3.2120 \(\int \frac {(a+b x) (a^2+2 a b x+b^2 x^2)^{5/2}}{(d+e x)^{15/2}} \, dx\)

Optimal. Leaf size=370 \[ -\frac {10 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4}{3 e^7 (a+b x) (d+e x)^{9/2}}+\frac {12 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5}{11 e^7 (a+b x) (d+e x)^{11/2}}-\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^6}{13 e^7 (a+b x) (d+e x)^{13/2}}-\frac {2 b^6 \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x) \sqrt {d+e x}}+\frac {4 b^5 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}{e^7 (a+b x) (d+e x)^{3/2}}-\frac {6 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}{e^7 (a+b x) (d+e x)^{5/2}}+\frac {40 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3}{7 e^7 (a+b x) (d+e x)^{7/2}} \]

[Out]

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

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Rubi [A]  time = 0.14, 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} \[ -\frac {2 b^6 \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x) \sqrt {d+e x}}+\frac {4 b^5 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}{e^7 (a+b x) (d+e x)^{3/2}}-\frac {6 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}{e^7 (a+b x) (d+e x)^{5/2}}+\frac {40 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3}{7 e^7 (a+b x) (d+e x)^{7/2}}-\frac {10 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4}{3 e^7 (a+b x) (d+e x)^{9/2}}+\frac {12 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5}{11 e^7 (a+b x) (d+e x)^{11/2}}-\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^6}{13 e^7 (a+b x) (d+e x)^{13/2}} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^(15/2),x]

[Out]

(-2*(b*d - a*e)^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(13*e^7*(a + b*x)*(d + e*x)^(13/2)) + (12*b*(b*d - a*e)^5*Sqr
t[a^2 + 2*a*b*x + b^2*x^2])/(11*e^7*(a + b*x)*(d + e*x)^(11/2)) - (10*b^2*(b*d - a*e)^4*Sqrt[a^2 + 2*a*b*x + b
^2*x^2])/(3*e^7*(a + b*x)*(d + e*x)^(9/2)) + (40*b^3*(b*d - a*e)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*e^7*(a +
b*x)*(d + e*x)^(7/2)) - (6*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)^(5/2)) +
(4*b^5*(b*d - a*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(a + b*x)*(d + e*x)^(3/2)) - (2*b^6*Sqrt[a^2 + 2*a*b*x
+ b^2*x^2])/(e^7*(a + b*x)*Sqrt[d + e*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)^{15/2}} \, 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)^{15/2}} \, 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)^{15/2}} \, 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)^{15/2}}-\frac {6 b (b d-a e)^5}{e^6 (d+e x)^{13/2}}+\frac {15 b^2 (b d-a e)^4}{e^6 (d+e x)^{11/2}}-\frac {20 b^3 (b d-a e)^3}{e^6 (d+e x)^{9/2}}+\frac {15 b^4 (b d-a e)^2}{e^6 (d+e x)^{7/2}}-\frac {6 b^5 (b d-a e)}{e^6 (d+e x)^{5/2}}+\frac {b^6}{e^6 (d+e x)^{3/2}}\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}}{13 e^7 (a+b x) (d+e x)^{13/2}}+\frac {12 b (b d-a e)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{11 e^7 (a+b x) (d+e x)^{11/2}}-\frac {10 b^2 (b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^7 (a+b x) (d+e x)^{9/2}}+\frac {40 b^3 (b d-a e)^3 \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^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)^{5/2}}+\frac {4 b^5 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x) (d+e x)^{3/2}}-\frac {2 b^6 \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x) \sqrt {d+e x}}\\ \end {align*}

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Mathematica [A]  time = 0.16, size = 163, normalized size = 0.44 \[ \frac {2 \sqrt {(a+b x)^2} \left (6006 b^5 (d+e x)^5 (b d-a e)-9009 b^4 (d+e x)^4 (b d-a e)^2+8580 b^3 (d+e x)^3 (b d-a e)^3-5005 b^2 (d+e x)^2 (b d-a e)^4+1638 b (d+e x) (b d-a e)^5-231 (b d-a e)^6-3003 b^6 (d+e x)^6\right )}{3003 e^7 (a+b x) (d+e x)^{13/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^(15/2),x]

[Out]

(2*Sqrt[(a + b*x)^2]*(-231*(b*d - a*e)^6 + 1638*b*(b*d - a*e)^5*(d + e*x) - 5005*b^2*(b*d - a*e)^4*(d + e*x)^2
+ 8580*b^3*(b*d - a*e)^3*(d + e*x)^3 - 9009*b^4*(b*d - a*e)^2*(d + e*x)^4 + 6006*b^5*(b*d - a*e)*(d + e*x)^5
- 3003*b^6*(d + e*x)^6))/(3003*e^7*(a + b*x)*(d + e*x)^(13/2))

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fricas [A]  time = 0.87, size = 431, normalized size = 1.16 \[ -\frac {2 \, {\left (3003 \, b^{6} e^{6} x^{6} + 1024 \, b^{6} d^{6} + 512 \, a b^{5} d^{5} e + 384 \, a^{2} b^{4} d^{4} e^{2} + 320 \, a^{3} b^{3} d^{3} e^{3} + 280 \, a^{4} b^{2} d^{2} e^{4} + 252 \, a^{5} b d e^{5} + 231 \, a^{6} e^{6} + 6006 \, {\left (2 \, b^{6} d e^{5} + a b^{5} e^{6}\right )} x^{5} + 3003 \, {\left (8 \, b^{6} d^{2} e^{4} + 4 \, a b^{5} d e^{5} + 3 \, a^{2} b^{4} e^{6}\right )} x^{4} + 1716 \, {\left (16 \, b^{6} d^{3} e^{3} + 8 \, a b^{5} d^{2} e^{4} + 6 \, a^{2} b^{4} d e^{5} + 5 \, a^{3} b^{3} e^{6}\right )} x^{3} + 143 \, {\left (128 \, b^{6} d^{4} e^{2} + 64 \, a b^{5} d^{3} e^{3} + 48 \, a^{2} b^{4} d^{2} e^{4} + 40 \, a^{3} b^{3} d e^{5} + 35 \, a^{4} b^{2} e^{6}\right )} x^{2} + 26 \, {\left (256 \, b^{6} d^{5} e + 128 \, a b^{5} d^{4} e^{2} + 96 \, a^{2} b^{4} d^{3} e^{3} + 80 \, a^{3} b^{3} d^{2} e^{4} + 70 \, a^{4} b^{2} d e^{5} + 63 \, a^{5} b e^{6}\right )} x\right )} \sqrt {e x + d}}{3003 \, {\left (e^{14} x^{7} + 7 \, d e^{13} x^{6} + 21 \, d^{2} e^{12} x^{5} + 35 \, d^{3} e^{11} x^{4} + 35 \, d^{4} e^{10} x^{3} + 21 \, d^{5} e^{9} x^{2} + 7 \, d^{6} e^{8} x + d^{7} e^{7}\right )}} \]

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)^(15/2),x, algorithm="fricas")

[Out]

-2/3003*(3003*b^6*e^6*x^6 + 1024*b^6*d^6 + 512*a*b^5*d^5*e + 384*a^2*b^4*d^4*e^2 + 320*a^3*b^3*d^3*e^3 + 280*a
^4*b^2*d^2*e^4 + 252*a^5*b*d*e^5 + 231*a^6*e^6 + 6006*(2*b^6*d*e^5 + a*b^5*e^6)*x^5 + 3003*(8*b^6*d^2*e^4 + 4*
a*b^5*d*e^5 + 3*a^2*b^4*e^6)*x^4 + 1716*(16*b^6*d^3*e^3 + 8*a*b^5*d^2*e^4 + 6*a^2*b^4*d*e^5 + 5*a^3*b^3*e^6)*x
^3 + 143*(128*b^6*d^4*e^2 + 64*a*b^5*d^3*e^3 + 48*a^2*b^4*d^2*e^4 + 40*a^3*b^3*d*e^5 + 35*a^4*b^2*e^6)*x^2 + 2
6*(256*b^6*d^5*e + 128*a*b^5*d^4*e^2 + 96*a^2*b^4*d^3*e^3 + 80*a^3*b^3*d^2*e^4 + 70*a^4*b^2*d*e^5 + 63*a^5*b*e
^6)*x)*sqrt(e*x + d)/(e^14*x^7 + 7*d*e^13*x^6 + 21*d^2*e^12*x^5 + 35*d^3*e^11*x^4 + 35*d^4*e^10*x^3 + 21*d^5*e
^9*x^2 + 7*d^6*e^8*x + d^7*e^7)

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giac [B]  time = 0.29, size = 614, normalized size = 1.66 \[ -\frac {2 \, {\left (3003 \, {\left (x e + d\right )}^{6} b^{6} \mathrm {sgn}\left (b x + a\right ) - 6006 \, {\left (x e + d\right )}^{5} b^{6} d \mathrm {sgn}\left (b x + a\right ) + 9009 \, {\left (x e + d\right )}^{4} b^{6} d^{2} \mathrm {sgn}\left (b x + a\right ) - 8580 \, {\left (x e + d\right )}^{3} b^{6} d^{3} \mathrm {sgn}\left (b x + a\right ) + 5005 \, {\left (x e + d\right )}^{2} b^{6} d^{4} \mathrm {sgn}\left (b x + a\right ) - 1638 \, {\left (x e + d\right )} b^{6} d^{5} \mathrm {sgn}\left (b x + a\right ) + 231 \, b^{6} d^{6} \mathrm {sgn}\left (b x + a\right ) + 6006 \, {\left (x e + d\right )}^{5} a b^{5} e \mathrm {sgn}\left (b x + a\right ) - 18018 \, {\left (x e + d\right )}^{4} a b^{5} d e \mathrm {sgn}\left (b x + a\right ) + 25740 \, {\left (x e + d\right )}^{3} a b^{5} d^{2} e \mathrm {sgn}\left (b x + a\right ) - 20020 \, {\left (x e + d\right )}^{2} a b^{5} d^{3} e \mathrm {sgn}\left (b x + a\right ) + 8190 \, {\left (x e + d\right )} a b^{5} d^{4} e \mathrm {sgn}\left (b x + a\right ) - 1386 \, a b^{5} d^{5} e \mathrm {sgn}\left (b x + a\right ) + 9009 \, {\left (x e + d\right )}^{4} a^{2} b^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) - 25740 \, {\left (x e + d\right )}^{3} a^{2} b^{4} d e^{2} \mathrm {sgn}\left (b x + a\right ) + 30030 \, {\left (x e + d\right )}^{2} a^{2} b^{4} d^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) - 16380 \, {\left (x e + d\right )} a^{2} b^{4} d^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) + 3465 \, a^{2} b^{4} d^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) + 8580 \, {\left (x e + d\right )}^{3} a^{3} b^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) - 20020 \, {\left (x e + d\right )}^{2} a^{3} b^{3} d e^{3} \mathrm {sgn}\left (b x + a\right ) + 16380 \, {\left (x e + d\right )} a^{3} b^{3} d^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) - 4620 \, a^{3} b^{3} d^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) + 5005 \, {\left (x e + d\right )}^{2} a^{4} b^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) - 8190 \, {\left (x e + d\right )} a^{4} b^{2} d e^{4} \mathrm {sgn}\left (b x + a\right ) + 3465 \, a^{4} b^{2} d^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) + 1638 \, {\left (x e + d\right )} a^{5} b e^{5} \mathrm {sgn}\left (b x + a\right ) - 1386 \, a^{5} b d e^{5} \mathrm {sgn}\left (b x + a\right ) + 231 \, a^{6} e^{6} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-7\right )}}{3003 \, {\left (x e + d\right )}^{\frac {13}{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)^(15/2),x, algorithm="giac")

[Out]

-2/3003*(3003*(x*e + d)^6*b^6*sgn(b*x + a) - 6006*(x*e + d)^5*b^6*d*sgn(b*x + a) + 9009*(x*e + d)^4*b^6*d^2*sg
n(b*x + a) - 8580*(x*e + d)^3*b^6*d^3*sgn(b*x + a) + 5005*(x*e + d)^2*b^6*d^4*sgn(b*x + a) - 1638*(x*e + d)*b^
6*d^5*sgn(b*x + a) + 231*b^6*d^6*sgn(b*x + a) + 6006*(x*e + d)^5*a*b^5*e*sgn(b*x + a) - 18018*(x*e + d)^4*a*b^
5*d*e*sgn(b*x + a) + 25740*(x*e + d)^3*a*b^5*d^2*e*sgn(b*x + a) - 20020*(x*e + d)^2*a*b^5*d^3*e*sgn(b*x + a) +
8190*(x*e + d)*a*b^5*d^4*e*sgn(b*x + a) - 1386*a*b^5*d^5*e*sgn(b*x + a) + 9009*(x*e + d)^4*a^2*b^4*e^2*sgn(b*
x + a) - 25740*(x*e + d)^3*a^2*b^4*d*e^2*sgn(b*x + a) + 30030*(x*e + d)^2*a^2*b^4*d^2*e^2*sgn(b*x + a) - 16380
*(x*e + d)*a^2*b^4*d^3*e^2*sgn(b*x + a) + 3465*a^2*b^4*d^4*e^2*sgn(b*x + a) + 8580*(x*e + d)^3*a^3*b^3*e^3*sgn
(b*x + a) - 20020*(x*e + d)^2*a^3*b^3*d*e^3*sgn(b*x + a) + 16380*(x*e + d)*a^3*b^3*d^2*e^3*sgn(b*x + a) - 4620
*a^3*b^3*d^3*e^3*sgn(b*x + a) + 5005*(x*e + d)^2*a^4*b^2*e^4*sgn(b*x + a) - 8190*(x*e + d)*a^4*b^2*d*e^4*sgn(b
*x + a) + 3465*a^4*b^2*d^2*e^4*sgn(b*x + a) + 1638*(x*e + d)*a^5*b*e^5*sgn(b*x + a) - 1386*a^5*b*d*e^5*sgn(b*x
+ a) + 231*a^6*e^6*sgn(b*x + a))*e^(-7)/(x*e + d)^(13/2)

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maple [A]  time = 0.05, size = 393, normalized size = 1.06 \[ -\frac {2 \left (3003 b^{6} e^{6} x^{6}+6006 a \,b^{5} e^{6} x^{5}+12012 b^{6} d \,e^{5} x^{5}+9009 a^{2} b^{4} e^{6} x^{4}+12012 a \,b^{5} d \,e^{5} x^{4}+24024 b^{6} d^{2} e^{4} x^{4}+8580 a^{3} b^{3} e^{6} x^{3}+10296 a^{2} b^{4} d \,e^{5} x^{3}+13728 a \,b^{5} d^{2} e^{4} x^{3}+27456 b^{6} d^{3} e^{3} x^{3}+5005 a^{4} b^{2} e^{6} x^{2}+5720 a^{3} b^{3} d \,e^{5} x^{2}+6864 a^{2} b^{4} d^{2} e^{4} x^{2}+9152 a \,b^{5} d^{3} e^{3} x^{2}+18304 b^{6} d^{4} e^{2} x^{2}+1638 a^{5} b \,e^{6} x +1820 a^{4} b^{2} d \,e^{5} x +2080 a^{3} b^{3} d^{2} e^{4} x +2496 a^{2} b^{4} d^{3} e^{3} x +3328 a \,b^{5} d^{4} e^{2} x +6656 b^{6} d^{5} e x +231 a^{6} e^{6}+252 a^{5} b d \,e^{5}+280 a^{4} b^{2} d^{2} e^{4}+320 a^{3} b^{3} d^{3} e^{3}+384 a^{2} b^{4} d^{4} e^{2}+512 a \,b^{5} d^{5} e +1024 b^{6} d^{6}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{3003 \left (e x +d \right )^{\frac {13}{2}} \left (b x +a \right )^{5} e^{7}} \]

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)^(15/2),x)

[Out]

-2/3003/(e*x+d)^(13/2)*(3003*b^6*e^6*x^6+6006*a*b^5*e^6*x^5+12012*b^6*d*e^5*x^5+9009*a^2*b^4*e^6*x^4+12012*a*b
^5*d*e^5*x^4+24024*b^6*d^2*e^4*x^4+8580*a^3*b^3*e^6*x^3+10296*a^2*b^4*d*e^5*x^3+13728*a*b^5*d^2*e^4*x^3+27456*
b^6*d^3*e^3*x^3+5005*a^4*b^2*e^6*x^2+5720*a^3*b^3*d*e^5*x^2+6864*a^2*b^4*d^2*e^4*x^2+9152*a*b^5*d^3*e^3*x^2+18
304*b^6*d^4*e^2*x^2+1638*a^5*b*e^6*x+1820*a^4*b^2*d*e^5*x+2080*a^3*b^3*d^2*e^4*x+2496*a^2*b^4*d^3*e^3*x+3328*a
*b^5*d^4*e^2*x+6656*b^6*d^5*e*x+231*a^6*e^6+252*a^5*b*d*e^5+280*a^4*b^2*d^2*e^4+320*a^3*b^3*d^3*e^3+384*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

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maxima [B]  time = 0.90, size = 735, normalized size = 1.99 \[ -\frac {2 \, {\left (3003 \, b^{5} e^{5} x^{5} + 256 \, b^{5} d^{5} + 384 \, a b^{4} d^{4} e + 480 \, a^{2} b^{3} d^{3} e^{2} + 560 \, a^{3} b^{2} d^{2} e^{3} + 630 \, a^{4} b d e^{4} + 693 \, a^{5} e^{5} + 3003 \, {\left (2 \, b^{5} d e^{4} + 3 \, a b^{4} e^{5}\right )} x^{4} + 858 \, {\left (8 \, b^{5} d^{2} e^{3} + 12 \, a b^{4} d e^{4} + 15 \, a^{2} b^{3} e^{5}\right )} x^{3} + 286 \, {\left (16 \, b^{5} d^{3} e^{2} + 24 \, a b^{4} d^{2} e^{3} + 30 \, a^{2} b^{3} d e^{4} + 35 \, a^{3} b^{2} e^{5}\right )} x^{2} + 13 \, {\left (128 \, b^{5} d^{4} e + 192 \, a b^{4} d^{3} e^{2} + 240 \, a^{2} b^{3} d^{2} e^{3} + 280 \, a^{3} b^{2} d e^{4} + 315 \, a^{4} b e^{5}\right )} x\right )} a}{9009 \, {\left (e^{12} x^{6} + 6 \, d e^{11} x^{5} + 15 \, d^{2} e^{10} x^{4} + 20 \, d^{3} e^{9} x^{3} + 15 \, d^{4} e^{8} x^{2} + 6 \, d^{5} e^{7} x + d^{6} e^{6}\right )} \sqrt {e x + d}} - \frac {2 \, {\left (9009 \, b^{5} e^{6} x^{6} + 3072 \, b^{5} d^{6} + 1280 \, a b^{4} d^{5} e + 768 \, a^{2} b^{3} d^{4} e^{2} + 480 \, a^{3} b^{2} d^{3} e^{3} + 280 \, a^{4} b d^{2} e^{4} + 126 \, a^{5} d e^{5} + 3003 \, {\left (12 \, b^{5} d e^{5} + 5 \, a b^{4} e^{6}\right )} x^{5} + 6006 \, {\left (12 \, b^{5} d^{2} e^{4} + 5 \, a b^{4} d e^{5} + 3 \, a^{2} b^{3} e^{6}\right )} x^{4} + 858 \, {\left (96 \, b^{5} d^{3} e^{3} + 40 \, a b^{4} d^{2} e^{4} + 24 \, a^{2} b^{3} d e^{5} + 15 \, a^{3} b^{2} e^{6}\right )} x^{3} + 143 \, {\left (384 \, b^{5} d^{4} e^{2} + 160 \, a b^{4} d^{3} e^{3} + 96 \, a^{2} b^{3} d^{2} e^{4} + 60 \, a^{3} b^{2} d e^{5} + 35 \, a^{4} b e^{6}\right )} x^{2} + 13 \, {\left (1536 \, b^{5} d^{5} e + 640 \, a b^{4} d^{4} e^{2} + 384 \, a^{2} b^{3} d^{3} e^{3} + 240 \, a^{3} b^{2} d^{2} e^{4} + 140 \, a^{4} b d e^{5} + 63 \, a^{5} e^{6}\right )} x\right )} b}{9009 \, {\left (e^{13} x^{6} + 6 \, d e^{12} x^{5} + 15 \, d^{2} e^{11} x^{4} + 20 \, d^{3} e^{10} x^{3} + 15 \, d^{4} e^{9} x^{2} + 6 \, d^{5} e^{8} x + d^{6} e^{7}\right )} \sqrt {e x + d}} \]

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)^(15/2),x, algorithm="maxima")

[Out]

-2/9009*(3003*b^5*e^5*x^5 + 256*b^5*d^5 + 384*a*b^4*d^4*e + 480*a^2*b^3*d^3*e^2 + 560*a^3*b^2*d^2*e^3 + 630*a^
4*b*d*e^4 + 693*a^5*e^5 + 3003*(2*b^5*d*e^4 + 3*a*b^4*e^5)*x^4 + 858*(8*b^5*d^2*e^3 + 12*a*b^4*d*e^4 + 15*a^2*
b^3*e^5)*x^3 + 286*(16*b^5*d^3*e^2 + 24*a*b^4*d^2*e^3 + 30*a^2*b^3*d*e^4 + 35*a^3*b^2*e^5)*x^2 + 13*(128*b^5*d
^4*e + 192*a*b^4*d^3*e^2 + 240*a^2*b^3*d^2*e^3 + 280*a^3*b^2*d*e^4 + 315*a^4*b*e^5)*x)*a/((e^12*x^6 + 6*d*e^11
*x^5 + 15*d^2*e^10*x^4 + 20*d^3*e^9*x^3 + 15*d^4*e^8*x^2 + 6*d^5*e^7*x + d^6*e^6)*sqrt(e*x + d)) - 2/9009*(900
9*b^5*e^6*x^6 + 3072*b^5*d^6 + 1280*a*b^4*d^5*e + 768*a^2*b^3*d^4*e^2 + 480*a^3*b^2*d^3*e^3 + 280*a^4*b*d^2*e^
4 + 126*a^5*d*e^5 + 3003*(12*b^5*d*e^5 + 5*a*b^4*e^6)*x^5 + 6006*(12*b^5*d^2*e^4 + 5*a*b^4*d*e^5 + 3*a^2*b^3*e
^6)*x^4 + 858*(96*b^5*d^3*e^3 + 40*a*b^4*d^2*e^4 + 24*a^2*b^3*d*e^5 + 15*a^3*b^2*e^6)*x^3 + 143*(384*b^5*d^4*e
^2 + 160*a*b^4*d^3*e^3 + 96*a^2*b^3*d^2*e^4 + 60*a^3*b^2*d*e^5 + 35*a^4*b*e^6)*x^2 + 13*(1536*b^5*d^5*e + 640*
a*b^4*d^4*e^2 + 384*a^2*b^3*d^3*e^3 + 240*a^3*b^2*d^2*e^4 + 140*a^4*b*d*e^5 + 63*a^5*e^6)*x)*b/((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)*sqrt(e*x + d))

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mupad [B]  time = 3.49, size = 561, normalized size = 1.52 \[ -\frac {\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}\,\left (\frac {462\,a^6\,e^6+504\,a^5\,b\,d\,e^5+560\,a^4\,b^2\,d^2\,e^4+640\,a^3\,b^3\,d^3\,e^3+768\,a^2\,b^4\,d^4\,e^2+1024\,a\,b^5\,d^5\,e+2048\,b^6\,d^6}{3003\,b\,e^{13}}+\frac {2\,b^5\,x^6}{e^7}+\frac {x\,\left (3276\,a^5\,b\,e^6+3640\,a^4\,b^2\,d\,e^5+4160\,a^3\,b^3\,d^2\,e^4+4992\,a^2\,b^4\,d^3\,e^3+6656\,a\,b^5\,d^4\,e^2+13312\,b^6\,d^5\,e\right )}{3003\,b\,e^{13}}+\frac {8\,b^2\,x^3\,\left (5\,a^3\,e^3+6\,a^2\,b\,d\,e^2+8\,a\,b^2\,d^2\,e+16\,b^3\,d^3\right )}{7\,e^{10}}+\frac {2\,b\,x^2\,\left (35\,a^4\,e^4+40\,a^3\,b\,d\,e^3+48\,a^2\,b^2\,d^2\,e^2+64\,a\,b^3\,d^3\,e+128\,b^4\,d^4\right )}{21\,e^{11}}+\frac {4\,b^4\,x^5\,\left (a\,e+2\,b\,d\right )}{e^8}+\frac {2\,b^3\,x^4\,\left (3\,a^2\,e^2+4\,a\,b\,d\,e+8\,b^2\,d^2\right )}{e^9}\right )}{x^7\,\sqrt {d+e\,x}+\frac {a\,d^6\,\sqrt {d+e\,x}}{b\,e^6}+\frac {x^6\,\left (a\,e+6\,b\,d\right )\,\sqrt {d+e\,x}}{b\,e}+\frac {3\,d\,x^5\,\left (2\,a\,e+5\,b\,d\right )\,\sqrt {d+e\,x}}{b\,e^2}+\frac {d^5\,x\,\left (6\,a\,e+b\,d\right )\,\sqrt {d+e\,x}}{b\,e^6}+\frac {5\,d^2\,x^4\,\left (3\,a\,e+4\,b\,d\right )\,\sqrt {d+e\,x}}{b\,e^3}+\frac {5\,d^3\,x^3\,\left (4\,a\,e+3\,b\,d\right )\,\sqrt {d+e\,x}}{b\,e^4}+\frac {3\,d^4\,x^2\,\left (5\,a\,e+2\,b\,d\right )\,\sqrt {d+e\,x}}{b\,e^5}} \]

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)^(15/2),x)

[Out]

-((a^2 + b^2*x^2 + 2*a*b*x)^(1/2)*((462*a^6*e^6 + 2048*b^6*d^6 + 768*a^2*b^4*d^4*e^2 + 640*a^3*b^3*d^3*e^3 + 5
60*a^4*b^2*d^2*e^4 + 1024*a*b^5*d^5*e + 504*a^5*b*d*e^5)/(3003*b*e^13) + (2*b^5*x^6)/e^7 + (x*(3276*a^5*b*e^6
+ 13312*b^6*d^5*e + 6656*a*b^5*d^4*e^2 + 3640*a^4*b^2*d*e^5 + 4992*a^2*b^4*d^3*e^3 + 4160*a^3*b^3*d^2*e^4))/(3
003*b*e^13) + (8*b^2*x^3*(5*a^3*e^3 + 16*b^3*d^3 + 8*a*b^2*d^2*e + 6*a^2*b*d*e^2))/(7*e^10) + (2*b*x^2*(35*a^4
*e^4 + 128*b^4*d^4 + 48*a^2*b^2*d^2*e^2 + 64*a*b^3*d^3*e + 40*a^3*b*d*e^3))/(21*e^11) + (4*b^4*x^5*(a*e + 2*b*
d))/e^8 + (2*b^3*x^4*(3*a^2*e^2 + 8*b^2*d^2 + 4*a*b*d*e))/e^9))/(x^7*(d + e*x)^(1/2) + (a*d^6*(d + e*x)^(1/2))
/(b*e^6) + (x^6*(a*e + 6*b*d)*(d + e*x)^(1/2))/(b*e) + (3*d*x^5*(2*a*e + 5*b*d)*(d + e*x)^(1/2))/(b*e^2) + (d^
5*x*(6*a*e + b*d)*(d + e*x)^(1/2))/(b*e^6) + (5*d^2*x^4*(3*a*e + 4*b*d)*(d + e*x)^(1/2))/(b*e^3) + (5*d^3*x^3*
(4*a*e + 3*b*d)*(d + e*x)^(1/2))/(b*e^4) + (3*d^4*x^2*(5*a*e + 2*b*d)*(d + e*x)^(1/2))/(b*e^5))

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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)**(15/2),x)

[Out]

Timed out

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