3.2065 \(\int \frac{(a+b x) (a^2+2 a b x+b^2 x^2)^3}{(d+e x)^{7/2}} \, dx\)

Optimal. Leaf size=210 \[ -\frac{2 b^6 (d+e x)^{7/2} (b d-a e)}{e^8}+\frac{42 b^5 (d+e x)^{5/2} (b d-a e)^2}{5 e^8}-\frac{70 b^4 (d+e x)^{3/2} (b d-a e)^3}{3 e^8}+\frac{70 b^3 \sqrt{d+e x} (b d-a e)^4}{e^8}+\frac{42 b^2 (b d-a e)^5}{e^8 \sqrt{d+e x}}-\frac{14 b (b d-a e)^6}{3 e^8 (d+e x)^{3/2}}+\frac{2 (b d-a e)^7}{5 e^8 (d+e x)^{5/2}}+\frac{2 b^7 (d+e x)^{9/2}}{9 e^8} \]

[Out]

(2*(b*d - a*e)^7)/(5*e^8*(d + e*x)^(5/2)) - (14*b*(b*d - a*e)^6)/(3*e^8*(d + e*x)^(3/2)) + (42*b^2*(b*d - a*e)
^5)/(e^8*Sqrt[d + e*x]) + (70*b^3*(b*d - a*e)^4*Sqrt[d + e*x])/e^8 - (70*b^4*(b*d - a*e)^3*(d + e*x)^(3/2))/(3
*e^8) + (42*b^5*(b*d - a*e)^2*(d + e*x)^(5/2))/(5*e^8) - (2*b^6*(b*d - a*e)*(d + e*x)^(7/2))/e^8 + (2*b^7*(d +
e*x)^(9/2))/(9*e^8)

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Rubi [A]  time = 0.0790765, antiderivative size = 210, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.061, Rules used = {27, 43} \[ -\frac{2 b^6 (d+e x)^{7/2} (b d-a e)}{e^8}+\frac{42 b^5 (d+e x)^{5/2} (b d-a e)^2}{5 e^8}-\frac{70 b^4 (d+e x)^{3/2} (b d-a e)^3}{3 e^8}+\frac{70 b^3 \sqrt{d+e x} (b d-a e)^4}{e^8}+\frac{42 b^2 (b d-a e)^5}{e^8 \sqrt{d+e x}}-\frac{14 b (b d-a e)^6}{3 e^8 (d+e x)^{3/2}}+\frac{2 (b d-a e)^7}{5 e^8 (d+e x)^{5/2}}+\frac{2 b^7 (d+e x)^{9/2}}{9 e^8} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^3)/(d + e*x)^(7/2),x]

[Out]

(2*(b*d - a*e)^7)/(5*e^8*(d + e*x)^(5/2)) - (14*b*(b*d - a*e)^6)/(3*e^8*(d + e*x)^(3/2)) + (42*b^2*(b*d - a*e)
^5)/(e^8*Sqrt[d + e*x]) + (70*b^3*(b*d - a*e)^4*Sqrt[d + e*x])/e^8 - (70*b^4*(b*d - a*e)^3*(d + e*x)^(3/2))/(3
*e^8) + (42*b^5*(b*d - a*e)^2*(d + e*x)^(5/2))/(5*e^8) - (2*b^6*(b*d - a*e)*(d + e*x)^(7/2))/e^8 + (2*b^7*(d +
e*x)^(9/2))/(9*e^8)

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

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 )^3}{(d+e x)^{7/2}} \, dx &=\int \frac{(a+b x)^7}{(d+e x)^{7/2}} \, dx\\ &=\int \left (\frac{(-b d+a e)^7}{e^7 (d+e x)^{7/2}}+\frac{7 b (b d-a e)^6}{e^7 (d+e x)^{5/2}}-\frac{21 b^2 (b d-a e)^5}{e^7 (d+e x)^{3/2}}+\frac{35 b^3 (b d-a e)^4}{e^7 \sqrt{d+e x}}-\frac{35 b^4 (b d-a e)^3 \sqrt{d+e x}}{e^7}+\frac{21 b^5 (b d-a e)^2 (d+e x)^{3/2}}{e^7}-\frac{7 b^6 (b d-a e) (d+e x)^{5/2}}{e^7}+\frac{b^7 (d+e x)^{7/2}}{e^7}\right ) \, dx\\ &=\frac{2 (b d-a e)^7}{5 e^8 (d+e x)^{5/2}}-\frac{14 b (b d-a e)^6}{3 e^8 (d+e x)^{3/2}}+\frac{42 b^2 (b d-a e)^5}{e^8 \sqrt{d+e x}}+\frac{70 b^3 (b d-a e)^4 \sqrt{d+e x}}{e^8}-\frac{70 b^4 (b d-a e)^3 (d+e x)^{3/2}}{3 e^8}+\frac{42 b^5 (b d-a e)^2 (d+e x)^{5/2}}{5 e^8}-\frac{2 b^6 (b d-a e) (d+e x)^{7/2}}{e^8}+\frac{2 b^7 (d+e x)^{9/2}}{9 e^8}\\ \end{align*}

Mathematica [A]  time = 0.109769, size = 167, normalized size = 0.8 \[ \frac{2 \left (945 b^2 (d+e x)^2 (b d-a e)^5+1575 b^3 (d+e x)^3 (b d-a e)^4-525 b^4 (d+e x)^4 (b d-a e)^3+189 b^5 (d+e x)^5 (b d-a e)^2-45 b^6 (d+e x)^6 (b d-a e)-105 b (d+e x) (b d-a e)^6+9 (b d-a e)^7+5 b^7 (d+e x)^7\right )}{45 e^8 (d+e x)^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^3)/(d + e*x)^(7/2),x]

[Out]

(2*(9*(b*d - a*e)^7 - 105*b*(b*d - a*e)^6*(d + e*x) + 945*b^2*(b*d - a*e)^5*(d + e*x)^2 + 1575*b^3*(b*d - a*e)
^4*(d + e*x)^3 - 525*b^4*(b*d - a*e)^3*(d + e*x)^4 + 189*b^5*(b*d - a*e)^2*(d + e*x)^5 - 45*b^6*(b*d - a*e)*(d
+ e*x)^6 + 5*b^7*(d + e*x)^7))/(45*e^8*(d + e*x)^(5/2))

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Maple [B]  time = 0.007, size = 498, normalized size = 2.4 \begin{align*} -{\frac{-10\,{b}^{7}{x}^{7}{e}^{7}-90\,a{b}^{6}{e}^{7}{x}^{6}+20\,{b}^{7}d{e}^{6}{x}^{6}-378\,{a}^{2}{b}^{5}{e}^{7}{x}^{5}+216\,a{b}^{6}d{e}^{6}{x}^{5}-48\,{b}^{7}{d}^{2}{e}^{5}{x}^{5}-1050\,{a}^{3}{b}^{4}{e}^{7}{x}^{4}+1260\,{a}^{2}{b}^{5}d{e}^{6}{x}^{4}-720\,a{b}^{6}{d}^{2}{e}^{5}{x}^{4}+160\,{b}^{7}{d}^{3}{e}^{4}{x}^{4}-3150\,{a}^{4}{b}^{3}{e}^{7}{x}^{3}+8400\,{a}^{3}{b}^{4}d{e}^{6}{x}^{3}-10080\,{a}^{2}{b}^{5}{d}^{2}{e}^{5}{x}^{3}+5760\,a{b}^{6}{d}^{3}{e}^{4}{x}^{3}-1280\,{b}^{7}{d}^{4}{e}^{3}{x}^{3}+1890\,{a}^{5}{b}^{2}{e}^{7}{x}^{2}-18900\,{a}^{4}{b}^{3}d{e}^{6}{x}^{2}+50400\,{a}^{3}{b}^{4}{d}^{2}{e}^{5}{x}^{2}-60480\,{a}^{2}{b}^{5}{d}^{3}{e}^{4}{x}^{2}+34560\,a{b}^{6}{d}^{4}{e}^{3}{x}^{2}-7680\,{b}^{7}{d}^{5}{e}^{2}{x}^{2}+210\,{a}^{6}b{e}^{7}x+2520\,{a}^{5}{b}^{2}d{e}^{6}x-25200\,{a}^{4}{b}^{3}{d}^{2}{e}^{5}x+67200\,{a}^{3}{b}^{4}{d}^{3}{e}^{4}x-80640\,{a}^{2}{b}^{5}{d}^{4}{e}^{3}x+46080\,a{b}^{6}{d}^{5}{e}^{2}x-10240\,{b}^{7}{d}^{6}ex+18\,{a}^{7}{e}^{7}+84\,{a}^{6}bd{e}^{6}+1008\,{a}^{5}{b}^{2}{d}^{2}{e}^{5}-10080\,{a}^{4}{b}^{3}{d}^{3}{e}^{4}+26880\,{a}^{3}{b}^{4}{d}^{4}{e}^{3}-32256\,{a}^{2}{b}^{5}{d}^{5}{e}^{2}+18432\,a{b}^{6}{d}^{6}e-4096\,{b}^{7}{d}^{7}}{45\,{e}^{8}} \left ( ex+d \right ) ^{-{\frac{5}{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)^3/(e*x+d)^(7/2),x)

[Out]

-2/45*(-5*b^7*e^7*x^7-45*a*b^6*e^7*x^6+10*b^7*d*e^6*x^6-189*a^2*b^5*e^7*x^5+108*a*b^6*d*e^6*x^5-24*b^7*d^2*e^5
*x^5-525*a^3*b^4*e^7*x^4+630*a^2*b^5*d*e^6*x^4-360*a*b^6*d^2*e^5*x^4+80*b^7*d^3*e^4*x^4-1575*a^4*b^3*e^7*x^3+4
200*a^3*b^4*d*e^6*x^3-5040*a^2*b^5*d^2*e^5*x^3+2880*a*b^6*d^3*e^4*x^3-640*b^7*d^4*e^3*x^3+945*a^5*b^2*e^7*x^2-
9450*a^4*b^3*d*e^6*x^2+25200*a^3*b^4*d^2*e^5*x^2-30240*a^2*b^5*d^3*e^4*x^2+17280*a*b^6*d^4*e^3*x^2-3840*b^7*d^
5*e^2*x^2+105*a^6*b*e^7*x+1260*a^5*b^2*d*e^6*x-12600*a^4*b^3*d^2*e^5*x+33600*a^3*b^4*d^3*e^4*x-40320*a^2*b^5*d
^4*e^3*x+23040*a*b^6*d^5*e^2*x-5120*b^7*d^6*e*x+9*a^7*e^7+42*a^6*b*d*e^6+504*a^5*b^2*d^2*e^5-5040*a^4*b^3*d^3*
e^4+13440*a^3*b^4*d^4*e^3-16128*a^2*b^5*d^5*e^2+9216*a*b^6*d^6*e-2048*b^7*d^7)/(e*x+d)^(5/2)/e^8

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Maxima [B]  time = 0.989395, size = 625, normalized size = 2.98 \begin{align*} \frac{2 \,{\left (\frac{5 \,{\left (e x + d\right )}^{\frac{9}{2}} b^{7} - 45 \,{\left (b^{7} d - a b^{6} e\right )}{\left (e x + d\right )}^{\frac{7}{2}} + 189 \,{\left (b^{7} d^{2} - 2 \, a b^{6} d e + a^{2} b^{5} e^{2}\right )}{\left (e x + d\right )}^{\frac{5}{2}} - 525 \,{\left (b^{7} d^{3} - 3 \, a b^{6} d^{2} e + 3 \, a^{2} b^{5} d e^{2} - a^{3} b^{4} e^{3}\right )}{\left (e x + d\right )}^{\frac{3}{2}} + 1575 \,{\left (b^{7} d^{4} - 4 \, a b^{6} d^{3} e + 6 \, a^{2} b^{5} d^{2} e^{2} - 4 \, a^{3} b^{4} d e^{3} + a^{4} b^{3} e^{4}\right )} \sqrt{e x + d}}{e^{7}} + \frac{3 \,{\left (3 \, b^{7} d^{7} - 21 \, a b^{6} d^{6} e + 63 \, a^{2} b^{5} d^{5} e^{2} - 105 \, a^{3} b^{4} d^{4} e^{3} + 105 \, a^{4} b^{3} d^{3} e^{4} - 63 \, a^{5} b^{2} d^{2} e^{5} + 21 \, a^{6} b d e^{6} - 3 \, a^{7} e^{7} + 315 \,{\left (b^{7} d^{5} - 5 \, a b^{6} d^{4} e + 10 \, a^{2} b^{5} d^{3} e^{2} - 10 \, a^{3} b^{4} d^{2} e^{3} + 5 \, a^{4} b^{3} d e^{4} - a^{5} b^{2} e^{5}\right )}{\left (e x + d\right )}^{2} - 35 \,{\left (b^{7} d^{6} - 6 \, a b^{6} d^{5} e + 15 \, a^{2} b^{5} d^{4} e^{2} - 20 \, a^{3} b^{4} d^{3} e^{3} + 15 \, a^{4} b^{3} d^{2} e^{4} - 6 \, a^{5} b^{2} d e^{5} + a^{6} b e^{6}\right )}{\left (e x + d\right )}\right )}}{{\left (e x + d\right )}^{\frac{5}{2}} e^{7}}\right )}}{45 \, e} \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)^3/(e*x+d)^(7/2),x, algorithm="maxima")

[Out]

2/45*((5*(e*x + d)^(9/2)*b^7 - 45*(b^7*d - a*b^6*e)*(e*x + d)^(7/2) + 189*(b^7*d^2 - 2*a*b^6*d*e + a^2*b^5*e^2
)*(e*x + d)^(5/2) - 525*(b^7*d^3 - 3*a*b^6*d^2*e + 3*a^2*b^5*d*e^2 - a^3*b^4*e^3)*(e*x + d)^(3/2) + 1575*(b^7*
d^4 - 4*a*b^6*d^3*e + 6*a^2*b^5*d^2*e^2 - 4*a^3*b^4*d*e^3 + a^4*b^3*e^4)*sqrt(e*x + d))/e^7 + 3*(3*b^7*d^7 - 2
1*a*b^6*d^6*e + 63*a^2*b^5*d^5*e^2 - 105*a^3*b^4*d^4*e^3 + 105*a^4*b^3*d^3*e^4 - 63*a^5*b^2*d^2*e^5 + 21*a^6*b
*d*e^6 - 3*a^7*e^7 + 315*(b^7*d^5 - 5*a*b^6*d^4*e + 10*a^2*b^5*d^3*e^2 - 10*a^3*b^4*d^2*e^3 + 5*a^4*b^3*d*e^4
- a^5*b^2*e^5)*(e*x + d)^2 - 35*(b^7*d^6 - 6*a*b^6*d^5*e + 15*a^2*b^5*d^4*e^2 - 20*a^3*b^4*d^3*e^3 + 15*a^4*b^
3*d^2*e^4 - 6*a^5*b^2*d*e^5 + a^6*b*e^6)*(e*x + d))/((e*x + d)^(5/2)*e^7))/e

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Fricas [B]  time = 1.08436, size = 1085, normalized size = 5.17 \begin{align*} \frac{2 \,{\left (5 \, b^{7} e^{7} x^{7} + 2048 \, b^{7} d^{7} - 9216 \, a b^{6} d^{6} e + 16128 \, a^{2} b^{5} d^{5} e^{2} - 13440 \, a^{3} b^{4} d^{4} e^{3} + 5040 \, a^{4} b^{3} d^{3} e^{4} - 504 \, a^{5} b^{2} d^{2} e^{5} - 42 \, a^{6} b d e^{6} - 9 \, a^{7} e^{7} - 5 \,{\left (2 \, b^{7} d e^{6} - 9 \, a b^{6} e^{7}\right )} x^{6} + 3 \,{\left (8 \, b^{7} d^{2} e^{5} - 36 \, a b^{6} d e^{6} + 63 \, a^{2} b^{5} e^{7}\right )} x^{5} - 5 \,{\left (16 \, b^{7} d^{3} e^{4} - 72 \, a b^{6} d^{2} e^{5} + 126 \, a^{2} b^{5} d e^{6} - 105 \, a^{3} b^{4} e^{7}\right )} x^{4} + 5 \,{\left (128 \, b^{7} d^{4} e^{3} - 576 \, a b^{6} d^{3} e^{4} + 1008 \, a^{2} b^{5} d^{2} e^{5} - 840 \, a^{3} b^{4} d e^{6} + 315 \, a^{4} b^{3} e^{7}\right )} x^{3} + 15 \,{\left (256 \, b^{7} d^{5} e^{2} - 1152 \, a b^{6} d^{4} e^{3} + 2016 \, a^{2} b^{5} d^{3} e^{4} - 1680 \, a^{3} b^{4} d^{2} e^{5} + 630 \, a^{4} b^{3} d e^{6} - 63 \, a^{5} b^{2} e^{7}\right )} x^{2} + 5 \,{\left (1024 \, b^{7} d^{6} e - 4608 \, a b^{6} d^{5} e^{2} + 8064 \, a^{2} b^{5} d^{4} e^{3} - 6720 \, a^{3} b^{4} d^{3} e^{4} + 2520 \, a^{4} b^{3} d^{2} e^{5} - 252 \, a^{5} b^{2} d e^{6} - 21 \, a^{6} b e^{7}\right )} x\right )} \sqrt{e x + d}}{45 \,{\left (e^{11} x^{3} + 3 \, d e^{10} x^{2} + 3 \, d^{2} e^{9} x + d^{3} e^{8}\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)^3/(e*x+d)^(7/2),x, algorithm="fricas")

[Out]

2/45*(5*b^7*e^7*x^7 + 2048*b^7*d^7 - 9216*a*b^6*d^6*e + 16128*a^2*b^5*d^5*e^2 - 13440*a^3*b^4*d^4*e^3 + 5040*a
^4*b^3*d^3*e^4 - 504*a^5*b^2*d^2*e^5 - 42*a^6*b*d*e^6 - 9*a^7*e^7 - 5*(2*b^7*d*e^6 - 9*a*b^6*e^7)*x^6 + 3*(8*b
^7*d^2*e^5 - 36*a*b^6*d*e^6 + 63*a^2*b^5*e^7)*x^5 - 5*(16*b^7*d^3*e^4 - 72*a*b^6*d^2*e^5 + 126*a^2*b^5*d*e^6 -
105*a^3*b^4*e^7)*x^4 + 5*(128*b^7*d^4*e^3 - 576*a*b^6*d^3*e^4 + 1008*a^2*b^5*d^2*e^5 - 840*a^3*b^4*d*e^6 + 31
5*a^4*b^3*e^7)*x^3 + 15*(256*b^7*d^5*e^2 - 1152*a*b^6*d^4*e^3 + 2016*a^2*b^5*d^3*e^4 - 1680*a^3*b^4*d^2*e^5 +
630*a^4*b^3*d*e^6 - 63*a^5*b^2*e^7)*x^2 + 5*(1024*b^7*d^6*e - 4608*a*b^6*d^5*e^2 + 8064*a^2*b^5*d^4*e^3 - 6720
*a^3*b^4*d^3*e^4 + 2520*a^4*b^3*d^2*e^5 - 252*a^5*b^2*d*e^6 - 21*a^6*b*e^7)*x)*sqrt(e*x + d)/(e^11*x^3 + 3*d*e
^10*x^2 + 3*d^2*e^9*x + d^3*e^8)

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Sympy [A]  time = 121.882, size = 298, normalized size = 1.42 \begin{align*} \frac{2 b^{7} \left (d + e x\right )^{\frac{9}{2}}}{9 e^{8}} - \frac{42 b^{2} \left (a e - b d\right )^{5}}{e^{8} \sqrt{d + e x}} - \frac{14 b \left (a e - b d\right )^{6}}{3 e^{8} \left (d + e x\right )^{\frac{3}{2}}} + \frac{\left (d + e x\right )^{\frac{7}{2}} \left (14 a b^{6} e - 14 b^{7} d\right )}{7 e^{8}} + \frac{\left (d + e x\right )^{\frac{5}{2}} \left (42 a^{2} b^{5} e^{2} - 84 a b^{6} d e + 42 b^{7} d^{2}\right )}{5 e^{8}} + \frac{\left (d + e x\right )^{\frac{3}{2}} \left (70 a^{3} b^{4} e^{3} - 210 a^{2} b^{5} d e^{2} + 210 a b^{6} d^{2} e - 70 b^{7} d^{3}\right )}{3 e^{8}} + \frac{\sqrt{d + e x} \left (70 a^{4} b^{3} e^{4} - 280 a^{3} b^{4} d e^{3} + 420 a^{2} b^{5} d^{2} e^{2} - 280 a b^{6} d^{3} e + 70 b^{7} d^{4}\right )}{e^{8}} - \frac{2 \left (a e - b d\right )^{7}}{5 e^{8} \left (d + e x\right )^{\frac{5}{2}}} \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)**3/(e*x+d)**(7/2),x)

[Out]

2*b**7*(d + e*x)**(9/2)/(9*e**8) - 42*b**2*(a*e - b*d)**5/(e**8*sqrt(d + e*x)) - 14*b*(a*e - b*d)**6/(3*e**8*(
d + e*x)**(3/2)) + (d + e*x)**(7/2)*(14*a*b**6*e - 14*b**7*d)/(7*e**8) + (d + e*x)**(5/2)*(42*a**2*b**5*e**2 -
84*a*b**6*d*e + 42*b**7*d**2)/(5*e**8) + (d + e*x)**(3/2)*(70*a**3*b**4*e**3 - 210*a**2*b**5*d*e**2 + 210*a*b
**6*d**2*e - 70*b**7*d**3)/(3*e**8) + sqrt(d + e*x)*(70*a**4*b**3*e**4 - 280*a**3*b**4*d*e**3 + 420*a**2*b**5*
d**2*e**2 - 280*a*b**6*d**3*e + 70*b**7*d**4)/e**8 - 2*(a*e - b*d)**7/(5*e**8*(d + e*x)**(5/2))

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Giac [B]  time = 1.17251, size = 821, normalized size = 3.91 \begin{align*} \frac{2}{45} \,{\left (5 \,{\left (x e + d\right )}^{\frac{9}{2}} b^{7} e^{64} - 45 \,{\left (x e + d\right )}^{\frac{7}{2}} b^{7} d e^{64} + 189 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{7} d^{2} e^{64} - 525 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{7} d^{3} e^{64} + 1575 \, \sqrt{x e + d} b^{7} d^{4} e^{64} + 45 \,{\left (x e + d\right )}^{\frac{7}{2}} a b^{6} e^{65} - 378 \,{\left (x e + d\right )}^{\frac{5}{2}} a b^{6} d e^{65} + 1575 \,{\left (x e + d\right )}^{\frac{3}{2}} a b^{6} d^{2} e^{65} - 6300 \, \sqrt{x e + d} a b^{6} d^{3} e^{65} + 189 \,{\left (x e + d\right )}^{\frac{5}{2}} a^{2} b^{5} e^{66} - 1575 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{2} b^{5} d e^{66} + 9450 \, \sqrt{x e + d} a^{2} b^{5} d^{2} e^{66} + 525 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{3} b^{4} e^{67} - 6300 \, \sqrt{x e + d} a^{3} b^{4} d e^{67} + 1575 \, \sqrt{x e + d} a^{4} b^{3} e^{68}\right )} e^{\left (-72\right )} + \frac{2 \,{\left (315 \,{\left (x e + d\right )}^{2} b^{7} d^{5} - 35 \,{\left (x e + d\right )} b^{7} d^{6} + 3 \, b^{7} d^{7} - 1575 \,{\left (x e + d\right )}^{2} a b^{6} d^{4} e + 210 \,{\left (x e + d\right )} a b^{6} d^{5} e - 21 \, a b^{6} d^{6} e + 3150 \,{\left (x e + d\right )}^{2} a^{2} b^{5} d^{3} e^{2} - 525 \,{\left (x e + d\right )} a^{2} b^{5} d^{4} e^{2} + 63 \, a^{2} b^{5} d^{5} e^{2} - 3150 \,{\left (x e + d\right )}^{2} a^{3} b^{4} d^{2} e^{3} + 700 \,{\left (x e + d\right )} a^{3} b^{4} d^{3} e^{3} - 105 \, a^{3} b^{4} d^{4} e^{3} + 1575 \,{\left (x e + d\right )}^{2} a^{4} b^{3} d e^{4} - 525 \,{\left (x e + d\right )} a^{4} b^{3} d^{2} e^{4} + 105 \, a^{4} b^{3} d^{3} e^{4} - 315 \,{\left (x e + d\right )}^{2} a^{5} b^{2} e^{5} + 210 \,{\left (x e + d\right )} a^{5} b^{2} d e^{5} - 63 \, a^{5} b^{2} d^{2} e^{5} - 35 \,{\left (x e + d\right )} a^{6} b e^{6} + 21 \, a^{6} b d e^{6} - 3 \, a^{7} e^{7}\right )} e^{\left (-8\right )}}{15 \,{\left (x e + d\right )}^{\frac{5}{2}}} \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)^3/(e*x+d)^(7/2),x, algorithm="giac")

[Out]

2/45*(5*(x*e + d)^(9/2)*b^7*e^64 - 45*(x*e + d)^(7/2)*b^7*d*e^64 + 189*(x*e + d)^(5/2)*b^7*d^2*e^64 - 525*(x*e
+ d)^(3/2)*b^7*d^3*e^64 + 1575*sqrt(x*e + d)*b^7*d^4*e^64 + 45*(x*e + d)^(7/2)*a*b^6*e^65 - 378*(x*e + d)^(5/
2)*a*b^6*d*e^65 + 1575*(x*e + d)^(3/2)*a*b^6*d^2*e^65 - 6300*sqrt(x*e + d)*a*b^6*d^3*e^65 + 189*(x*e + d)^(5/2
)*a^2*b^5*e^66 - 1575*(x*e + d)^(3/2)*a^2*b^5*d*e^66 + 9450*sqrt(x*e + d)*a^2*b^5*d^2*e^66 + 525*(x*e + d)^(3/
2)*a^3*b^4*e^67 - 6300*sqrt(x*e + d)*a^3*b^4*d*e^67 + 1575*sqrt(x*e + d)*a^4*b^3*e^68)*e^(-72) + 2/15*(315*(x*
e + d)^2*b^7*d^5 - 35*(x*e + d)*b^7*d^6 + 3*b^7*d^7 - 1575*(x*e + d)^2*a*b^6*d^4*e + 210*(x*e + d)*a*b^6*d^5*e
- 21*a*b^6*d^6*e + 3150*(x*e + d)^2*a^2*b^5*d^3*e^2 - 525*(x*e + d)*a^2*b^5*d^4*e^2 + 63*a^2*b^5*d^5*e^2 - 31
50*(x*e + d)^2*a^3*b^4*d^2*e^3 + 700*(x*e + d)*a^3*b^4*d^3*e^3 - 105*a^3*b^4*d^4*e^3 + 1575*(x*e + d)^2*a^4*b^
3*d*e^4 - 525*(x*e + d)*a^4*b^3*d^2*e^4 + 105*a^4*b^3*d^3*e^4 - 315*(x*e + d)^2*a^5*b^2*e^5 + 210*(x*e + d)*a^
5*b^2*d*e^5 - 63*a^5*b^2*d^2*e^5 - 35*(x*e + d)*a^6*b*e^6 + 21*a^6*b*d*e^6 - 3*a^7*e^7)*e^(-8)/(x*e + d)^(5/2)