3.21.65 \(\int \frac {(a+b x) (a^2+2 a b x+b^2 x^2)^3}{(d+e x)^{7/2}} \, dx\) [2065]

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

[Out]

2/5*(-a*e+b*d)^7/e^8/(e*x+d)^(5/2)-14/3*b*(-a*e+b*d)^6/e^8/(e*x+d)^(3/2)-70/3*b^4*(-a*e+b*d)^3*(e*x+d)^(3/2)/e
^8+42/5*b^5*(-a*e+b*d)^2*(e*x+d)^(5/2)/e^8-2*b^6*(-a*e+b*d)*(e*x+d)^(7/2)/e^8+2/9*b^7*(e*x+d)^(9/2)/e^8+42*b^2
*(-a*e+b*d)^5/e^8/(e*x+d)^(1/2)+70*b^3*(-a*e+b*d)^4*(e*x+d)^(1/2)/e^8

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Rubi [A]
time = 0.05, antiderivative size = 210, normalized size of antiderivative = 1.00, 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, 45} \begin {gather*} -\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} \end {gather*}

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 45

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*}

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Mathematica [A]
time = 0.18, size = 376, normalized size = 1.79 \begin {gather*} \frac {2 \left (-9 a^7 e^7-21 a^6 b e^6 (2 d+5 e x)-63 a^5 b^2 e^5 \left (8 d^2+20 d e x+15 e^2 x^2\right )+315 a^4 b^3 e^4 \left (16 d^3+40 d^2 e x+30 d e^2 x^2+5 e^3 x^3\right )-105 a^3 b^4 e^3 \left (128 d^4+320 d^3 e x+240 d^2 e^2 x^2+40 d e^3 x^3-5 e^4 x^4\right )+63 a^2 b^5 e^2 \left (256 d^5+640 d^4 e x+480 d^3 e^2 x^2+80 d^2 e^3 x^3-10 d e^4 x^4+3 e^5 x^5\right )-9 a b^6 e \left (1024 d^6+2560 d^5 e x+1920 d^4 e^2 x^2+320 d^3 e^3 x^3-40 d^2 e^4 x^4+12 d e^5 x^5-5 e^6 x^6\right )+b^7 \left (2048 d^7+5120 d^6 e x+3840 d^5 e^2 x^2+640 d^4 e^3 x^3-80 d^3 e^4 x^4+24 d^2 e^5 x^5-10 d e^6 x^6+5 e^7 x^7\right )\right )}{45 e^8 (d+e x)^{5/2}} \end {gather*}

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*a^7*e^7 - 21*a^6*b*e^6*(2*d + 5*e*x) - 63*a^5*b^2*e^5*(8*d^2 + 20*d*e*x + 15*e^2*x^2) + 315*a^4*b^3*e^4
*(16*d^3 + 40*d^2*e*x + 30*d*e^2*x^2 + 5*e^3*x^3) - 105*a^3*b^4*e^3*(128*d^4 + 320*d^3*e*x + 240*d^2*e^2*x^2 +
40*d*e^3*x^3 - 5*e^4*x^4) + 63*a^2*b^5*e^2*(256*d^5 + 640*d^4*e*x + 480*d^3*e^2*x^2 + 80*d^2*e^3*x^3 - 10*d*e
^4*x^4 + 3*e^5*x^5) - 9*a*b^6*e*(1024*d^6 + 2560*d^5*e*x + 1920*d^4*e^2*x^2 + 320*d^3*e^3*x^3 - 40*d^2*e^4*x^4
+ 12*d*e^5*x^5 - 5*e^6*x^6) + b^7*(2048*d^7 + 5120*d^6*e*x + 3840*d^5*e^2*x^2 + 640*d^4*e^3*x^3 - 80*d^3*e^4*
x^4 + 24*d^2*e^5*x^5 - 10*d*e^6*x^6 + 5*e^7*x^7)))/(45*e^8*(d + e*x)^(5/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(515\) vs. \(2(184)=368\).
time = 0.07, size = 516, normalized size = 2.46 Too large to display

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,method=_RETURNVERBOSE)

[Out]

2/e^8*(1/9*b^7*(e*x+d)^(9/2)+a*b^6*e*(e*x+d)^(7/2)-b^7*d*(e*x+d)^(7/2)+21/5*a^2*b^5*e^2*(e*x+d)^(5/2)-42/5*a*b
^6*d*e*(e*x+d)^(5/2)+21/5*b^7*d^2*(e*x+d)^(5/2)+35/3*a^3*b^4*e^3*(e*x+d)^(3/2)-35*a^2*b^5*d*e^2*(e*x+d)^(3/2)+
35*a*b^6*d^2*e*(e*x+d)^(3/2)-35/3*b^7*d^3*(e*x+d)^(3/2)+35*a^4*b^3*e^4*(e*x+d)^(1/2)-140*a^3*b^4*d*e^3*(e*x+d)
^(1/2)+210*a^2*b^5*d^2*e^2*(e*x+d)^(1/2)-140*a*b^6*d^3*e*(e*x+d)^(1/2)+35*b^7*d^4*(e*x+d)^(1/2)-21*b^2*(a^5*e^
5-5*a^4*b*d*e^4+10*a^3*b^2*d^2*e^3-10*a^2*b^3*d^3*e^2+5*a*b^4*d^4*e-b^5*d^5)/(e*x+d)^(1/2)-7/3*b*(a^6*e^6-6*a^
5*b*d*e^5+15*a^4*b^2*d^2*e^4-20*a^3*b^3*d^3*e^3+15*a^2*b^4*d^4*e^2-6*a*b^5*d^5*e+b^6*d^6)/(e*x+d)^(3/2)-1/5*(a
^7*e^7-7*a^6*b*d*e^6+21*a^5*b^2*d^2*e^5-35*a^4*b^3*d^3*e^4+35*a^3*b^4*d^4*e^3-21*a^2*b^5*d^5*e^2+7*a*b^6*d^6*e
-b^7*d^7)/(e*x+d)^(5/2))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 454 vs. \(2 (191) = 382\).
time = 0.32, size = 454, normalized size = 2.16 \begin {gather*} \frac {2}{45} \, {\left ({\left (5 \, {\left (x e + d\right )}^{\frac {9}{2}} b^{7} - 45 \, {\left (b^{7} d - a b^{6} e\right )} {\left (x e + 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 (x e + 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 (x e + 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 {x e + d}\right )} e^{\left (-7\right )} + \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 (x e + 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 (x e + d\right )}\right )} e^{\left (-7\right )}}{{\left (x e + d\right )}^{\frac {5}{2}}}\right )} e^{\left (-1\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)^3/(e*x+d)^(7/2),x, algorithm="maxima")

[Out]

2/45*((5*(x*e + d)^(9/2)*b^7 - 45*(b^7*d - a*b^6*e)*(x*e + d)^(7/2) + 189*(b^7*d^2 - 2*a*b^6*d*e + a^2*b^5*e^2
)*(x*e + 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)*(x*e + 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(x*e + d))*e^(-7) + 3*(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*(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)*(x*e + 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)*(x*e + d))*e^(-7)/(x*e + d)^(5/2))*e^(-1)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 456 vs. \(2 (191) = 382\).
time = 3.47, size = 456, normalized size = 2.17 \begin {gather*} \frac {2 \, {\left (2048 \, b^{7} d^{7} + {\left (5 \, b^{7} x^{7} + 45 \, a b^{6} x^{6} + 189 \, a^{2} b^{5} x^{5} + 525 \, a^{3} b^{4} x^{4} + 1575 \, a^{4} b^{3} x^{3} - 945 \, a^{5} b^{2} x^{2} - 105 \, a^{6} b x - 9 \, a^{7}\right )} e^{7} - 2 \, {\left (5 \, b^{7} d x^{6} + 54 \, a b^{6} d x^{5} + 315 \, a^{2} b^{5} d x^{4} + 2100 \, a^{3} b^{4} d x^{3} - 4725 \, a^{4} b^{3} d x^{2} + 630 \, a^{5} b^{2} d x + 21 \, a^{6} b d\right )} e^{6} + 24 \, {\left (b^{7} d^{2} x^{5} + 15 \, a b^{6} d^{2} x^{4} + 210 \, a^{2} b^{5} d^{2} x^{3} - 1050 \, a^{3} b^{4} d^{2} x^{2} + 525 \, a^{4} b^{3} d^{2} x - 21 \, a^{5} b^{2} d^{2}\right )} e^{5} - 80 \, {\left (b^{7} d^{3} x^{4} + 36 \, a b^{6} d^{3} x^{3} - 378 \, a^{2} b^{5} d^{3} x^{2} + 420 \, a^{3} b^{4} d^{3} x - 63 \, a^{4} b^{3} d^{3}\right )} e^{4} + 640 \, {\left (b^{7} d^{4} x^{3} - 27 \, a b^{6} d^{4} x^{2} + 63 \, a^{2} b^{5} d^{4} x - 21 \, a^{3} b^{4} d^{4}\right )} e^{3} + 768 \, {\left (5 \, b^{7} d^{5} x^{2} - 30 \, a b^{6} d^{5} x + 21 \, a^{2} b^{5} d^{5}\right )} e^{2} + 1024 \, {\left (5 \, b^{7} d^{6} x - 9 \, a b^{6} d^{6}\right )} e\right )} \sqrt {x e + d}}{45 \, {\left (x^{3} e^{11} + 3 \, d x^{2} e^{10} + 3 \, d^{2} x e^{9} + d^{3} e^{8}\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)^3/(e*x+d)^(7/2),x, algorithm="fricas")

[Out]

2/45*(2048*b^7*d^7 + (5*b^7*x^7 + 45*a*b^6*x^6 + 189*a^2*b^5*x^5 + 525*a^3*b^4*x^4 + 1575*a^4*b^3*x^3 - 945*a^
5*b^2*x^2 - 105*a^6*b*x - 9*a^7)*e^7 - 2*(5*b^7*d*x^6 + 54*a*b^6*d*x^5 + 315*a^2*b^5*d*x^4 + 2100*a^3*b^4*d*x^
3 - 4725*a^4*b^3*d*x^2 + 630*a^5*b^2*d*x + 21*a^6*b*d)*e^6 + 24*(b^7*d^2*x^5 + 15*a*b^6*d^2*x^4 + 210*a^2*b^5*
d^2*x^3 - 1050*a^3*b^4*d^2*x^2 + 525*a^4*b^3*d^2*x - 21*a^5*b^2*d^2)*e^5 - 80*(b^7*d^3*x^4 + 36*a*b^6*d^3*x^3
- 378*a^2*b^5*d^3*x^2 + 420*a^3*b^4*d^3*x - 63*a^4*b^3*d^3)*e^4 + 640*(b^7*d^4*x^3 - 27*a*b^6*d^4*x^2 + 63*a^2
*b^5*d^4*x - 21*a^3*b^4*d^4)*e^3 + 768*(5*b^7*d^5*x^2 - 30*a*b^6*d^5*x + 21*a^2*b^5*d^5)*e^2 + 1024*(5*b^7*d^6
*x - 9*a*b^6*d^6)*e)*sqrt(x*e + d)/(x^3*e^11 + 3*d*x^2*e^10 + 3*d^2*x*e^9 + d^3*e^8)

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Sympy [A]
time = 72.04, size = 298, normalized size = 1.42 \begin {gather*} \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}} \cdot \left (14 a b^{6} e - 14 b^{7} d\right )}{7 e^{8}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \cdot \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}} \cdot \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 {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)**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] Leaf count of result is larger than twice the leaf count of optimal. 608 vs. \(2 (191) = 382\).
time = 1.23, size = 608, normalized size = 2.90 \begin {gather*} \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 {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)^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)

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Mupad [B]
time = 0.08, size = 388, normalized size = 1.85 \begin {gather*} \frac {2\,b^7\,{\left (d+e\,x\right )}^{9/2}}{9\,e^8}-\frac {\left (14\,b^7\,d-14\,a\,b^6\,e\right )\,{\left (d+e\,x\right )}^{7/2}}{7\,e^8}+\frac {{\left (d+e\,x\right )}^2\,\left (-42\,a^5\,b^2\,e^5+210\,a^4\,b^3\,d\,e^4-420\,a^3\,b^4\,d^2\,e^3+420\,a^2\,b^5\,d^3\,e^2-210\,a\,b^6\,d^4\,e+42\,b^7\,d^5\right )-\left (d+e\,x\right )\,\left (\frac {14\,a^6\,b\,e^6}{3}-28\,a^5\,b^2\,d\,e^5+70\,a^4\,b^3\,d^2\,e^4-\frac {280\,a^3\,b^4\,d^3\,e^3}{3}+70\,a^2\,b^5\,d^4\,e^2-28\,a\,b^6\,d^5\,e+\frac {14\,b^7\,d^6}{3}\right )-\frac {2\,a^7\,e^7}{5}+\frac {2\,b^7\,d^7}{5}+\frac {42\,a^2\,b^5\,d^5\,e^2}{5}-14\,a^3\,b^4\,d^4\,e^3+14\,a^4\,b^3\,d^3\,e^4-\frac {42\,a^5\,b^2\,d^2\,e^5}{5}-\frac {14\,a\,b^6\,d^6\,e}{5}+\frac {14\,a^6\,b\,d\,e^6}{5}}{e^8\,{\left (d+e\,x\right )}^{5/2}}+\frac {70\,b^3\,{\left (a\,e-b\,d\right )}^4\,\sqrt {d+e\,x}}{e^8}+\frac {70\,b^4\,{\left (a\,e-b\,d\right )}^3\,{\left (d+e\,x\right )}^{3/2}}{3\,e^8}+\frac {42\,b^5\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{5/2}}{5\,e^8} \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)^3)/(d + e*x)^(7/2),x)

[Out]

(2*b^7*(d + e*x)^(9/2))/(9*e^8) - ((14*b^7*d - 14*a*b^6*e)*(d + e*x)^(7/2))/(7*e^8) + ((d + e*x)^2*(42*b^7*d^5
- 42*a^5*b^2*e^5 + 210*a^4*b^3*d*e^4 + 420*a^2*b^5*d^3*e^2 - 420*a^3*b^4*d^2*e^3 - 210*a*b^6*d^4*e) - (d + e*
x)*((14*b^7*d^6)/3 + (14*a^6*b*e^6)/3 - 28*a^5*b^2*d*e^5 + 70*a^2*b^5*d^4*e^2 - (280*a^3*b^4*d^3*e^3)/3 + 70*a
^4*b^3*d^2*e^4 - 28*a*b^6*d^5*e) - (2*a^7*e^7)/5 + (2*b^7*d^7)/5 + (42*a^2*b^5*d^5*e^2)/5 - 14*a^3*b^4*d^4*e^3
+ 14*a^4*b^3*d^3*e^4 - (42*a^5*b^2*d^2*e^5)/5 - (14*a*b^6*d^6*e)/5 + (14*a^6*b*d*e^6)/5)/(e^8*(d + e*x)^(5/2)
) + (70*b^3*(a*e - b*d)^4*(d + e*x)^(1/2))/e^8 + (70*b^4*(a*e - b*d)^3*(d + e*x)^(3/2))/(3*e^8) + (42*b^5*(a*e
- b*d)^2*(d + e*x)^(5/2))/(5*e^8)

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