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

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

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Rubi [A]  time = 0.08, 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, 43} \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 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*}

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Mathematica [A]  time = 0.10, size = 167, normalized size = 0.80 \begin {gather*} \frac {2 \left (-45 b^6 (d+e x)^6 (b d-a e)+189 b^5 (d+e x)^5 (b d-a e)^2-525 b^4 (d+e x)^4 (b d-a e)^3+1575 b^3 (d+e x)^3 (b d-a e)^4+945 b^2 (d+e x)^2 (b d-a e)^5-105 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}} \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*(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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IntegrateAlgebraic [B]  time = 0.12, size = 582, normalized size = 2.77 \begin {gather*} \frac {2 \left (-9 a^7 e^7-105 a^6 b e^6 (d+e x)+63 a^6 b d e^6-189 a^5 b^2 d^2 e^5-945 a^5 b^2 e^5 (d+e x)^2+630 a^5 b^2 d e^5 (d+e x)+315 a^4 b^3 d^3 e^4-1575 a^4 b^3 d^2 e^4 (d+e x)+1575 a^4 b^3 e^4 (d+e x)^3+4725 a^4 b^3 d e^4 (d+e x)^2-315 a^3 b^4 d^4 e^3+2100 a^3 b^4 d^3 e^3 (d+e x)-9450 a^3 b^4 d^2 e^3 (d+e x)^2+525 a^3 b^4 e^3 (d+e x)^4-6300 a^3 b^4 d e^3 (d+e x)^3+189 a^2 b^5 d^5 e^2-1575 a^2 b^5 d^4 e^2 (d+e x)+9450 a^2 b^5 d^3 e^2 (d+e x)^2+9450 a^2 b^5 d^2 e^2 (d+e x)^3+189 a^2 b^5 e^2 (d+e x)^5-1575 a^2 b^5 d e^2 (d+e x)^4-63 a b^6 d^6 e+630 a b^6 d^5 e (d+e x)-4725 a b^6 d^4 e (d+e x)^2-6300 a b^6 d^3 e (d+e x)^3+1575 a b^6 d^2 e (d+e x)^4+45 a b^6 e (d+e x)^6-378 a b^6 d e (d+e x)^5+9 b^7 d^7-105 b^7 d^6 (d+e x)+945 b^7 d^5 (d+e x)^2+1575 b^7 d^4 (d+e x)^3-525 b^7 d^3 (d+e x)^4+189 b^7 d^2 (d+e x)^5+5 b^7 (d+e x)^7-45 b^7 d (d+e x)^6\right )}{45 e^8 (d+e x)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(9*b^7*d^7 - 63*a*b^6*d^6*e + 189*a^2*b^5*d^5*e^2 - 315*a^3*b^4*d^4*e^3 + 315*a^4*b^3*d^3*e^4 - 189*a^5*b^2
*d^2*e^5 + 63*a^6*b*d*e^6 - 9*a^7*e^7 - 105*b^7*d^6*(d + e*x) + 630*a*b^6*d^5*e*(d + e*x) - 1575*a^2*b^5*d^4*e
^2*(d + e*x) + 2100*a^3*b^4*d^3*e^3*(d + e*x) - 1575*a^4*b^3*d^2*e^4*(d + e*x) + 630*a^5*b^2*d*e^5*(d + e*x) -
105*a^6*b*e^6*(d + e*x) + 945*b^7*d^5*(d + e*x)^2 - 4725*a*b^6*d^4*e*(d + e*x)^2 + 9450*a^2*b^5*d^3*e^2*(d +
e*x)^2 - 9450*a^3*b^4*d^2*e^3*(d + e*x)^2 + 4725*a^4*b^3*d*e^4*(d + e*x)^2 - 945*a^5*b^2*e^5*(d + e*x)^2 + 157
5*b^7*d^4*(d + e*x)^3 - 6300*a*b^6*d^3*e*(d + e*x)^3 + 9450*a^2*b^5*d^2*e^2*(d + e*x)^3 - 6300*a^3*b^4*d*e^3*(
d + e*x)^3 + 1575*a^4*b^3*e^4*(d + e*x)^3 - 525*b^7*d^3*(d + e*x)^4 + 1575*a*b^6*d^2*e*(d + e*x)^4 - 1575*a^2*
b^5*d*e^2*(d + e*x)^4 + 525*a^3*b^4*e^3*(d + e*x)^4 + 189*b^7*d^2*(d + e*x)^5 - 378*a*b^6*d*e*(d + e*x)^5 + 18
9*a^2*b^5*e^2*(d + e*x)^5 - 45*b^7*d*(d + e*x)^6 + 45*a*b^6*e*(d + e*x)^6 + 5*b^7*(d + e*x)^7))/(45*e^8*(d + e
*x)^(5/2))

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fricas [B]  time = 0.43, size = 496, normalized size = 2.36 \begin {gather*} \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 {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*(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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giac [B]  time = 0.23, 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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maple [B]  time = 0.05, size = 498, normalized size = 2.37 \begin {gather*} -\frac {2 \left (-5 b^{7} x^{7} e^{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}+4200 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}\right )}{45 \left (e x +d \right )^{\frac {5}{2}} e^{8}} \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)^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 = 0.55, size = 463, normalized size = 2.20 \begin {gather*} \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 {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*(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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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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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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]

Timed out

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