2} b^{4} e^{6}\right )} x^{4} + 140 \, {\left (16 \, b^{6} d^{3} e^{3} - 40 \, a b^{5} d^{2} e^{4} + 30 \, a^{2} b^{4} d e^{5} - 5 \, a^{3} b^{3} e^{6}\right )} x^{3} + 35 \, {\left (128 \, b^{6} d^{4} e^{2} - 320 \, a b^{5} d^{3} e^{3} + 240 \, a^{2} b^{4} d^{2} e^{4} - 40 \, a^{3} b^{3} d e^{5} - 5 \, a^{4} b^{2} e^{6}\right )} x^{2} + 14 \, {\left (256 \, b^{6} d^{5} e - 640 \, a b^{5} d^{4} e^{2} + 480 \, a^{2} b^{4} d^{3} e^{3} - 80 \, a^{3} b^{3} d^{2} e^{4} - 10 \, a^{4} b^{2} d e^{5} - 3 \, a^{5} b e^{6}\right )} x\right )} \sqrt {e x + d}}{35 \, {\left (e^{11} x^{4} + 4 \, d e^{10} x^{3} + 6 \, d^{2} e^{9} x^{2} + 4 \, d^{3} e^{8} x + d^{4} e^{7}\right )}} \]

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)^(9/2),x, algorithm="fricas")

[Out]

2/35*(7*b^6*e^6*x^6 + 1024*b^6*d^6 - 2560*a*b^5*d^5*e + 1920*a^2*b^4*d^4*e^2 - 320*a^3*b^3*d^3*e^3 - 40*a^4*b^
2*d^2*e^4 - 12*a^5*b*d*e^5 - 5*a^6*e^6 - 14*(2*b^6*d*e^5 - 5*a*b^5*e^6)*x^5 + 35*(8*b^6*d^2*e^4 - 20*a*b^5*d*e
^5 + 15*a^2*b^4*e^6)*x^4 + 140*(16*b^6*d^3*e^3 - 40*a*b^5*d^2*e^4 + 30*a^2*b^4*d*e^5 - 5*a^3*b^3*e^6)*x^3 + 35
*(128*b^6*d^4*e^2 - 320*a*b^5*d^3*e^3 + 240*a^2*b^4*d^2*e^4 - 40*a^3*b^3*d*e^5 - 5*a^4*b^2*e^6)*x^2 + 14*(256*
b^6*d^5*e - 640*a*b^5*d^4*e^2 + 480*a^2*b^4*d^3*e^3 - 80*a^3*b^3*d^2*e^4 - 10*a^4*b^2*d*e^5 - 3*a^5*b*e^6)*x)*
sqrt(e*x + d)/(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)

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giac [B]  time = 0.28, size = 625, normalized size = 1.70 \[ \frac {2}{5} \, {\left ({\left (x e + d\right )}^{\frac {5}{2}} b^{6} e^{28} \mathrm {sgn}\left (b x + a\right ) - 10 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{6} d e^{28} \mathrm {sgn}\left (b x + a\right ) + 75 \, \sqrt {x e + d} b^{6} d^{2} e^{28} \mathrm {sgn}\left (b x + a\right ) + 10 \, {\left (x e + d\right )}^{\frac {3}{2}} a b^{5} e^{29} \mathrm {sgn}\left (b x + a\right ) - 150 \, \sqrt {x e + d} a b^{5} d e^{29} \mathrm {sgn}\left (b x + a\right ) + 75 \, \sqrt {x e + d} a^{2} b^{4} e^{30} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-35\right )} + \frac {2 \, {\left (700 \, {\left (x e + d\right )}^{3} b^{6} d^{3} \mathrm {sgn}\left (b x + a\right ) - 175 \, {\left (x e + d\right )}^{2} b^{6} d^{4} \mathrm {sgn}\left (b x + a\right ) + 42 \, {\left (x e + d\right )} b^{6} d^{5} \mathrm {sgn}\left (b x + a\right ) - 5 \, b^{6} d^{6} \mathrm {sgn}\left (b x + a\right ) - 2100 \, {\left (x e + d\right )}^{3} a b^{5} d^{2} e \mathrm {sgn}\left (b x + a\right ) + 700 \, {\left (x e + d\right )}^{2} a b^{5} d^{3} e \mathrm {sgn}\left (b x + a\right ) - 210 \, {\left (x e + d\right )} a b^{5} d^{4} e \mathrm {sgn}\left (b x + a\right ) + 30 \, a b^{5} d^{5} e \mathrm {sgn}\left (b x + a\right ) + 2100 \, {\left (x e + d\right )}^{3} a^{2} b^{4} d e^{2} \mathrm {sgn}\left (b x + a\right ) - 1050 \, {\left (x e + d\right )}^{2} a^{2} b^{4} d^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) + 420 \, {\left (x e + d\right )} a^{2} b^{4} d^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) - 75 \, a^{2} b^{4} d^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) - 700 \, {\left (x e + d\right )}^{3} a^{3} b^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) + 700 \, {\left (x e + d\right )}^{2} a^{3} b^{3} d e^{3} \mathrm {sgn}\left (b x + a\right ) - 420 \, {\left (x e + d\right )} a^{3} b^{3} d^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) + 100 \, a^{3} b^{3} d^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) - 175 \, {\left (x e + d\right )}^{2} a^{4} b^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) + 210 \, {\left (x e + d\right )} a^{4} b^{2} d e^{4} \mathrm {sgn}\left (b x + a\right ) - 75 \, a^{4} b^{2} d^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) - 42 \, {\left (x e + d\right )} a^{5} b e^{5} \mathrm {sgn}\left (b x + a\right ) + 30 \, a^{5} b d e^{5} \mathrm {sgn}\left (b x + a\right ) - 5 \, a^{6} e^{6} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-7\right )}}{35 \, {\left (x e + d\right )}^{\frac {7}{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)^(9/2),x, algorithm="giac")

[Out]

2/5*((x*e + d)^(5/2)*b^6*e^28*sgn(b*x + a) - 10*(x*e + d)^(3/2)*b^6*d*e^28*sgn(b*x + a) + 75*sqrt(x*e + d)*b^6
*d^2*e^28*sgn(b*x + a) + 10*(x*e + d)^(3/2)*a*b^5*e^29*sgn(b*x + a) - 150*sqrt(x*e + d)*a*b^5*d*e^29*sgn(b*x +
a) + 75*sqrt(x*e + d)*a^2*b^4*e^30*sgn(b*x + a))*e^(-35) + 2/35*(700*(x*e + d)^3*b^6*d^3*sgn(b*x + a) - 175*(
x*e + d)^2*b^6*d^4*sgn(b*x + a) + 42*(x*e + d)*b^6*d^5*sgn(b*x + a) - 5*b^6*d^6*sgn(b*x + a) - 2100*(x*e + d)^
3*a*b^5*d^2*e*sgn(b*x + a) + 700*(x*e + d)^2*a*b^5*d^3*e*sgn(b*x + a) - 210*(x*e + d)*a*b^5*d^4*e*sgn(b*x + a)
+ 30*a*b^5*d^5*e*sgn(b*x + a) + 2100*(x*e + d)^3*a^2*b^4*d*e^2*sgn(b*x + a) - 1050*(x*e + d)^2*a^2*b^4*d^2*e^
2*sgn(b*x + a) + 420*(x*e + d)*a^2*b^4*d^3*e^2*sgn(b*x + a) - 75*a^2*b^4*d^4*e^2*sgn(b*x + a) - 700*(x*e + d)^
3*a^3*b^3*e^3*sgn(b*x + a) + 700*(x*e + d)^2*a^3*b^3*d*e^3*sgn(b*x + a) - 420*(x*e + d)*a^3*b^3*d^2*e^3*sgn(b*
x + a) + 100*a^3*b^3*d^3*e^3*sgn(b*x + a) - 175*(x*e + d)^2*a^4*b^2*e^4*sgn(b*x + a) + 210*(x*e + d)*a^4*b^2*d
*e^4*sgn(b*x + a) - 75*a^4*b^2*d^2*e^4*sgn(b*x + a) - 42*(x*e + d)*a^5*b*e^5*sgn(b*x + a) + 30*a^5*b*d*e^5*sgn
(b*x + a) - 5*a^6*e^6*sgn(b*x + a))*e^(-7)/(x*e + d)^(7/2)

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maple [A]  time = 0.05, size = 393, normalized size = 1.07 \[ -\frac {2 \left (-7 b^{6} e^{6} x^{6}-70 a \,b^{5} e^{6} x^{5}+28 b^{6} d \,e^{5} x^{5}-525 a^{2} b^{4} e^{6} x^{4}+700 a \,b^{5} d \,e^{5} x^{4}-280 b^{6} d^{2} e^{4} x^{4}+700 a^{3} b^{3} e^{6} x^{3}-4200 a^{2} b^{4} d \,e^{5} x^{3}+5600 a \,b^{5} d^{2} e^{4} x^{3}-2240 b^{6} d^{3} e^{3} x^{3}+175 a^{4} b^{2} e^{6} x^{2}+1400 a^{3} b^{3} d \,e^{5} x^{2}-8400 a^{2} b^{4} d^{2} e^{4} x^{2}+11200 a \,b^{5} d^{3} e^{3} x^{2}-4480 b^{6} d^{4} e^{2} x^{2}+42 a^{5} b \,e^{6} x +140 a^{4} b^{2} d \,e^{5} x +1120 a^{3} b^{3} d^{2} e^{4} x -6720 a^{2} b^{4} d^{3} e^{3} x +8960 a \,b^{5} d^{4} e^{2} x -3584 b^{6} d^{5} e x +5 a^{6} e^{6}+12 a^{5} b d \,e^{5}+40 a^{4} b^{2} d^{2} e^{4}+320 a^{3} b^{3} d^{3} e^{3}-1920 a^{2} b^{4} d^{4} e^{2}+2560 a \,b^{5} d^{5} e -1024 b^{6} d^{6}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{35 \left (e x +d \right )^{\frac {7}{2}} \left (b x +a \right )^{5} e^{7}} \]

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)^(9/2),x)

[Out]

-2/35/(e*x+d)^(7/2)*(-7*b^6*e^6*x^6-70*a*b^5*e^6*x^5+28*b^6*d*e^5*x^5-525*a^2*b^4*e^6*x^4+700*a*b^5*d*e^5*x^4-
280*b^6*d^2*e^4*x^4+700*a^3*b^3*e^6*x^3-4200*a^2*b^4*d*e^5*x^3+5600*a*b^5*d^2*e^4*x^3-2240*b^6*d^3*e^3*x^3+175
*a^4*b^2*e^6*x^2+1400*a^3*b^3*d*e^5*x^2-8400*a^2*b^4*d^2*e^4*x^2+11200*a*b^5*d^3*e^3*x^2-4480*b^6*d^4*e^2*x^2+
42*a^5*b*e^6*x+140*a^4*b^2*d*e^5*x+1120*a^3*b^3*d^2*e^4*x-6720*a^2*b^4*d^3*e^3*x+8960*a*b^5*d^4*e^2*x-3584*b^6
*d^5*e*x+5*a^6*e^6+12*a^5*b*d*e^5+40*a^4*b^2*d^2*e^4+320*a^3*b^3*d^3*e^3-1920*a^2*b^4*d^4*e^2+2560*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 = 0.82, size = 668, normalized size = 1.82 \[ \frac {2 \, {\left (7 \, b^{5} e^{5} x^{5} - 256 \, b^{5} d^{5} + 384 \, a b^{4} d^{4} e - 96 \, a^{2} b^{3} d^{3} e^{2} - 16 \, a^{3} b^{2} d^{2} e^{3} - 6 \, a^{4} b d e^{4} - 3 \, a^{5} e^{5} - 35 \, {\left (2 \, b^{5} d e^{4} - 3 \, a b^{4} e^{5}\right )} x^{4} - 70 \, {\left (8 \, b^{5} d^{2} e^{3} - 12 \, a b^{4} d e^{4} + 3 \, a^{2} b^{3} e^{5}\right )} x^{3} - 70 \, {\left (16 \, b^{5} d^{3} e^{2} - 24 \, a b^{4} d^{2} e^{3} + 6 \, a^{2} b^{3} d e^{4} + a^{3} b^{2} e^{5}\right )} x^{2} - 7 \, {\left (128 \, b^{5} d^{4} e - 192 \, a b^{4} d^{3} e^{2} + 48 \, a^{2} b^{3} d^{2} e^{3} + 8 \, a^{3} b^{2} d e^{4} + 3 \, a^{4} b e^{5}\right )} x\right )} a}{21 \, {\left (e^{9} x^{3} + 3 \, d e^{8} x^{2} + 3 \, d^{2} e^{7} x + d^{3} e^{6}\right )} \sqrt {e x + d}} + \frac {2 \, {\left (21 \, b^{5} e^{6} x^{6} + 3072 \, b^{5} d^{6} - 6400 \, a b^{4} d^{5} e + 3840 \, a^{2} b^{3} d^{4} e^{2} - 480 \, a^{3} b^{2} d^{3} e^{3} - 40 \, a^{4} b d^{2} e^{4} - 6 \, a^{5} d e^{5} - 7 \, {\left (12 \, b^{5} d e^{5} - 25 \, a b^{4} e^{6}\right )} x^{5} + 70 \, {\left (12 \, b^{5} d^{2} e^{4} - 25 \, a b^{4} d e^{5} + 15 \, a^{2} b^{3} e^{6}\right )} x^{4} + 70 \, {\left (96 \, b^{5} d^{3} e^{3} - 200 \, a b^{4} d^{2} e^{4} + 120 \, a^{2} b^{3} d e^{5} - 15 \, a^{3} b^{2} e^{6}\right )} x^{3} + 35 \, {\left (384 \, b^{5} d^{4} e^{2} - 800 \, a b^{4} d^{3} e^{3} + 480 \, a^{2} b^{3} d^{2} e^{4} - 60 \, a^{3} b^{2} d e^{5} - 5 \, a^{4} b e^{6}\right )} x^{2} + 7 \, {\left (1536 \, b^{5} d^{5} e - 3200 \, a b^{4} d^{4} e^{2} + 1920 \, a^{2} b^{3} d^{3} e^{3} - 240 \, a^{3} b^{2} d^{2} e^{4} - 20 \, a^{4} b d e^{5} - 3 \, a^{5} e^{6}\right )} x\right )} b}{105 \, {\left (e^{10} x^{3} + 3 \, d e^{9} x^{2} + 3 \, d^{2} e^{8} x + d^{3} e^{7}\right )} \sqrt {e x + d}} \]

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)^(9/2),x, algorithm="maxima")

[Out]

2/21*(7*b^5*e^5*x^5 - 256*b^5*d^5 + 384*a*b^4*d^4*e - 96*a^2*b^3*d^3*e^2 - 16*a^3*b^2*d^2*e^3 - 6*a^4*b*d*e^4
- 3*a^5*e^5 - 35*(2*b^5*d*e^4 - 3*a*b^4*e^5)*x^4 - 70*(8*b^5*d^2*e^3 - 12*a*b^4*d*e^4 + 3*a^2*b^3*e^5)*x^3 - 7
0*(16*b^5*d^3*e^2 - 24*a*b^4*d^2*e^3 + 6*a^2*b^3*d*e^4 + a^3*b^2*e^5)*x^2 - 7*(128*b^5*d^4*e - 192*a*b^4*d^3*e
^2 + 48*a^2*b^3*d^2*e^3 + 8*a^3*b^2*d*e^4 + 3*a^4*b*e^5)*x)*a/((e^9*x^3 + 3*d*e^8*x^2 + 3*d^2*e^7*x + d^3*e^6)
*sqrt(e*x + d)) + 2/105*(21*b^5*e^6*x^6 + 3072*b^5*d^6 - 6400*a*b^4*d^5*e + 3840*a^2*b^3*d^4*e^2 - 480*a^3*b^2
*d^3*e^3 - 40*a^4*b*d^2*e^4 - 6*a^5*d*e^5 - 7*(12*b^5*d*e^5 - 25*a*b^4*e^6)*x^5 + 70*(12*b^5*d^2*e^4 - 25*a*b^
4*d*e^5 + 15*a^2*b^3*e^6)*x^4 + 70*(96*b^5*d^3*e^3 - 200*a*b^4*d^2*e^4 + 120*a^2*b^3*d*e^5 - 15*a^3*b^2*e^6)*x
^3 + 35*(384*b^5*d^4*e^2 - 800*a*b^4*d^3*e^3 + 480*a^2*b^3*d^2*e^4 - 60*a^3*b^2*d*e^5 - 5*a^4*b*e^6)*x^2 + 7*(
1536*b^5*d^5*e - 3200*a*b^4*d^4*e^2 + 1920*a^2*b^3*d^3*e^3 - 240*a^3*b^2*d^2*e^4 - 20*a^4*b*d*e^5 - 3*a^5*e^6)
*x)*b/((e^10*x^3 + 3*d*e^9*x^2 + 3*d^2*e^8*x + d^3*e^7)*sqrt(e*x + d))

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mupad [B]  time = 3.23, size = 489, normalized size = 1.33 \[ -\frac {\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}\,\left (\frac {10\,a^6\,e^6+24\,a^5\,b\,d\,e^5+80\,a^4\,b^2\,d^2\,e^4+640\,a^3\,b^3\,d^3\,e^3-3840\,a^2\,b^4\,d^4\,e^2+5120\,a\,b^5\,d^5\,e-2048\,b^6\,d^6}{35\,b\,e^{10}}-\frac {2\,b^5\,x^6}{5\,e^4}+\frac {x\,\left (84\,a^5\,b\,e^6+280\,a^4\,b^2\,d\,e^5+2240\,a^3\,b^3\,d^2\,e^4-13440\,a^2\,b^4\,d^3\,e^3+17920\,a\,b^5\,d^4\,e^2-7168\,b^6\,d^5\,e\right )}{35\,b\,e^{10}}+\frac {8\,b^2\,x^3\,\left (5\,a^3\,e^3-30\,a^2\,b\,d\,e^2+40\,a\,b^2\,d^2\,e-16\,b^3\,d^3\right )}{e^7}-\frac {4\,b^4\,x^5\,\left (5\,a\,e-2\,b\,d\right )}{5\,e^5}-\frac {2\,b^3\,x^4\,\left (15\,a^2\,e^2-20\,a\,b\,d\,e+8\,b^2\,d^2\right )}{e^6}+\frac {x^2\,\left (350\,a^4\,b^2\,e^6+2800\,a^3\,b^3\,d\,e^5-16800\,a^2\,b^4\,d^2\,e^4+22400\,a\,b^5\,d^3\,e^3-8960\,b^6\,d^4\,e^2\right )}{35\,b\,e^{10}}\right )}{x^4\,\sqrt {d+e\,x}+\frac {a\,d^3\,\sqrt {d+e\,x}}{b\,e^3}+\frac {x^3\,\left (35\,a\,e^{10}+105\,b\,d\,e^9\right )\,\sqrt {d+e\,x}}{35\,b\,e^{10}}+\frac {3\,d\,x^2\,\left (a\,e+b\,d\right )\,\sqrt {d+e\,x}}{b\,e^2}+\frac {d^2\,x\,\left (3\,a\,e+b\,d\right )\,\sqrt {d+e\,x}}{b\,e^3}} \]

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)^(9/2),x)

[Out]

-((a^2 + b^2*x^2 + 2*a*b*x)^(1/2)*((10*a^6*e^6 - 2048*b^6*d^6 - 3840*a^2*b^4*d^4*e^2 + 640*a^3*b^3*d^3*e^3 + 8
0*a^4*b^2*d^2*e^4 + 5120*a*b^5*d^5*e + 24*a^5*b*d*e^5)/(35*b*e^10) - (2*b^5*x^6)/(5*e^4) + (x*(84*a^5*b*e^6 -
7168*b^6*d^5*e + 17920*a*b^5*d^4*e^2 + 280*a^4*b^2*d*e^5 - 13440*a^2*b^4*d^3*e^3 + 2240*a^3*b^3*d^2*e^4))/(35*
b*e^10) + (8*b^2*x^3*(5*a^3*e^3 - 16*b^3*d^3 + 40*a*b^2*d^2*e - 30*a^2*b*d*e^2))/e^7 - (4*b^4*x^5*(5*a*e - 2*b
*d))/(5*e^5) - (2*b^3*x^4*(15*a^2*e^2 + 8*b^2*d^2 - 20*a*b*d*e))/e^6 + (x^2*(350*a^4*b^2*e^6 - 8960*b^6*d^4*e^
2 + 22400*a*b^5*d^3*e^3 + 2800*a^3*b^3*d*e^5 - 16800*a^2*b^4*d^2*e^4))/(35*b*e^10)))/(x^4*(d + e*x)^(1/2) + (a
*d^3*(d + e*x)^(1/2))/(b*e^3) + (x^3*(35*a*e^10 + 105*b*d*e^9)*(d + e*x)^(1/2))/(35*b*e^10) + (3*d*x^2*(a*e +
b*d)*(d + e*x)^(1/2))/(b*e^2) + (d^2*x*(3*a*e + 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)**(9/2),x)

[Out]

Timed out

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