3.19.96 \(\int \frac {(a+b x) (a^2+2 a b x+b^2 x^2)^{5/2}}{(d+e x)^{17/2}} \, dx\)

Optimal. Leaf size=376 \[ -\frac {30 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4}{11 e^7 (a+b x) (d+e x)^{11/2}}+\frac {12 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5}{13 e^7 (a+b x) (d+e x)^{13/2}}-\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^6}{15 e^7 (a+b x) (d+e x)^{15/2}}-\frac {2 b^6 \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^7 (a+b x) (d+e x)^{3/2}}+\frac {12 b^5 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}{5 e^7 (a+b x) (d+e x)^{5/2}}-\frac {30 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}{7 e^7 (a+b x) (d+e x)^{7/2}}+\frac {40 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3}{9 e^7 (a+b x) (d+e x)^{9/2}} \]

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Rubi [A]  time = 0.14, antiderivative size = 376, 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}}{3 e^7 (a+b x) (d+e x)^{3/2}}+\frac {12 b^5 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}{5 e^7 (a+b x) (d+e x)^{5/2}}-\frac {30 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}{7 e^7 (a+b x) (d+e x)^{7/2}}+\frac {40 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3}{9 e^7 (a+b x) (d+e x)^{9/2}}-\frac {30 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4}{11 e^7 (a+b x) (d+e x)^{11/2}}+\frac {12 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5}{13 e^7 (a+b x) (d+e x)^{13/2}}-\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^6}{15 e^7 (a+b x) (d+e x)^{15/2}} \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)^(17/2),x]

[Out]

(-2*(b*d - a*e)^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(15*e^7*(a + b*x)*(d + e*x)^(15/2)) + (12*b*(b*d - a*e)^5*Sqr
t[a^2 + 2*a*b*x + b^2*x^2])/(13*e^7*(a + b*x)*(d + e*x)^(13/2)) - (30*b^2*(b*d - a*e)^4*Sqrt[a^2 + 2*a*b*x + b
^2*x^2])/(11*e^7*(a + b*x)*(d + e*x)^(11/2)) + (40*b^3*(b*d - a*e)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*e^7*(a
+ b*x)*(d + e*x)^(9/2)) - (30*b^4*(b*d - a*e)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*e^7*(a + b*x)*(d + e*x)^(7/2
)) + (12*b^5*(b*d - a*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*e^7*(a + b*x)*(d + e*x)^(5/2)) - (2*b^6*Sqrt[a^2 +
2*a*b*x + b^2*x^2])/(3*e^7*(a + b*x)*(d + e*x)^(3/2))

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)^{17/2}} \, 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)^{17/2}} \, 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)^{17/2}} \, 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)^{17/2}}-\frac {6 b (b d-a e)^5}{e^6 (d+e x)^{15/2}}+\frac {15 b^2 (b d-a e)^4}{e^6 (d+e x)^{13/2}}-\frac {20 b^3 (b d-a e)^3}{e^6 (d+e x)^{11/2}}+\frac {15 b^4 (b d-a e)^2}{e^6 (d+e x)^{9/2}}-\frac {6 b^5 (b d-a e)}{e^6 (d+e x)^{7/2}}+\frac {b^6}{e^6 (d+e x)^{5/2}}\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}}{15 e^7 (a+b x) (d+e x)^{15/2}}+\frac {12 b (b d-a e)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{13 e^7 (a+b x) (d+e x)^{13/2}}-\frac {30 b^2 (b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{11 e^7 (a+b x) (d+e x)^{11/2}}+\frac {40 b^3 (b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{9 e^7 (a+b x) (d+e x)^{9/2}}-\frac {30 b^4 (b d-a e)^2 \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^5 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^7 (a+b x) (d+e x)^{5/2}}-\frac {2 b^6 \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^7 (a+b x) (d+e x)^{3/2}}\\ \end {align*}

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Mathematica [A]  time = 0.16, size = 163, normalized size = 0.43 \begin {gather*} \frac {2 \sqrt {(a+b x)^2} \left (54054 b^5 (d+e x)^5 (b d-a e)-96525 b^4 (d+e x)^4 (b d-a e)^2+100100 b^3 (d+e x)^3 (b d-a e)^3-61425 b^2 (d+e x)^2 (b d-a e)^4+20790 b (d+e x) (b d-a e)^5-3003 (b d-a e)^6-15015 b^6 (d+e x)^6\right )}{45045 e^7 (a+b x) (d+e x)^{15/2}} \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)^(17/2),x]

[Out]

(2*Sqrt[(a + b*x)^2]*(-3003*(b*d - a*e)^6 + 20790*b*(b*d - a*e)^5*(d + e*x) - 61425*b^2*(b*d - a*e)^4*(d + e*x
)^2 + 100100*b^3*(b*d - a*e)^3*(d + e*x)^3 - 96525*b^4*(b*d - a*e)^2*(d + e*x)^4 + 54054*b^5*(b*d - a*e)*(d +
e*x)^5 - 15015*b^6*(d + e*x)^6))/(45045*e^7*(a + b*x)*(d + e*x)^(15/2))

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IntegrateAlgebraic [A]  time = 0.32, size = 398, normalized size = 1.06 \begin {gather*} -\frac {2 \sqrt {(a+b x)^2} \left (3003 a^6 e^6+2772 a^5 b d e^5+20790 a^5 b e^6 x+2520 a^4 b^2 d^2 e^4+18900 a^4 b^2 d e^5 x+61425 a^4 b^2 e^6 x^2+2240 a^3 b^3 d^3 e^3+16800 a^3 b^3 d^2 e^4 x+54600 a^3 b^3 d e^5 x^2+100100 a^3 b^3 e^6 x^3+1920 a^2 b^4 d^4 e^2+14400 a^2 b^4 d^3 e^3 x+46800 a^2 b^4 d^2 e^4 x^2+85800 a^2 b^4 d e^5 x^3+96525 a^2 b^4 e^6 x^4+1536 a b^5 d^5 e+11520 a b^5 d^4 e^2 x+37440 a b^5 d^3 e^3 x^2+68640 a b^5 d^2 e^4 x^3+77220 a b^5 d e^5 x^4+54054 a b^5 e^6 x^5+1024 b^6 d^6+7680 b^6 d^5 e x+24960 b^6 d^4 e^2 x^2+45760 b^6 d^3 e^3 x^3+51480 b^6 d^2 e^4 x^4+36036 b^6 d e^5 x^5+15015 b^6 e^6 x^6\right )}{45045 e^7 (a+b x) (d+e x)^{15/2}} \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)^(17/2),x]

[Out]

(-2*Sqrt[(a + b*x)^2]*(1024*b^6*d^6 + 1536*a*b^5*d^5*e + 1920*a^2*b^4*d^4*e^2 + 2240*a^3*b^3*d^3*e^3 + 2520*a^
4*b^2*d^2*e^4 + 2772*a^5*b*d*e^5 + 3003*a^6*e^6 + 7680*b^6*d^5*e*x + 11520*a*b^5*d^4*e^2*x + 14400*a^2*b^4*d^3
*e^3*x + 16800*a^3*b^3*d^2*e^4*x + 18900*a^4*b^2*d*e^5*x + 20790*a^5*b*e^6*x + 24960*b^6*d^4*e^2*x^2 + 37440*a
*b^5*d^3*e^3*x^2 + 46800*a^2*b^4*d^2*e^4*x^2 + 54600*a^3*b^3*d*e^5*x^2 + 61425*a^4*b^2*e^6*x^2 + 45760*b^6*d^3
*e^3*x^3 + 68640*a*b^5*d^2*e^4*x^3 + 85800*a^2*b^4*d*e^5*x^3 + 100100*a^3*b^3*e^6*x^3 + 51480*b^6*d^2*e^4*x^4
+ 77220*a*b^5*d*e^5*x^4 + 96525*a^2*b^4*e^6*x^4 + 36036*b^6*d*e^5*x^5 + 54054*a*b^5*e^6*x^5 + 15015*b^6*e^6*x^
6))/(45045*e^7*(a + b*x)*(d + e*x)^(15/2))

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fricas [A]  time = 0.43, size = 443, normalized size = 1.18 \begin {gather*} -\frac {2 \, {\left (15015 \, b^{6} e^{6} x^{6} + 1024 \, b^{6} d^{6} + 1536 \, a b^{5} d^{5} e + 1920 \, a^{2} b^{4} d^{4} e^{2} + 2240 \, a^{3} b^{3} d^{3} e^{3} + 2520 \, a^{4} b^{2} d^{2} e^{4} + 2772 \, a^{5} b d e^{5} + 3003 \, a^{6} e^{6} + 18018 \, {\left (2 \, b^{6} d e^{5} + 3 \, a b^{5} e^{6}\right )} x^{5} + 6435 \, {\left (8 \, b^{6} d^{2} e^{4} + 12 \, a b^{5} d e^{5} + 15 \, a^{2} b^{4} e^{6}\right )} x^{4} + 2860 \, {\left (16 \, b^{6} d^{3} e^{3} + 24 \, a b^{5} d^{2} e^{4} + 30 \, a^{2} b^{4} d e^{5} + 35 \, a^{3} b^{3} e^{6}\right )} x^{3} + 195 \, {\left (128 \, b^{6} d^{4} e^{2} + 192 \, a b^{5} d^{3} e^{3} + 240 \, a^{2} b^{4} d^{2} e^{4} + 280 \, a^{3} b^{3} d e^{5} + 315 \, a^{4} b^{2} e^{6}\right )} x^{2} + 30 \, {\left (256 \, b^{6} d^{5} e + 384 \, a b^{5} d^{4} e^{2} + 480 \, a^{2} b^{4} d^{3} e^{3} + 560 \, a^{3} b^{3} d^{2} e^{4} + 630 \, a^{4} b^{2} d e^{5} + 693 \, a^{5} b e^{6}\right )} x\right )} \sqrt {e x + d}}{45045 \, {\left (e^{15} x^{8} + 8 \, d e^{14} x^{7} + 28 \, d^{2} e^{13} x^{6} + 56 \, d^{3} e^{12} x^{5} + 70 \, d^{4} e^{11} x^{4} + 56 \, d^{5} e^{10} x^{3} + 28 \, d^{6} e^{9} x^{2} + 8 \, d^{7} e^{8} x + d^{8} 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)^(17/2),x, algorithm="fricas")

[Out]

-2/45045*(15015*b^6*e^6*x^6 + 1024*b^6*d^6 + 1536*a*b^5*d^5*e + 1920*a^2*b^4*d^4*e^2 + 2240*a^3*b^3*d^3*e^3 +
2520*a^4*b^2*d^2*e^4 + 2772*a^5*b*d*e^5 + 3003*a^6*e^6 + 18018*(2*b^6*d*e^5 + 3*a*b^5*e^6)*x^5 + 6435*(8*b^6*d
^2*e^4 + 12*a*b^5*d*e^5 + 15*a^2*b^4*e^6)*x^4 + 2860*(16*b^6*d^3*e^3 + 24*a*b^5*d^2*e^4 + 30*a^2*b^4*d*e^5 + 3
5*a^3*b^3*e^6)*x^3 + 195*(128*b^6*d^4*e^2 + 192*a*b^5*d^3*e^3 + 240*a^2*b^4*d^2*e^4 + 280*a^3*b^3*d*e^5 + 315*
a^4*b^2*e^6)*x^2 + 30*(256*b^6*d^5*e + 384*a*b^5*d^4*e^2 + 480*a^2*b^4*d^3*e^3 + 560*a^3*b^3*d^2*e^4 + 630*a^4
*b^2*d*e^5 + 693*a^5*b*e^6)*x)*sqrt(e*x + d)/(e^15*x^8 + 8*d*e^14*x^7 + 28*d^2*e^13*x^6 + 56*d^3*e^12*x^5 + 70
*d^4*e^11*x^4 + 56*d^5*e^10*x^3 + 28*d^6*e^9*x^2 + 8*d^7*e^8*x + d^8*e^7)

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giac [B]  time = 0.29, size = 614, normalized size = 1.63 \begin {gather*} -\frac {2 \, {\left (15015 \, {\left (x e + d\right )}^{6} b^{6} \mathrm {sgn}\left (b x + a\right ) - 54054 \, {\left (x e + d\right )}^{5} b^{6} d \mathrm {sgn}\left (b x + a\right ) + 96525 \, {\left (x e + d\right )}^{4} b^{6} d^{2} \mathrm {sgn}\left (b x + a\right ) - 100100 \, {\left (x e + d\right )}^{3} b^{6} d^{3} \mathrm {sgn}\left (b x + a\right ) + 61425 \, {\left (x e + d\right )}^{2} b^{6} d^{4} \mathrm {sgn}\left (b x + a\right ) - 20790 \, {\left (x e + d\right )} b^{6} d^{5} \mathrm {sgn}\left (b x + a\right ) + 3003 \, b^{6} d^{6} \mathrm {sgn}\left (b x + a\right ) + 54054 \, {\left (x e + d\right )}^{5} a b^{5} e \mathrm {sgn}\left (b x + a\right ) - 193050 \, {\left (x e + d\right )}^{4} a b^{5} d e \mathrm {sgn}\left (b x + a\right ) + 300300 \, {\left (x e + d\right )}^{3} a b^{5} d^{2} e \mathrm {sgn}\left (b x + a\right ) - 245700 \, {\left (x e + d\right )}^{2} a b^{5} d^{3} e \mathrm {sgn}\left (b x + a\right ) + 103950 \, {\left (x e + d\right )} a b^{5} d^{4} e \mathrm {sgn}\left (b x + a\right ) - 18018 \, a b^{5} d^{5} e \mathrm {sgn}\left (b x + a\right ) + 96525 \, {\left (x e + d\right )}^{4} a^{2} b^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) - 300300 \, {\left (x e + d\right )}^{3} a^{2} b^{4} d e^{2} \mathrm {sgn}\left (b x + a\right ) + 368550 \, {\left (x e + d\right )}^{2} a^{2} b^{4} d^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) - 207900 \, {\left (x e + d\right )} a^{2} b^{4} d^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) + 45045 \, a^{2} b^{4} d^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) + 100100 \, {\left (x e + d\right )}^{3} a^{3} b^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) - 245700 \, {\left (x e + d\right )}^{2} a^{3} b^{3} d e^{3} \mathrm {sgn}\left (b x + a\right ) + 207900 \, {\left (x e + d\right )} a^{3} b^{3} d^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) - 60060 \, a^{3} b^{3} d^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) + 61425 \, {\left (x e + d\right )}^{2} a^{4} b^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) - 103950 \, {\left (x e + d\right )} a^{4} b^{2} d e^{4} \mathrm {sgn}\left (b x + a\right ) + 45045 \, a^{4} b^{2} d^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) + 20790 \, {\left (x e + d\right )} a^{5} b e^{5} \mathrm {sgn}\left (b x + a\right ) - 18018 \, a^{5} b d e^{5} \mathrm {sgn}\left (b x + a\right ) + 3003 \, a^{6} e^{6} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-7\right )}}{45045 \, {\left (x e + d\right )}^{\frac {15}{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)^(17/2),x, algorithm="giac")

[Out]

-2/45045*(15015*(x*e + d)^6*b^6*sgn(b*x + a) - 54054*(x*e + d)^5*b^6*d*sgn(b*x + a) + 96525*(x*e + d)^4*b^6*d^
2*sgn(b*x + a) - 100100*(x*e + d)^3*b^6*d^3*sgn(b*x + a) + 61425*(x*e + d)^2*b^6*d^4*sgn(b*x + a) - 20790*(x*e
+ d)*b^6*d^5*sgn(b*x + a) + 3003*b^6*d^6*sgn(b*x + a) + 54054*(x*e + d)^5*a*b^5*e*sgn(b*x + a) - 193050*(x*e
+ d)^4*a*b^5*d*e*sgn(b*x + a) + 300300*(x*e + d)^3*a*b^5*d^2*e*sgn(b*x + a) - 245700*(x*e + d)^2*a*b^5*d^3*e*s
gn(b*x + a) + 103950*(x*e + d)*a*b^5*d^4*e*sgn(b*x + a) - 18018*a*b^5*d^5*e*sgn(b*x + a) + 96525*(x*e + d)^4*a
^2*b^4*e^2*sgn(b*x + a) - 300300*(x*e + d)^3*a^2*b^4*d*e^2*sgn(b*x + a) + 368550*(x*e + d)^2*a^2*b^4*d^2*e^2*s
gn(b*x + a) - 207900*(x*e + d)*a^2*b^4*d^3*e^2*sgn(b*x + a) + 45045*a^2*b^4*d^4*e^2*sgn(b*x + a) + 100100*(x*e
+ d)^3*a^3*b^3*e^3*sgn(b*x + a) - 245700*(x*e + d)^2*a^3*b^3*d*e^3*sgn(b*x + a) + 207900*(x*e + d)*a^3*b^3*d^
2*e^3*sgn(b*x + a) - 60060*a^3*b^3*d^3*e^3*sgn(b*x + a) + 61425*(x*e + d)^2*a^4*b^2*e^4*sgn(b*x + a) - 103950*
(x*e + d)*a^4*b^2*d*e^4*sgn(b*x + a) + 45045*a^4*b^2*d^2*e^4*sgn(b*x + a) + 20790*(x*e + d)*a^5*b*e^5*sgn(b*x
+ a) - 18018*a^5*b*d*e^5*sgn(b*x + a) + 3003*a^6*e^6*sgn(b*x + a))*e^(-7)/(x*e + d)^(15/2)

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maple [A]  time = 0.05, size = 393, normalized size = 1.05 \begin {gather*} -\frac {2 \left (15015 b^{6} e^{6} x^{6}+54054 a \,b^{5} e^{6} x^{5}+36036 b^{6} d \,e^{5} x^{5}+96525 a^{2} b^{4} e^{6} x^{4}+77220 a \,b^{5} d \,e^{5} x^{4}+51480 b^{6} d^{2} e^{4} x^{4}+100100 a^{3} b^{3} e^{6} x^{3}+85800 a^{2} b^{4} d \,e^{5} x^{3}+68640 a \,b^{5} d^{2} e^{4} x^{3}+45760 b^{6} d^{3} e^{3} x^{3}+61425 a^{4} b^{2} e^{6} x^{2}+54600 a^{3} b^{3} d \,e^{5} x^{2}+46800 a^{2} b^{4} d^{2} e^{4} x^{2}+37440 a \,b^{5} d^{3} e^{3} x^{2}+24960 b^{6} d^{4} e^{2} x^{2}+20790 a^{5} b \,e^{6} x +18900 a^{4} b^{2} d \,e^{5} x +16800 a^{3} b^{3} d^{2} e^{4} x +14400 a^{2} b^{4} d^{3} e^{3} x +11520 a \,b^{5} d^{4} e^{2} x +7680 b^{6} d^{5} e x +3003 a^{6} e^{6}+2772 a^{5} b d \,e^{5}+2520 a^{4} b^{2} d^{2} e^{4}+2240 a^{3} b^{3} d^{3} e^{3}+1920 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}}}{45045 \left (e x +d \right )^{\frac {15}{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)^(17/2),x)

[Out]

-2/45045/(e*x+d)^(15/2)*(15015*b^6*e^6*x^6+54054*a*b^5*e^6*x^5+36036*b^6*d*e^5*x^5+96525*a^2*b^4*e^6*x^4+77220
*a*b^5*d*e^5*x^4+51480*b^6*d^2*e^4*x^4+100100*a^3*b^3*e^6*x^3+85800*a^2*b^4*d*e^5*x^3+68640*a*b^5*d^2*e^4*x^3+
45760*b^6*d^3*e^3*x^3+61425*a^4*b^2*e^6*x^2+54600*a^3*b^3*d*e^5*x^2+46800*a^2*b^4*d^2*e^4*x^2+37440*a*b^5*d^3*
e^3*x^2+24960*b^6*d^4*e^2*x^2+20790*a^5*b*e^6*x+18900*a^4*b^2*d*e^5*x+16800*a^3*b^3*d^2*e^4*x+14400*a^2*b^4*d^
3*e^3*x+11520*a*b^5*d^4*e^2*x+7680*b^6*d^5*e*x+3003*a^6*e^6+2772*a^5*b*d*e^5+2520*a^4*b^2*d^2*e^4+2240*a^3*b^3
*d^3*e^3+1920*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 = 0.97, size = 757, normalized size = 2.01 \begin {gather*} -\frac {2 \, {\left (9009 \, b^{5} e^{5} x^{5} + 256 \, b^{5} d^{5} + 640 \, a b^{4} d^{4} e + 1120 \, a^{2} b^{3} d^{3} e^{2} + 1680 \, a^{3} b^{2} d^{2} e^{3} + 2310 \, a^{4} b d e^{4} + 3003 \, a^{5} e^{5} + 6435 \, {\left (2 \, b^{5} d e^{4} + 5 \, a b^{4} e^{5}\right )} x^{4} + 1430 \, {\left (8 \, b^{5} d^{2} e^{3} + 20 \, a b^{4} d e^{4} + 35 \, a^{2} b^{3} e^{5}\right )} x^{3} + 390 \, {\left (16 \, b^{5} d^{3} e^{2} + 40 \, a b^{4} d^{2} e^{3} + 70 \, a^{2} b^{3} d e^{4} + 105 \, a^{3} b^{2} e^{5}\right )} x^{2} + 15 \, {\left (128 \, b^{5} d^{4} e + 320 \, a b^{4} d^{3} e^{2} + 560 \, a^{2} b^{3} d^{2} e^{3} + 840 \, a^{3} b^{2} d e^{4} + 1155 \, a^{4} b e^{5}\right )} x\right )} a}{45045 \, {\left (e^{13} x^{7} + 7 \, d e^{12} x^{6} + 21 \, d^{2} e^{11} x^{5} + 35 \, d^{3} e^{10} x^{4} + 35 \, d^{4} e^{9} x^{3} + 21 \, d^{5} e^{8} x^{2} + 7 \, d^{6} e^{7} x + d^{7} e^{6}\right )} \sqrt {e x + d}} - \frac {2 \, {\left (15015 \, b^{5} e^{6} x^{6} + 1024 \, b^{5} d^{6} + 1280 \, a b^{4} d^{5} e + 1280 \, a^{2} b^{3} d^{4} e^{2} + 1120 \, a^{3} b^{2} d^{3} e^{3} + 840 \, a^{4} b d^{2} e^{4} + 462 \, a^{5} d e^{5} + 9009 \, {\left (4 \, b^{5} d e^{5} + 5 \, a b^{4} e^{6}\right )} x^{5} + 12870 \, {\left (4 \, b^{5} d^{2} e^{4} + 5 \, a b^{4} d e^{5} + 5 \, a^{2} b^{3} e^{6}\right )} x^{4} + 1430 \, {\left (32 \, b^{5} d^{3} e^{3} + 40 \, a b^{4} d^{2} e^{4} + 40 \, a^{2} b^{3} d e^{5} + 35 \, a^{3} b^{2} e^{6}\right )} x^{3} + 195 \, {\left (128 \, b^{5} d^{4} e^{2} + 160 \, a b^{4} d^{3} e^{3} + 160 \, a^{2} b^{3} d^{2} e^{4} + 140 \, a^{3} b^{2} d e^{5} + 105 \, a^{4} b e^{6}\right )} x^{2} + 15 \, {\left (512 \, b^{5} d^{5} e + 640 \, a b^{4} d^{4} e^{2} + 640 \, a^{2} b^{3} d^{3} e^{3} + 560 \, a^{3} b^{2} d^{2} e^{4} + 420 \, a^{4} b d e^{5} + 231 \, a^{5} e^{6}\right )} x\right )} b}{45045 \, {\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 )} \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)^(17/2),x, algorithm="maxima")

[Out]

-2/45045*(9009*b^5*e^5*x^5 + 256*b^5*d^5 + 640*a*b^4*d^4*e + 1120*a^2*b^3*d^3*e^2 + 1680*a^3*b^2*d^2*e^3 + 231
0*a^4*b*d*e^4 + 3003*a^5*e^5 + 6435*(2*b^5*d*e^4 + 5*a*b^4*e^5)*x^4 + 1430*(8*b^5*d^2*e^3 + 20*a*b^4*d*e^4 + 3
5*a^2*b^3*e^5)*x^3 + 390*(16*b^5*d^3*e^2 + 40*a*b^4*d^2*e^3 + 70*a^2*b^3*d*e^4 + 105*a^3*b^2*e^5)*x^2 + 15*(12
8*b^5*d^4*e + 320*a*b^4*d^3*e^2 + 560*a^2*b^3*d^2*e^3 + 840*a^3*b^2*d*e^4 + 1155*a^4*b*e^5)*x)*a/((e^13*x^7 +
7*d*e^12*x^6 + 21*d^2*e^11*x^5 + 35*d^3*e^10*x^4 + 35*d^4*e^9*x^3 + 21*d^5*e^8*x^2 + 7*d^6*e^7*x + d^7*e^6)*sq
rt(e*x + d)) - 2/45045*(15015*b^5*e^6*x^6 + 1024*b^5*d^6 + 1280*a*b^4*d^5*e + 1280*a^2*b^3*d^4*e^2 + 1120*a^3*
b^2*d^3*e^3 + 840*a^4*b*d^2*e^4 + 462*a^5*d*e^5 + 9009*(4*b^5*d*e^5 + 5*a*b^4*e^6)*x^5 + 12870*(4*b^5*d^2*e^4
+ 5*a*b^4*d*e^5 + 5*a^2*b^3*e^6)*x^4 + 1430*(32*b^5*d^3*e^3 + 40*a*b^4*d^2*e^4 + 40*a^2*b^3*d*e^5 + 35*a^3*b^2
*e^6)*x^3 + 195*(128*b^5*d^4*e^2 + 160*a*b^4*d^3*e^3 + 160*a^2*b^3*d^2*e^4 + 140*a^3*b^2*d*e^5 + 105*a^4*b*e^6
)*x^2 + 15*(512*b^5*d^5*e + 640*a*b^4*d^4*e^2 + 640*a^2*b^3*d^3*e^3 + 560*a^3*b^2*d^2*e^4 + 420*a^4*b*d*e^5 +
231*a^5*e^6)*x)*b/((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)*sqrt(e*x + d))

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mupad [B]  time = 3.53, size = 588, normalized size = 1.56 \begin {gather*} -\frac {\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}\,\left (\frac {6006\,a^6\,e^6+5544\,a^5\,b\,d\,e^5+5040\,a^4\,b^2\,d^2\,e^4+4480\,a^3\,b^3\,d^3\,e^3+3840\,a^2\,b^4\,d^4\,e^2+3072\,a\,b^5\,d^5\,e+2048\,b^6\,d^6}{45045\,b\,e^{14}}+\frac {2\,b^5\,x^6}{3\,e^8}+\frac {x\,\left (41580\,a^5\,b\,e^6+37800\,a^4\,b^2\,d\,e^5+33600\,a^3\,b^3\,d^2\,e^4+28800\,a^2\,b^4\,d^3\,e^3+23040\,a\,b^5\,d^4\,e^2+15360\,b^6\,d^5\,e\right )}{45045\,b\,e^{14}}+\frac {8\,b^2\,x^3\,\left (35\,a^3\,e^3+30\,a^2\,b\,d\,e^2+24\,a\,b^2\,d^2\,e+16\,b^3\,d^3\right )}{63\,e^{11}}+\frac {2\,b\,x^2\,\left (315\,a^4\,e^4+280\,a^3\,b\,d\,e^3+240\,a^2\,b^2\,d^2\,e^2+192\,a\,b^3\,d^3\,e+128\,b^4\,d^4\right )}{231\,e^{12}}+\frac {4\,b^4\,x^5\,\left (3\,a\,e+2\,b\,d\right )}{5\,e^9}+\frac {2\,b^3\,x^4\,\left (15\,a^2\,e^2+12\,a\,b\,d\,e+8\,b^2\,d^2\right )}{7\,e^{10}}\right )}{x^8\,\sqrt {d+e\,x}+\frac {a\,d^7\,\sqrt {d+e\,x}}{b\,e^7}+\frac {x^7\,\left (a\,e+7\,b\,d\right )\,\sqrt {d+e\,x}}{b\,e}+\frac {7\,d\,x^6\,\left (a\,e+3\,b\,d\right )\,\sqrt {d+e\,x}}{b\,e^2}+\frac {d^6\,x\,\left (7\,a\,e+b\,d\right )\,\sqrt {d+e\,x}}{b\,e^7}+\frac {35\,d^3\,x^4\,\left (a\,e+b\,d\right )\,\sqrt {d+e\,x}}{b\,e^4}+\frac {7\,d^5\,x^2\,\left (3\,a\,e+b\,d\right )\,\sqrt {d+e\,x}}{b\,e^6}+\frac {7\,d^2\,x^5\,\left (3\,a\,e+5\,b\,d\right )\,\sqrt {d+e\,x}}{b\,e^3}+\frac {7\,d^4\,x^3\,\left (5\,a\,e+3\,b\,d\right )\,\sqrt {d+e\,x}}{b\,e^5}} \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)^(17/2),x)

[Out]

-((a^2 + b^2*x^2 + 2*a*b*x)^(1/2)*((6006*a^6*e^6 + 2048*b^6*d^6 + 3840*a^2*b^4*d^4*e^2 + 4480*a^3*b^3*d^3*e^3
+ 5040*a^4*b^2*d^2*e^4 + 3072*a*b^5*d^5*e + 5544*a^5*b*d*e^5)/(45045*b*e^14) + (2*b^5*x^6)/(3*e^8) + (x*(41580
*a^5*b*e^6 + 15360*b^6*d^5*e + 23040*a*b^5*d^4*e^2 + 37800*a^4*b^2*d*e^5 + 28800*a^2*b^4*d^3*e^3 + 33600*a^3*b
^3*d^2*e^4))/(45045*b*e^14) + (8*b^2*x^3*(35*a^3*e^3 + 16*b^3*d^3 + 24*a*b^2*d^2*e + 30*a^2*b*d*e^2))/(63*e^11
) + (2*b*x^2*(315*a^4*e^4 + 128*b^4*d^4 + 240*a^2*b^2*d^2*e^2 + 192*a*b^3*d^3*e + 280*a^3*b*d*e^3))/(231*e^12)
+ (4*b^4*x^5*(3*a*e + 2*b*d))/(5*e^9) + (2*b^3*x^4*(15*a^2*e^2 + 8*b^2*d^2 + 12*a*b*d*e))/(7*e^10)))/(x^8*(d
+ e*x)^(1/2) + (a*d^7*(d + e*x)^(1/2))/(b*e^7) + (x^7*(a*e + 7*b*d)*(d + e*x)^(1/2))/(b*e) + (7*d*x^6*(a*e + 3
*b*d)*(d + e*x)^(1/2))/(b*e^2) + (d^6*x*(7*a*e + b*d)*(d + e*x)^(1/2))/(b*e^7) + (35*d^3*x^4*(a*e + b*d)*(d +
e*x)^(1/2))/(b*e^4) + (7*d^5*x^2*(3*a*e + b*d)*(d + e*x)^(1/2))/(b*e^6) + (7*d^2*x^5*(3*a*e + 5*b*d)*(d + e*x)
^(1/2))/(b*e^3) + (7*d^4*x^3*(5*a*e + 3*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 \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)**(17/2),x)

[Out]

Timed out

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