3.2119 \(\int \frac{(a+b x) (a^2+2 a b x+b^2 x^2)^{5/2}}{(d+e x)^{13/2}} \, dx\)

Optimal. Leaf size=368 \[ \frac{2 b^6 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x}}{e^7 (a+b x)}+\frac{12 b^5 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{e^7 (a+b x) \sqrt{d+e x}}-\frac{10 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{e^7 (a+b x) (d+e x)^{3/2}}+\frac{8 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{e^7 (a+b x) (d+e x)^{5/2}}-\frac{30 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{7 e^7 (a+b x) (d+e x)^{7/2}}+\frac{4 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}{3 e^7 (a+b x) (d+e x)^{9/2}}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^6}{11 e^7 (a+b x) (d+e x)^{11/2}} \]

[Out]

(-2*(b*d - a*e)^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*e^7*(a + b*x)*(d + e*x)^(11/2)) + (4*b*(b*d - a*e)^5*Sqrt
[a^2 + 2*a*b*x + b^2*x^2])/(3*e^7*(a + b*x)*(d + e*x)^(9/2)) - (30*b^2*(b*d - a*e)^4*Sqrt[a^2 + 2*a*b*x + b^2*
x^2])/(7*e^7*(a + b*x)*(d + e*x)^(7/2)) + (8*b^3*(b*d - a*e)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(a + b*x)*(
d + e*x)^(5/2)) - (10*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)^(3/2)) + (12*b
^5*(b*d - a*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(a + b*x)*Sqrt[d + e*x]) + (2*b^6*Sqrt[d + e*x]*Sqrt[a^2 +
2*a*b*x + b^2*x^2])/(e^7*(a + b*x))

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Rubi [A]  time = 0.142529, antiderivative size = 368, normalized size of antiderivative = 1., 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} \[ \frac{2 b^6 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x}}{e^7 (a+b x)}+\frac{12 b^5 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{e^7 (a+b x) \sqrt{d+e x}}-\frac{10 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{e^7 (a+b x) (d+e x)^{3/2}}+\frac{8 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{e^7 (a+b x) (d+e x)^{5/2}}-\frac{30 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{7 e^7 (a+b x) (d+e x)^{7/2}}+\frac{4 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}{3 e^7 (a+b x) (d+e x)^{9/2}}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^6}{11 e^7 (a+b x) (d+e x)^{11/2}} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^(13/2),x]

[Out]

(-2*(b*d - a*e)^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*e^7*(a + b*x)*(d + e*x)^(11/2)) + (4*b*(b*d - a*e)^5*Sqrt
[a^2 + 2*a*b*x + b^2*x^2])/(3*e^7*(a + b*x)*(d + e*x)^(9/2)) - (30*b^2*(b*d - a*e)^4*Sqrt[a^2 + 2*a*b*x + b^2*
x^2])/(7*e^7*(a + b*x)*(d + e*x)^(7/2)) + (8*b^3*(b*d - a*e)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(a + b*x)*(
d + e*x)^(5/2)) - (10*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)^(3/2)) + (12*b
^5*(b*d - a*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(a + b*x)*Sqrt[d + e*x]) + (2*b^6*Sqrt[d + e*x]*Sqrt[a^2 +
2*a*b*x + b^2*x^2])/(e^7*(a + b*x))

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]

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])

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)^{13/2}} \, 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)^{13/2}} \, 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)^{13/2}} \, 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)^{13/2}}-\frac{6 b (b d-a e)^5}{e^6 (d+e x)^{11/2}}+\frac{15 b^2 (b d-a e)^4}{e^6 (d+e x)^{9/2}}-\frac{20 b^3 (b d-a e)^3}{e^6 (d+e x)^{7/2}}+\frac{15 b^4 (b d-a e)^2}{e^6 (d+e x)^{5/2}}-\frac{6 b^5 (b d-a e)}{e^6 (d+e x)^{3/2}}+\frac{b^6}{e^6 \sqrt{d+e x}}\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}}{11 e^7 (a+b x) (d+e x)^{11/2}}+\frac{4 b (b d-a e)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{3 e^7 (a+b x) (d+e x)^{9/2}}-\frac{30 b^2 (b d-a e)^4 \sqrt{a^2+2 a b x+b^2 x^2}}{7 e^7 (a+b x) (d+e x)^{7/2}}+\frac{8 b^3 (b d-a e)^3 \sqrt{a^2+2 a b x+b^2 x^2}}{e^7 (a+b x) (d+e x)^{5/2}}-\frac{10 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)^{3/2}}+\frac{12 b^5 (b d-a e) \sqrt{a^2+2 a b x+b^2 x^2}}{e^7 (a+b x) \sqrt{d+e x}}+\frac{2 b^6 \sqrt{d+e x} \sqrt{a^2+2 a b x+b^2 x^2}}{e^7 (a+b x)}\\ \end{align*}

Mathematica [A]  time = 0.168949, size = 163, normalized size = 0.44 \[ \frac{2 \sqrt{(a+b x)^2} \left (-495 b^2 (d+e x)^2 (b d-a e)^4+924 b^3 (d+e x)^3 (b d-a e)^3-1155 b^4 (d+e x)^4 (b d-a e)^2+1386 b^5 (d+e x)^5 (b d-a e)+154 b (d+e x) (b d-a e)^5-21 (b d-a e)^6+231 b^6 (d+e x)^6\right )}{231 e^7 (a+b x) (d+e x)^{11/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^(13/2),x]

[Out]

(2*Sqrt[(a + b*x)^2]*(-21*(b*d - a*e)^6 + 154*b*(b*d - a*e)^5*(d + e*x) - 495*b^2*(b*d - a*e)^4*(d + e*x)^2 +
924*b^3*(b*d - a*e)^3*(d + e*x)^3 - 1155*b^4*(b*d - a*e)^2*(d + e*x)^4 + 1386*b^5*(b*d - a*e)*(d + e*x)^5 + 23
1*b^6*(d + e*x)^6))/(231*e^7*(a + b*x)*(d + e*x)^(11/2))

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Maple [A]  time = 0.007, size = 393, normalized size = 1.1 \begin{align*} -{\frac{-462\,{x}^{6}{b}^{6}{e}^{6}+2772\,{x}^{5}a{b}^{5}{e}^{6}-5544\,{x}^{5}{b}^{6}d{e}^{5}+2310\,{x}^{4}{a}^{2}{b}^{4}{e}^{6}+9240\,{x}^{4}a{b}^{5}d{e}^{5}-18480\,{x}^{4}{b}^{6}{d}^{2}{e}^{4}+1848\,{x}^{3}{a}^{3}{b}^{3}{e}^{6}+3696\,{x}^{3}{a}^{2}{b}^{4}d{e}^{5}+14784\,{x}^{3}a{b}^{5}{d}^{2}{e}^{4}-29568\,{x}^{3}{b}^{6}{d}^{3}{e}^{3}+990\,{x}^{2}{a}^{4}{b}^{2}{e}^{6}+1584\,{x}^{2}{a}^{3}{b}^{3}d{e}^{5}+3168\,{x}^{2}{a}^{2}{b}^{4}{d}^{2}{e}^{4}+12672\,{x}^{2}a{b}^{5}{d}^{3}{e}^{3}-25344\,{x}^{2}{b}^{6}{d}^{4}{e}^{2}+308\,x{a}^{5}b{e}^{6}+440\,x{a}^{4}{b}^{2}d{e}^{5}+704\,x{a}^{3}{b}^{3}{d}^{2}{e}^{4}+1408\,x{a}^{2}{b}^{4}{d}^{3}{e}^{3}+5632\,xa{b}^{5}{d}^{4}{e}^{2}-11264\,x{b}^{6}{d}^{5}e+42\,{a}^{6}{e}^{6}+56\,d{e}^{5}{a}^{5}b+80\,{a}^{4}{b}^{2}{d}^{2}{e}^{4}+128\,{a}^{3}{b}^{3}{d}^{3}{e}^{3}+256\,{a}^{2}{b}^{4}{d}^{4}{e}^{2}+1024\,a{b}^{5}{d}^{5}e-2048\,{b}^{6}{d}^{6}}{231\, \left ( bx+a \right ) ^{5}{e}^{7}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}} \left ( ex+d \right ) ^{-{\frac{11}{2}}}} \end{align*}

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)^(13/2),x)

[Out]

-2/231/(e*x+d)^(11/2)*(-231*b^6*e^6*x^6+1386*a*b^5*e^6*x^5-2772*b^6*d*e^5*x^5+1155*a^2*b^4*e^6*x^4+4620*a*b^5*
d*e^5*x^4-9240*b^6*d^2*e^4*x^4+924*a^3*b^3*e^6*x^3+1848*a^2*b^4*d*e^5*x^3+7392*a*b^5*d^2*e^4*x^3-14784*b^6*d^3
*e^3*x^3+495*a^4*b^2*e^6*x^2+792*a^3*b^3*d*e^5*x^2+1584*a^2*b^4*d^2*e^4*x^2+6336*a*b^5*d^3*e^3*x^2-12672*b^6*d
^4*e^2*x^2+154*a^5*b*e^6*x+220*a^4*b^2*d*e^5*x+352*a^3*b^3*d^2*e^4*x+704*a^2*b^4*d^3*e^3*x+2816*a*b^5*d^4*e^2*
x-5632*b^6*d^5*e*x+21*a^6*e^6+28*a^5*b*d*e^5+40*a^4*b^2*d^2*e^4+64*a^3*b^3*d^3*e^3+128*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 = 1.25095, size = 961, normalized size = 2.61 \begin{align*} -\frac{2 \,{\left (693 \, b^{5} e^{5} x^{5} + 256 \, b^{5} d^{5} + 128 \, a b^{4} d^{4} e + 96 \, a^{2} b^{3} d^{3} e^{2} + 80 \, a^{3} b^{2} d^{2} e^{3} + 70 \, a^{4} b d e^{4} + 63 \, a^{5} e^{5} + 1155 \,{\left (2 \, b^{5} d e^{4} + a b^{4} e^{5}\right )} x^{4} + 462 \,{\left (8 \, b^{5} d^{2} e^{3} + 4 \, a b^{4} d e^{4} + 3 \, a^{2} b^{3} e^{5}\right )} x^{3} + 198 \,{\left (16 \, b^{5} d^{3} e^{2} + 8 \, a b^{4} d^{2} e^{3} + 6 \, a^{2} b^{3} d e^{4} + 5 \, a^{3} b^{2} e^{5}\right )} x^{2} + 11 \,{\left (128 \, b^{5} d^{4} e + 64 \, a b^{4} d^{3} e^{2} + 48 \, a^{2} b^{3} d^{2} e^{3} + 40 \, a^{3} b^{2} d e^{4} + 35 \, a^{4} b e^{5}\right )} x\right )} a}{693 \,{\left (e^{11} x^{5} + 5 \, d e^{10} x^{4} + 10 \, d^{2} e^{9} x^{3} + 10 \, d^{3} e^{8} x^{2} + 5 \, d^{4} e^{7} x + d^{5} e^{6}\right )} \sqrt{e x + d}} + \frac{2 \,{\left (693 \, b^{5} e^{6} x^{6} + 3072 \, b^{5} d^{6} - 1280 \, a b^{4} d^{5} e - 256 \, a^{2} b^{3} d^{4} e^{2} - 96 \, a^{3} b^{2} d^{3} e^{3} - 40 \, a^{4} b d^{2} e^{4} - 14 \, a^{5} d e^{5} + 693 \,{\left (12 \, b^{5} d e^{5} - 5 \, a b^{4} e^{6}\right )} x^{5} + 2310 \,{\left (12 \, b^{5} d^{2} e^{4} - 5 \, a b^{4} d e^{5} - a^{2} b^{3} e^{6}\right )} x^{4} + 462 \,{\left (96 \, b^{5} d^{3} e^{3} - 40 \, a b^{4} d^{2} e^{4} - 8 \, a^{2} b^{3} d e^{5} - 3 \, a^{3} b^{2} e^{6}\right )} x^{3} + 99 \,{\left (384 \, b^{5} d^{4} e^{2} - 160 \, a b^{4} d^{3} e^{3} - 32 \, a^{2} b^{3} d^{2} e^{4} - 12 \, a^{3} b^{2} d e^{5} - 5 \, a^{4} b e^{6}\right )} x^{2} + 11 \,{\left (1536 \, b^{5} d^{5} e - 640 \, a b^{4} d^{4} e^{2} - 128 \, a^{2} b^{3} d^{3} e^{3} - 48 \, a^{3} b^{2} d^{2} e^{4} - 20 \, a^{4} b d e^{5} - 7 \, a^{5} e^{6}\right )} x\right )} b}{693 \,{\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 )} \sqrt{e x + d}} \end{align*}

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)^(13/2),x, algorithm="maxima")

[Out]

-2/693*(693*b^5*e^5*x^5 + 256*b^5*d^5 + 128*a*b^4*d^4*e + 96*a^2*b^3*d^3*e^2 + 80*a^3*b^2*d^2*e^3 + 70*a^4*b*d
*e^4 + 63*a^5*e^5 + 1155*(2*b^5*d*e^4 + a*b^4*e^5)*x^4 + 462*(8*b^5*d^2*e^3 + 4*a*b^4*d*e^4 + 3*a^2*b^3*e^5)*x
^3 + 198*(16*b^5*d^3*e^2 + 8*a*b^4*d^2*e^3 + 6*a^2*b^3*d*e^4 + 5*a^3*b^2*e^5)*x^2 + 11*(128*b^5*d^4*e + 64*a*b
^4*d^3*e^2 + 48*a^2*b^3*d^2*e^3 + 40*a^3*b^2*d*e^4 + 35*a^4*b*e^5)*x)*a/((e^11*x^5 + 5*d*e^10*x^4 + 10*d^2*e^9
*x^3 + 10*d^3*e^8*x^2 + 5*d^4*e^7*x + d^5*e^6)*sqrt(e*x + d)) + 2/693*(693*b^5*e^6*x^6 + 3072*b^5*d^6 - 1280*a
*b^4*d^5*e - 256*a^2*b^3*d^4*e^2 - 96*a^3*b^2*d^3*e^3 - 40*a^4*b*d^2*e^4 - 14*a^5*d*e^5 + 693*(12*b^5*d*e^5 -
5*a*b^4*e^6)*x^5 + 2310*(12*b^5*d^2*e^4 - 5*a*b^4*d*e^5 - a^2*b^3*e^6)*x^4 + 462*(96*b^5*d^3*e^3 - 40*a*b^4*d^
2*e^4 - 8*a^2*b^3*d*e^5 - 3*a^3*b^2*e^6)*x^3 + 99*(384*b^5*d^4*e^2 - 160*a*b^4*d^3*e^3 - 32*a^2*b^3*d^2*e^4 -
12*a^3*b^2*d*e^5 - 5*a^4*b*e^6)*x^2 + 11*(1536*b^5*d^5*e - 640*a*b^4*d^4*e^2 - 128*a^2*b^3*d^3*e^3 - 48*a^3*b^
2*d^2*e^4 - 20*a^4*b*d*e^5 - 7*a^5*e^6)*x)*b/((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)*sqrt(e*x + d))

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Fricas [A]  time = 1.0083, size = 890, normalized size = 2.42 \begin{align*} \frac{2 \,{\left (231 \, b^{6} e^{6} x^{6} + 1024 \, b^{6} d^{6} - 512 \, a b^{5} d^{5} e - 128 \, a^{2} b^{4} d^{4} e^{2} - 64 \, a^{3} b^{3} d^{3} e^{3} - 40 \, a^{4} b^{2} d^{2} e^{4} - 28 \, a^{5} b d e^{5} - 21 \, a^{6} e^{6} + 1386 \,{\left (2 \, b^{6} d e^{5} - a b^{5} e^{6}\right )} x^{5} + 1155 \,{\left (8 \, b^{6} d^{2} e^{4} - 4 \, a b^{5} d e^{5} - a^{2} b^{4} e^{6}\right )} x^{4} + 924 \,{\left (16 \, b^{6} d^{3} e^{3} - 8 \, a b^{5} d^{2} e^{4} - 2 \, a^{2} b^{4} d e^{5} - a^{3} b^{3} e^{6}\right )} x^{3} + 99 \,{\left (128 \, b^{6} d^{4} e^{2} - 64 \, a b^{5} d^{3} e^{3} - 16 \, a^{2} b^{4} d^{2} e^{4} - 8 \, a^{3} b^{3} d e^{5} - 5 \, a^{4} b^{2} e^{6}\right )} x^{2} + 22 \,{\left (256 \, b^{6} d^{5} e - 128 \, a b^{5} d^{4} e^{2} - 32 \, a^{2} b^{4} d^{3} e^{3} - 16 \, a^{3} b^{3} d^{2} e^{4} - 10 \, a^{4} b^{2} d e^{5} - 7 \, a^{5} b e^{6}\right )} x\right )} \sqrt{e x + d}}{231 \,{\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 )}} \end{align*}

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)^(13/2),x, algorithm="fricas")

[Out]

2/231*(231*b^6*e^6*x^6 + 1024*b^6*d^6 - 512*a*b^5*d^5*e - 128*a^2*b^4*d^4*e^2 - 64*a^3*b^3*d^3*e^3 - 40*a^4*b^
2*d^2*e^4 - 28*a^5*b*d*e^5 - 21*a^6*e^6 + 1386*(2*b^6*d*e^5 - a*b^5*e^6)*x^5 + 1155*(8*b^6*d^2*e^4 - 4*a*b^5*d
*e^5 - a^2*b^4*e^6)*x^4 + 924*(16*b^6*d^3*e^3 - 8*a*b^5*d^2*e^4 - 2*a^2*b^4*d*e^5 - a^3*b^3*e^6)*x^3 + 99*(128
*b^6*d^4*e^2 - 64*a*b^5*d^3*e^3 - 16*a^2*b^4*d^2*e^4 - 8*a^3*b^3*d*e^5 - 5*a^4*b^2*e^6)*x^2 + 22*(256*b^6*d^5*
e - 128*a*b^5*d^4*e^2 - 32*a^2*b^4*d^3*e^3 - 16*a^3*b^3*d^2*e^4 - 10*a^4*b^2*d*e^5 - 7*a^5*b*e^6)*x)*sqrt(e*x
+ d)/(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)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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)**(13/2),x)

[Out]

Timed out

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Giac [B]  time = 1.19972, size = 833, normalized size = 2.26 \begin{align*} \text{result too large to display} \end{align*}

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)^(13/2),x, algorithm="giac")

[Out]

2*sqrt(x*e + d)*b^6*e^(-7)*sgn(b*x + a) + 2/231*(1386*(x*e + d)^5*b^6*d*sgn(b*x + a) - 1155*(x*e + d)^4*b^6*d^
2*sgn(b*x + a) + 924*(x*e + d)^3*b^6*d^3*sgn(b*x + a) - 495*(x*e + d)^2*b^6*d^4*sgn(b*x + a) + 154*(x*e + d)*b
^6*d^5*sgn(b*x + a) - 21*b^6*d^6*sgn(b*x + a) - 1386*(x*e + d)^5*a*b^5*e*sgn(b*x + a) + 2310*(x*e + d)^4*a*b^5
*d*e*sgn(b*x + a) - 2772*(x*e + d)^3*a*b^5*d^2*e*sgn(b*x + a) + 1980*(x*e + d)^2*a*b^5*d^3*e*sgn(b*x + a) - 77
0*(x*e + d)*a*b^5*d^4*e*sgn(b*x + a) + 126*a*b^5*d^5*e*sgn(b*x + a) - 1155*(x*e + d)^4*a^2*b^4*e^2*sgn(b*x + a
) + 2772*(x*e + d)^3*a^2*b^4*d*e^2*sgn(b*x + a) - 2970*(x*e + d)^2*a^2*b^4*d^2*e^2*sgn(b*x + a) + 1540*(x*e +
d)*a^2*b^4*d^3*e^2*sgn(b*x + a) - 315*a^2*b^4*d^4*e^2*sgn(b*x + a) - 924*(x*e + d)^3*a^3*b^3*e^3*sgn(b*x + a)
+ 1980*(x*e + d)^2*a^3*b^3*d*e^3*sgn(b*x + a) - 1540*(x*e + d)*a^3*b^3*d^2*e^3*sgn(b*x + a) + 420*a^3*b^3*d^3*
e^3*sgn(b*x + a) - 495*(x*e + d)^2*a^4*b^2*e^4*sgn(b*x + a) + 770*(x*e + d)*a^4*b^2*d*e^4*sgn(b*x + a) - 315*a
^4*b^2*d^2*e^4*sgn(b*x + a) - 154*(x*e + d)*a^5*b*e^5*sgn(b*x + a) + 126*a^5*b*d*e^5*sgn(b*x + a) - 21*a^6*e^6
*sgn(b*x + a))*e^(-7)/(x*e + d)^(11/2)