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

[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])

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Rubi [A]  time = 0.141162, antiderivative size = 370, 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} \[ -\frac{2 b^6 \sqrt{a^2+2 a b x+b^2 x^2}}{e^7 (a+b x) \sqrt{d+e x}}+\frac{4 b^5 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{e^7 (a+b x) (d+e x)^{3/2}}-\frac{6 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{e^7 (a+b x) (d+e x)^{5/2}}+\frac{40 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{7 e^7 (a+b x) (d+e x)^{7/2}}-\frac{10 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{3 e^7 (a+b x) (d+e x)^{9/2}}+\frac{12 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}{11 e^7 (a+b x) (d+e x)^{11/2}}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^6}{13 e^7 (a+b x) (d+e x)^{13/2}} \]

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 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)^{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*}

Mathematica [A]  time = 0.173134, size = 163, normalized size = 0.44 \[ \frac{2 \sqrt{(a+b x)^2} \left (-5005 b^2 (d+e x)^2 (b d-a e)^4+8580 b^3 (d+e x)^3 (b d-a e)^3-9009 b^4 (d+e x)^4 (b d-a e)^2+6006 b^5 (d+e x)^5 (b d-a e)+1638 b (d+e x) (b d-a e)^5-231 (b d-a e)^6-3003 b^6 (d+e x)^6\right )}{3003 e^7 (a+b x) (d+e x)^{13/2}} \]

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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Maple [A]  time = 0.008, size = 393, normalized size = 1.1 \begin{align*} -{\frac{6006\,{x}^{6}{b}^{6}{e}^{6}+12012\,{x}^{5}a{b}^{5}{e}^{6}+24024\,{x}^{5}{b}^{6}d{e}^{5}+18018\,{x}^{4}{a}^{2}{b}^{4}{e}^{6}+24024\,{x}^{4}a{b}^{5}d{e}^{5}+48048\,{x}^{4}{b}^{6}{d}^{2}{e}^{4}+17160\,{x}^{3}{a}^{3}{b}^{3}{e}^{6}+20592\,{x}^{3}{a}^{2}{b}^{4}d{e}^{5}+27456\,{x}^{3}a{b}^{5}{d}^{2}{e}^{4}+54912\,{x}^{3}{b}^{6}{d}^{3}{e}^{3}+10010\,{x}^{2}{a}^{4}{b}^{2}{e}^{6}+11440\,{x}^{2}{a}^{3}{b}^{3}d{e}^{5}+13728\,{x}^{2}{a}^{2}{b}^{4}{d}^{2}{e}^{4}+18304\,{x}^{2}a{b}^{5}{d}^{3}{e}^{3}+36608\,{x}^{2}{b}^{6}{d}^{4}{e}^{2}+3276\,x{a}^{5}b{e}^{6}+3640\,x{a}^{4}{b}^{2}d{e}^{5}+4160\,x{a}^{3}{b}^{3}{d}^{2}{e}^{4}+4992\,x{a}^{2}{b}^{4}{d}^{3}{e}^{3}+6656\,xa{b}^{5}{d}^{4}{e}^{2}+13312\,x{b}^{6}{d}^{5}e+462\,{a}^{6}{e}^{6}+504\,d{e}^{5}{a}^{5}b+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\, \left ( bx+a \right ) ^{5}{e}^{7}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}} \left ( ex+d \right ) ^{-{\frac{13}{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)^(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 = 1.25156, size = 992, normalized size = 2.68 \begin{align*} -\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}} \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)^(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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Fricas [A]  time = 0.997792, size = 936, normalized size = 2.53 \begin{align*} -\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 )}} \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)^(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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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)**(15/2),x)

[Out]

Timed out

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Giac [B]  time = 1.23936, size = 829, normalized size = 2.24 \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)^(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)