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

Optimal. Leaf size=362 \[ -\frac {15 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4}{14 e^7 (a+b x) (d+e x)^{14}}+\frac {2 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5}{5 e^7 (a+b x) (d+e x)^{15}}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^6}{16 e^7 (a+b x) (d+e x)^{16}}-\frac {b^6 \sqrt {a^2+2 a b x+b^2 x^2}}{10 e^7 (a+b x) (d+e x)^{10}}+\frac {6 b^5 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}{11 e^7 (a+b x) (d+e x)^{11}}-\frac {5 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}{4 e^7 (a+b x) (d+e x)^{12}}+\frac {20 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3}{13 e^7 (a+b x) (d+e x)^{13}} \]

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Rubi [A]  time = 0.20, antiderivative size = 362, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {770, 21, 43} \begin {gather*} -\frac {b^6 \sqrt {a^2+2 a b x+b^2 x^2}}{10 e^7 (a+b x) (d+e x)^{10}}+\frac {6 b^5 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}{11 e^7 (a+b x) (d+e x)^{11}}-\frac {5 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}{4 e^7 (a+b x) (d+e x)^{12}}+\frac {20 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3}{13 e^7 (a+b x) (d+e x)^{13}}-\frac {15 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4}{14 e^7 (a+b x) (d+e x)^{14}}+\frac {2 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)^{15}}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^6}{16 e^7 (a+b x) (d+e x)^{16}} \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,x]

[Out]

-((b*d - a*e)^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(16*e^7*(a + b*x)*(d + e*x)^16) + (2*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)^15) - (15*b^2*(b*d - a*e)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(14
*e^7*(a + b*x)*(d + e*x)^14) + (20*b^3*(b*d - a*e)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(13*e^7*(a + b*x)*(d + e*x
)^13) - (5*b^4*(b*d - a*e)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(4*e^7*(a + b*x)*(d + e*x)^12) + (6*b^5*(b*d - a*e
)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*e^7*(a + b*x)*(d + e*x)^11) - (b^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(10*e^7
*(a + b*x)*(d + e*x)^10)

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}} \, 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}} \, 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}} \, 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}}-\frac {6 b (b d-a e)^5}{e^6 (d+e x)^{16}}+\frac {15 b^2 (b d-a e)^4}{e^6 (d+e x)^{15}}-\frac {20 b^3 (b d-a e)^3}{e^6 (d+e x)^{14}}+\frac {15 b^4 (b d-a e)^2}{e^6 (d+e x)^{13}}-\frac {6 b^5 (b d-a e)}{e^6 (d+e x)^{12}}+\frac {b^6}{e^6 (d+e x)^{11}}\right ) \, dx}{a b+b^2 x}\\ &=-\frac {(b d-a e)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{16 e^7 (a+b x) (d+e x)^{16}}+\frac {2 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)^{15}}-\frac {15 b^2 (b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{14 e^7 (a+b x) (d+e x)^{14}}+\frac {20 b^3 (b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{13 e^7 (a+b x) (d+e x)^{13}}-\frac {5 b^4 (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{4 e^7 (a+b x) (d+e x)^{12}}+\frac {6 b^5 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}{11 e^7 (a+b x) (d+e x)^{11}}-\frac {b^6 \sqrt {a^2+2 a b x+b^2 x^2}}{10 e^7 (a+b x) (d+e x)^{10}}\\ \end {align*}

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Mathematica [A]  time = 0.10, size = 295, normalized size = 0.81 \begin {gather*} -\frac {\sqrt {(a+b x)^2} \left (5005 a^6 e^6+2002 a^5 b e^5 (d+16 e x)+715 a^4 b^2 e^4 \left (d^2+16 d e x+120 e^2 x^2\right )+220 a^3 b^3 e^3 \left (d^3+16 d^2 e x+120 d e^2 x^2+560 e^3 x^3\right )+55 a^2 b^4 e^2 \left (d^4+16 d^3 e x+120 d^2 e^2 x^2+560 d e^3 x^3+1820 e^4 x^4\right )+10 a b^5 e \left (d^5+16 d^4 e x+120 d^3 e^2 x^2+560 d^2 e^3 x^3+1820 d e^4 x^4+4368 e^5 x^5\right )+b^6 \left (d^6+16 d^5 e x+120 d^4 e^2 x^2+560 d^3 e^3 x^3+1820 d^2 e^4 x^4+4368 d e^5 x^5+8008 e^6 x^6\right )\right )}{80080 e^7 (a+b x) (d+e x)^{16}} \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,x]

[Out]

-1/80080*(Sqrt[(a + b*x)^2]*(5005*a^6*e^6 + 2002*a^5*b*e^5*(d + 16*e*x) + 715*a^4*b^2*e^4*(d^2 + 16*d*e*x + 12
0*e^2*x^2) + 220*a^3*b^3*e^3*(d^3 + 16*d^2*e*x + 120*d*e^2*x^2 + 560*e^3*x^3) + 55*a^2*b^4*e^2*(d^4 + 16*d^3*e
*x + 120*d^2*e^2*x^2 + 560*d*e^3*x^3 + 1820*e^4*x^4) + 10*a*b^5*e*(d^5 + 16*d^4*e*x + 120*d^3*e^2*x^2 + 560*d^
2*e^3*x^3 + 1820*d*e^4*x^4 + 4368*e^5*x^5) + b^6*(d^6 + 16*d^5*e*x + 120*d^4*e^2*x^2 + 560*d^3*e^3*x^3 + 1820*
d^2*e^4*x^4 + 4368*d*e^5*x^5 + 8008*e^6*x^6)))/(e^7*(a + b*x)*(d + e*x)^16)

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IntegrateAlgebraic [F]  time = 180.31, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^17,x]

[Out]

$Aborted

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fricas [A]  time = 0.42, size = 518, normalized size = 1.43 \begin {gather*} -\frac {8008 \, b^{6} e^{6} x^{6} + b^{6} d^{6} + 10 \, a b^{5} d^{5} e + 55 \, a^{2} b^{4} d^{4} e^{2} + 220 \, a^{3} b^{3} d^{3} e^{3} + 715 \, a^{4} b^{2} d^{2} e^{4} + 2002 \, a^{5} b d e^{5} + 5005 \, a^{6} e^{6} + 4368 \, {\left (b^{6} d e^{5} + 10 \, a b^{5} e^{6}\right )} x^{5} + 1820 \, {\left (b^{6} d^{2} e^{4} + 10 \, a b^{5} d e^{5} + 55 \, a^{2} b^{4} e^{6}\right )} x^{4} + 560 \, {\left (b^{6} d^{3} e^{3} + 10 \, a b^{5} d^{2} e^{4} + 55 \, a^{2} b^{4} d e^{5} + 220 \, a^{3} b^{3} e^{6}\right )} x^{3} + 120 \, {\left (b^{6} d^{4} e^{2} + 10 \, a b^{5} d^{3} e^{3} + 55 \, a^{2} b^{4} d^{2} e^{4} + 220 \, a^{3} b^{3} d e^{5} + 715 \, a^{4} b^{2} e^{6}\right )} x^{2} + 16 \, {\left (b^{6} d^{5} e + 10 \, a b^{5} d^{4} e^{2} + 55 \, a^{2} b^{4} d^{3} e^{3} + 220 \, a^{3} b^{3} d^{2} e^{4} + 715 \, a^{4} b^{2} d e^{5} + 2002 \, a^{5} b e^{6}\right )} x}{80080 \, {\left (e^{23} x^{16} + 16 \, d e^{22} x^{15} + 120 \, d^{2} e^{21} x^{14} + 560 \, d^{3} e^{20} x^{13} + 1820 \, d^{4} e^{19} x^{12} + 4368 \, d^{5} e^{18} x^{11} + 8008 \, d^{6} e^{17} x^{10} + 11440 \, d^{7} e^{16} x^{9} + 12870 \, d^{8} e^{15} x^{8} + 11440 \, d^{9} e^{14} x^{7} + 8008 \, d^{10} e^{13} x^{6} + 4368 \, d^{11} e^{12} x^{5} + 1820 \, d^{12} e^{11} x^{4} + 560 \, d^{13} e^{10} x^{3} + 120 \, d^{14} e^{9} x^{2} + 16 \, d^{15} e^{8} x + d^{16} 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,x, algorithm="fricas")

[Out]

-1/80080*(8008*b^6*e^6*x^6 + b^6*d^6 + 10*a*b^5*d^5*e + 55*a^2*b^4*d^4*e^2 + 220*a^3*b^3*d^3*e^3 + 715*a^4*b^2
*d^2*e^4 + 2002*a^5*b*d*e^5 + 5005*a^6*e^6 + 4368*(b^6*d*e^5 + 10*a*b^5*e^6)*x^5 + 1820*(b^6*d^2*e^4 + 10*a*b^
5*d*e^5 + 55*a^2*b^4*e^6)*x^4 + 560*(b^6*d^3*e^3 + 10*a*b^5*d^2*e^4 + 55*a^2*b^4*d*e^5 + 220*a^3*b^3*e^6)*x^3
+ 120*(b^6*d^4*e^2 + 10*a*b^5*d^3*e^3 + 55*a^2*b^4*d^2*e^4 + 220*a^3*b^3*d*e^5 + 715*a^4*b^2*e^6)*x^2 + 16*(b^
6*d^5*e + 10*a*b^5*d^4*e^2 + 55*a^2*b^4*d^3*e^3 + 220*a^3*b^3*d^2*e^4 + 715*a^4*b^2*d*e^5 + 2002*a^5*b*e^6)*x)
/(e^23*x^16 + 16*d*e^22*x^15 + 120*d^2*e^21*x^14 + 560*d^3*e^20*x^13 + 1820*d^4*e^19*x^12 + 4368*d^5*e^18*x^11
+ 8008*d^6*e^17*x^10 + 11440*d^7*e^16*x^9 + 12870*d^8*e^15*x^8 + 11440*d^9*e^14*x^7 + 8008*d^10*e^13*x^6 + 43
68*d^11*e^12*x^5 + 1820*d^12*e^11*x^4 + 560*d^13*e^10*x^3 + 120*d^14*e^9*x^2 + 16*d^15*e^8*x + d^16*e^7)

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giac [A]  time = 0.22, size = 520, normalized size = 1.44 \begin {gather*} -\frac {{\left (8008 \, b^{6} x^{6} e^{6} \mathrm {sgn}\left (b x + a\right ) + 4368 \, b^{6} d x^{5} e^{5} \mathrm {sgn}\left (b x + a\right ) + 1820 \, b^{6} d^{2} x^{4} e^{4} \mathrm {sgn}\left (b x + a\right ) + 560 \, b^{6} d^{3} x^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) + 120 \, b^{6} d^{4} x^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) + 16 \, b^{6} d^{5} x e \mathrm {sgn}\left (b x + a\right ) + b^{6} d^{6} \mathrm {sgn}\left (b x + a\right ) + 43680 \, a b^{5} x^{5} e^{6} \mathrm {sgn}\left (b x + a\right ) + 18200 \, a b^{5} d x^{4} e^{5} \mathrm {sgn}\left (b x + a\right ) + 5600 \, a b^{5} d^{2} x^{3} e^{4} \mathrm {sgn}\left (b x + a\right ) + 1200 \, a b^{5} d^{3} x^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) + 160 \, a b^{5} d^{4} x e^{2} \mathrm {sgn}\left (b x + a\right ) + 10 \, a b^{5} d^{5} e \mathrm {sgn}\left (b x + a\right ) + 100100 \, a^{2} b^{4} x^{4} e^{6} \mathrm {sgn}\left (b x + a\right ) + 30800 \, a^{2} b^{4} d x^{3} e^{5} \mathrm {sgn}\left (b x + a\right ) + 6600 \, a^{2} b^{4} d^{2} x^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) + 880 \, a^{2} b^{4} d^{3} x e^{3} \mathrm {sgn}\left (b x + a\right ) + 55 \, a^{2} b^{4} d^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) + 123200 \, a^{3} b^{3} x^{3} e^{6} \mathrm {sgn}\left (b x + a\right ) + 26400 \, a^{3} b^{3} d x^{2} e^{5} \mathrm {sgn}\left (b x + a\right ) + 3520 \, a^{3} b^{3} d^{2} x e^{4} \mathrm {sgn}\left (b x + a\right ) + 220 \, a^{3} b^{3} d^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) + 85800 \, a^{4} b^{2} x^{2} e^{6} \mathrm {sgn}\left (b x + a\right ) + 11440 \, a^{4} b^{2} d x e^{5} \mathrm {sgn}\left (b x + a\right ) + 715 \, a^{4} b^{2} d^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) + 32032 \, a^{5} b x e^{6} \mathrm {sgn}\left (b x + a\right ) + 2002 \, a^{5} b d e^{5} \mathrm {sgn}\left (b x + a\right ) + 5005 \, a^{6} e^{6} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-7\right )}}{80080 \, {\left (x e + d\right )}^{16}} \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,x, algorithm="giac")

[Out]

-1/80080*(8008*b^6*x^6*e^6*sgn(b*x + a) + 4368*b^6*d*x^5*e^5*sgn(b*x + a) + 1820*b^6*d^2*x^4*e^4*sgn(b*x + a)
+ 560*b^6*d^3*x^3*e^3*sgn(b*x + a) + 120*b^6*d^4*x^2*e^2*sgn(b*x + a) + 16*b^6*d^5*x*e*sgn(b*x + a) + b^6*d^6*
sgn(b*x + a) + 43680*a*b^5*x^5*e^6*sgn(b*x + a) + 18200*a*b^5*d*x^4*e^5*sgn(b*x + a) + 5600*a*b^5*d^2*x^3*e^4*
sgn(b*x + a) + 1200*a*b^5*d^3*x^2*e^3*sgn(b*x + a) + 160*a*b^5*d^4*x*e^2*sgn(b*x + a) + 10*a*b^5*d^5*e*sgn(b*x
+ a) + 100100*a^2*b^4*x^4*e^6*sgn(b*x + a) + 30800*a^2*b^4*d*x^3*e^5*sgn(b*x + a) + 6600*a^2*b^4*d^2*x^2*e^4*
sgn(b*x + a) + 880*a^2*b^4*d^3*x*e^3*sgn(b*x + a) + 55*a^2*b^4*d^4*e^2*sgn(b*x + a) + 123200*a^3*b^3*x^3*e^6*s
gn(b*x + a) + 26400*a^3*b^3*d*x^2*e^5*sgn(b*x + a) + 3520*a^3*b^3*d^2*x*e^4*sgn(b*x + a) + 220*a^3*b^3*d^3*e^3
*sgn(b*x + a) + 85800*a^4*b^2*x^2*e^6*sgn(b*x + a) + 11440*a^4*b^2*d*x*e^5*sgn(b*x + a) + 715*a^4*b^2*d^2*e^4*
sgn(b*x + a) + 32032*a^5*b*x*e^6*sgn(b*x + a) + 2002*a^5*b*d*e^5*sgn(b*x + a) + 5005*a^6*e^6*sgn(b*x + a))*e^(
-7)/(x*e + d)^16

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maple [A]  time = 0.06, size = 392, normalized size = 1.08 \begin {gather*} -\frac {\left (8008 b^{6} e^{6} x^{6}+43680 a \,b^{5} e^{6} x^{5}+4368 b^{6} d \,e^{5} x^{5}+100100 a^{2} b^{4} e^{6} x^{4}+18200 a \,b^{5} d \,e^{5} x^{4}+1820 b^{6} d^{2} e^{4} x^{4}+123200 a^{3} b^{3} e^{6} x^{3}+30800 a^{2} b^{4} d \,e^{5} x^{3}+5600 a \,b^{5} d^{2} e^{4} x^{3}+560 b^{6} d^{3} e^{3} x^{3}+85800 a^{4} b^{2} e^{6} x^{2}+26400 a^{3} b^{3} d \,e^{5} x^{2}+6600 a^{2} b^{4} d^{2} e^{4} x^{2}+1200 a \,b^{5} d^{3} e^{3} x^{2}+120 b^{6} d^{4} e^{2} x^{2}+32032 a^{5} b \,e^{6} x +11440 a^{4} b^{2} d \,e^{5} x +3520 a^{3} b^{3} d^{2} e^{4} x +880 a^{2} b^{4} d^{3} e^{3} x +160 a \,b^{5} d^{4} e^{2} x +16 b^{6} d^{5} e x +5005 a^{6} e^{6}+2002 a^{5} b d \,e^{5}+715 a^{4} b^{2} d^{2} e^{4}+220 a^{3} b^{3} d^{3} e^{3}+55 a^{2} b^{4} d^{4} e^{2}+10 a \,b^{5} d^{5} e +b^{6} d^{6}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{80080 \left (e x +d \right )^{16} \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,x)

[Out]

-1/80080/e^7*(8008*b^6*e^6*x^6+43680*a*b^5*e^6*x^5+4368*b^6*d*e^5*x^5+100100*a^2*b^4*e^6*x^4+18200*a*b^5*d*e^5
*x^4+1820*b^6*d^2*e^4*x^4+123200*a^3*b^3*e^6*x^3+30800*a^2*b^4*d*e^5*x^3+5600*a*b^5*d^2*e^4*x^3+560*b^6*d^3*e^
3*x^3+85800*a^4*b^2*e^6*x^2+26400*a^3*b^3*d*e^5*x^2+6600*a^2*b^4*d^2*e^4*x^2+1200*a*b^5*d^3*e^3*x^2+120*b^6*d^
4*e^2*x^2+32032*a^5*b*e^6*x+11440*a^4*b^2*d*e^5*x+3520*a^3*b^3*d^2*e^4*x+880*a^2*b^4*d^3*e^3*x+160*a*b^5*d^4*e
^2*x+16*b^6*d^5*e*x+5005*a^6*e^6+2002*a^5*b*d*e^5+715*a^4*b^2*d^2*e^4+220*a^3*b^3*d^3*e^3+55*a^2*b^4*d^4*e^2+1
0*a*b^5*d^5*e+b^6*d^6)*((b*x+a)^2)^(5/2)/(e*x+d)^16/(b*x+a)^5

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \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,x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(a*e-b*d>0)', see `assume?` for
more details)Is a*e-b*d zero or nonzero?

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mupad [B]  time = 2.45, size = 1010, normalized size = 2.79 \begin {gather*} \frac {\left (\frac {-6\,a^5\,b\,e^5+15\,a^4\,b^2\,d\,e^4-20\,a^3\,b^3\,d^2\,e^3+15\,a^2\,b^4\,d^3\,e^2-6\,a\,b^5\,d^4\,e+b^6\,d^5}{15\,e^7}+\frac {d\,\left (\frac {15\,a^4\,b^2\,e^5-20\,a^3\,b^3\,d\,e^4+15\,a^2\,b^4\,d^2\,e^3-6\,a\,b^5\,d^3\,e^2+b^6\,d^4\,e}{15\,e^7}-\frac {d\,\left (\frac {20\,a^3\,b^3\,e^5-15\,a^2\,b^4\,d\,e^4+6\,a\,b^5\,d^2\,e^3-b^6\,d^3\,e^2}{15\,e^7}-\frac {d\,\left (\frac {d\,\left (\frac {b^6\,d}{15\,e^3}-\frac {b^5\,\left (6\,a\,e-b\,d\right )}{15\,e^3}\right )}{e}+\frac {b^4\,\left (15\,a^2\,e^2-6\,a\,b\,d\,e+b^2\,d^2\right )}{15\,e^4}\right )}{e}\right )}{e}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^{15}}-\frac {\left (\frac {15\,a^2\,b^4\,e^2-24\,a\,b^5\,d\,e+10\,b^6\,d^2}{12\,e^7}+\frac {d\,\left (\frac {b^6\,d}{12\,e^6}-\frac {b^5\,\left (3\,a\,e-2\,b\,d\right )}{6\,e^6}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^{12}}-\frac {\left (\frac {a^6}{16\,e}-\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {3\,a\,b^5}{8\,e}-\frac {b^6\,d}{16\,e^2}\right )}{e}-\frac {15\,a^2\,b^4}{16\,e}\right )}{e}+\frac {5\,a^3\,b^3}{4\,e}\right )}{e}-\frac {15\,a^4\,b^2}{16\,e}\right )}{e}+\frac {3\,a^5\,b}{8\,e}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^{16}}-\frac {\left (\frac {15\,a^4\,b^2\,e^4-40\,a^3\,b^3\,d\,e^3+45\,a^2\,b^4\,d^2\,e^2-24\,a\,b^5\,d^3\,e+5\,b^6\,d^4}{14\,e^7}+\frac {d\,\left (\frac {-20\,a^3\,b^3\,e^4+30\,a^2\,b^4\,d\,e^3-18\,a\,b^5\,d^2\,e^2+4\,b^6\,d^3\,e}{14\,e^7}+\frac {d\,\left (\frac {d\,\left (\frac {b^6\,d}{14\,e^4}-\frac {b^5\,\left (3\,a\,e-b\,d\right )}{7\,e^4}\right )}{e}+\frac {3\,b^4\,\left (5\,a^2\,e^2-4\,a\,b\,d\,e+b^2\,d^2\right )}{14\,e^5}\right )}{e}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^{14}}+\frac {\left (\frac {5\,b^6\,d-6\,a\,b^5\,e}{11\,e^7}+\frac {b^6\,d}{11\,e^7}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^{11}}+\frac {\left (\frac {-20\,a^3\,b^3\,e^3+45\,a^2\,b^4\,d\,e^2-36\,a\,b^5\,d^2\,e+10\,b^6\,d^3}{13\,e^7}+\frac {d\,\left (\frac {d\,\left (\frac {b^6\,d}{13\,e^5}-\frac {3\,b^5\,\left (2\,a\,e-b\,d\right )}{13\,e^5}\right )}{e}+\frac {3\,b^4\,\left (5\,a^2\,e^2-6\,a\,b\,d\,e+2\,b^2\,d^2\right )}{13\,e^6}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^{13}}-\frac {b^6\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{10\,e^7\,\left (a+b\,x\right )\,{\left (d+e\,x\right )}^{10}} \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,x)

[Out]

(((b^6*d^5 - 6*a^5*b*e^5 + 15*a^4*b^2*d*e^4 + 15*a^2*b^4*d^3*e^2 - 20*a^3*b^3*d^2*e^3 - 6*a*b^5*d^4*e)/(15*e^7
) + (d*((b^6*d^4*e + 15*a^4*b^2*e^5 - 6*a*b^5*d^3*e^2 - 20*a^3*b^3*d*e^4 + 15*a^2*b^4*d^2*e^3)/(15*e^7) - (d*(
(20*a^3*b^3*e^5 - b^6*d^3*e^2 + 6*a*b^5*d^2*e^3 - 15*a^2*b^4*d*e^4)/(15*e^7) - (d*((d*((b^6*d)/(15*e^3) - (b^5
*(6*a*e - b*d))/(15*e^3)))/e + (b^4*(15*a^2*e^2 + b^2*d^2 - 6*a*b*d*e))/(15*e^4)))/e))/e))/e)*(a^2 + b^2*x^2 +
2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^15) - (((10*b^6*d^2 + 15*a^2*b^4*e^2 - 24*a*b^5*d*e)/(12*e^7) + (d*((b^6
*d)/(12*e^6) - (b^5*(3*a*e - 2*b*d))/(6*e^6)))/e)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^12) -
((a^6/(16*e) - (d*((d*((d*((d*((d*((3*a*b^5)/(8*e) - (b^6*d)/(16*e^2)))/e - (15*a^2*b^4)/(16*e)))/e + (5*a^3*b
^3)/(4*e)))/e - (15*a^4*b^2)/(16*e)))/e + (3*a^5*b)/(8*e)))/e)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a + b*x)*(d
+ e*x)^16) - (((5*b^6*d^4 + 15*a^4*b^2*e^4 - 40*a^3*b^3*d*e^3 + 45*a^2*b^4*d^2*e^2 - 24*a*b^5*d^3*e)/(14*e^7)
+ (d*((4*b^6*d^3*e - 20*a^3*b^3*e^4 - 18*a*b^5*d^2*e^2 + 30*a^2*b^4*d*e^3)/(14*e^7) + (d*((d*((b^6*d)/(14*e^4)
- (b^5*(3*a*e - b*d))/(7*e^4)))/e + (3*b^4*(5*a^2*e^2 + b^2*d^2 - 4*a*b*d*e))/(14*e^5)))/e))/e)*(a^2 + b^2*x^
2 + 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^14) + (((5*b^6*d - 6*a*b^5*e)/(11*e^7) + (b^6*d)/(11*e^7))*(a^2 + b^2
*x^2 + 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^11) + (((10*b^6*d^3 - 20*a^3*b^3*e^3 + 45*a^2*b^4*d*e^2 - 36*a*b^5
*d^2*e)/(13*e^7) + (d*((d*((b^6*d)/(13*e^5) - (3*b^5*(2*a*e - b*d))/(13*e^5)))/e + (3*b^4*(5*a^2*e^2 + 2*b^2*d
^2 - 6*a*b*d*e))/(13*e^6)))/e)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^13) - (b^6*(a^2 + b^2*x^2
+ 2*a*b*x)^(1/2))/(10*e^7*(a + b*x)*(d + e*x)^10)

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sympy [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: HeuristicGCDFailed} \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,x)

[Out]

Exception raised: HeuristicGCDFailed

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