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

Optimal. Leaf size=206 \[ -\frac {14 b^6 (d+e x)^{11/2} (b d-a e)}{11 e^8}+\frac {14 b^5 (d+e x)^{9/2} (b d-a e)^2}{3 e^8}-\frac {10 b^4 (d+e x)^{7/2} (b d-a e)^3}{e^8}+\frac {14 b^3 (d+e x)^{5/2} (b d-a e)^4}{e^8}-\frac {14 b^2 (d+e x)^{3/2} (b d-a e)^5}{e^8}+\frac {14 b \sqrt {d+e x} (b d-a e)^6}{e^8}+\frac {2 (b d-a e)^7}{e^8 \sqrt {d+e x}}+\frac {2 b^7 (d+e x)^{13/2}}{13 e^8} \]

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Rubi [A]  time = 0.08, antiderivative size = 206, 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 {14 b^6 (d+e x)^{11/2} (b d-a e)}{11 e^8}+\frac {14 b^5 (d+e x)^{9/2} (b d-a e)^2}{3 e^8}-\frac {10 b^4 (d+e x)^{7/2} (b d-a e)^3}{e^8}+\frac {14 b^3 (d+e x)^{5/2} (b d-a e)^4}{e^8}-\frac {14 b^2 (d+e x)^{3/2} (b d-a e)^5}{e^8}+\frac {14 b \sqrt {d+e x} (b d-a e)^6}{e^8}+\frac {2 (b d-a e)^7}{e^8 \sqrt {d+e x}}+\frac {2 b^7 (d+e x)^{13/2}}{13 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)^(3/2),x]

[Out]

(2*(b*d - a*e)^7)/(e^8*Sqrt[d + e*x]) + (14*b*(b*d - a*e)^6*Sqrt[d + e*x])/e^8 - (14*b^2*(b*d - a*e)^5*(d + e*
x)^(3/2))/e^8 + (14*b^3*(b*d - a*e)^4*(d + e*x)^(5/2))/e^8 - (10*b^4*(b*d - a*e)^3*(d + e*x)^(7/2))/e^8 + (14*
b^5*(b*d - a*e)^2*(d + e*x)^(9/2))/(3*e^8) - (14*b^6*(b*d - a*e)*(d + e*x)^(11/2))/(11*e^8) + (2*b^7*(d + e*x)
^(13/2))/(13*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)^{3/2}} \, dx &=\int \frac {(a+b x)^7}{(d+e x)^{3/2}} \, dx\\ &=\int \left (\frac {(-b d+a e)^7}{e^7 (d+e x)^{3/2}}+\frac {7 b (b d-a e)^6}{e^7 \sqrt {d+e x}}-\frac {21 b^2 (b d-a e)^5 \sqrt {d+e x}}{e^7}+\frac {35 b^3 (b d-a e)^4 (d+e x)^{3/2}}{e^7}-\frac {35 b^4 (b d-a e)^3 (d+e x)^{5/2}}{e^7}+\frac {21 b^5 (b d-a e)^2 (d+e x)^{7/2}}{e^7}-\frac {7 b^6 (b d-a e) (d+e x)^{9/2}}{e^7}+\frac {b^7 (d+e x)^{11/2}}{e^7}\right ) \, dx\\ &=\frac {2 (b d-a e)^7}{e^8 \sqrt {d+e x}}+\frac {14 b (b d-a e)^6 \sqrt {d+e x}}{e^8}-\frac {14 b^2 (b d-a e)^5 (d+e x)^{3/2}}{e^8}+\frac {14 b^3 (b d-a e)^4 (d+e x)^{5/2}}{e^8}-\frac {10 b^4 (b d-a e)^3 (d+e x)^{7/2}}{e^8}+\frac {14 b^5 (b d-a e)^2 (d+e x)^{9/2}}{3 e^8}-\frac {14 b^6 (b d-a e) (d+e x)^{11/2}}{11 e^8}+\frac {2 b^7 (d+e x)^{13/2}}{13 e^8}\\ \end {align*}

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Mathematica [A]  time = 0.10, size = 167, normalized size = 0.81 \begin {gather*} \frac {2 \left (-273 b^6 (d+e x)^6 (b d-a e)+1001 b^5 (d+e x)^5 (b d-a e)^2-2145 b^4 (d+e x)^4 (b d-a e)^3+3003 b^3 (d+e x)^3 (b d-a e)^4-3003 b^2 (d+e x)^2 (b d-a e)^5+3003 b (d+e x) (b d-a e)^6+429 (b d-a e)^7+33 b^7 (d+e x)^7\right )}{429 e^8 \sqrt {d+e x}} \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)^(3/2),x]

[Out]

(2*(429*(b*d - a*e)^7 + 3003*b*(b*d - a*e)^6*(d + e*x) - 3003*b^2*(b*d - a*e)^5*(d + e*x)^2 + 3003*b^3*(b*d -
a*e)^4*(d + e*x)^3 - 2145*b^4*(b*d - a*e)^3*(d + e*x)^4 + 1001*b^5*(b*d - a*e)^2*(d + e*x)^5 - 273*b^6*(b*d -
a*e)*(d + e*x)^6 + 33*b^7*(d + e*x)^7))/(429*e^8*Sqrt[d + e*x])

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IntegrateAlgebraic [B]  time = 0.11, size = 582, normalized size = 2.83 \begin {gather*} \frac {2 \left (-429 a^7 e^7+3003 a^6 b e^6 (d+e x)+3003 a^6 b d e^6-9009 a^5 b^2 d^2 e^5+3003 a^5 b^2 e^5 (d+e x)^2-18018 a^5 b^2 d e^5 (d+e x)+15015 a^4 b^3 d^3 e^4+45045 a^4 b^3 d^2 e^4 (d+e x)+3003 a^4 b^3 e^4 (d+e x)^3-15015 a^4 b^3 d e^4 (d+e x)^2-15015 a^3 b^4 d^4 e^3-60060 a^3 b^4 d^3 e^3 (d+e x)+30030 a^3 b^4 d^2 e^3 (d+e x)^2+2145 a^3 b^4 e^3 (d+e x)^4-12012 a^3 b^4 d e^3 (d+e x)^3+9009 a^2 b^5 d^5 e^2+45045 a^2 b^5 d^4 e^2 (d+e x)-30030 a^2 b^5 d^3 e^2 (d+e x)^2+18018 a^2 b^5 d^2 e^2 (d+e x)^3+1001 a^2 b^5 e^2 (d+e x)^5-6435 a^2 b^5 d e^2 (d+e x)^4-3003 a b^6 d^6 e-18018 a b^6 d^5 e (d+e x)+15015 a b^6 d^4 e (d+e x)^2-12012 a b^6 d^3 e (d+e x)^3+6435 a b^6 d^2 e (d+e x)^4+273 a b^6 e (d+e x)^6-2002 a b^6 d e (d+e x)^5+429 b^7 d^7+3003 b^7 d^6 (d+e x)-3003 b^7 d^5 (d+e x)^2+3003 b^7 d^4 (d+e x)^3-2145 b^7 d^3 (d+e x)^4+1001 b^7 d^2 (d+e x)^5+33 b^7 (d+e x)^7-273 b^7 d (d+e x)^6\right )}{429 e^8 \sqrt {d+e x}} \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)^(3/2),x]

[Out]

(2*(429*b^7*d^7 - 3003*a*b^6*d^6*e + 9009*a^2*b^5*d^5*e^2 - 15015*a^3*b^4*d^4*e^3 + 15015*a^4*b^3*d^3*e^4 - 90
09*a^5*b^2*d^2*e^5 + 3003*a^6*b*d*e^6 - 429*a^7*e^7 + 3003*b^7*d^6*(d + e*x) - 18018*a*b^6*d^5*e*(d + e*x) + 4
5045*a^2*b^5*d^4*e^2*(d + e*x) - 60060*a^3*b^4*d^3*e^3*(d + e*x) + 45045*a^4*b^3*d^2*e^4*(d + e*x) - 18018*a^5
*b^2*d*e^5*(d + e*x) + 3003*a^6*b*e^6*(d + e*x) - 3003*b^7*d^5*(d + e*x)^2 + 15015*a*b^6*d^4*e*(d + e*x)^2 - 3
0030*a^2*b^5*d^3*e^2*(d + e*x)^2 + 30030*a^3*b^4*d^2*e^3*(d + e*x)^2 - 15015*a^4*b^3*d*e^4*(d + e*x)^2 + 3003*
a^5*b^2*e^5*(d + e*x)^2 + 3003*b^7*d^4*(d + e*x)^3 - 12012*a*b^6*d^3*e*(d + e*x)^3 + 18018*a^2*b^5*d^2*e^2*(d
+ e*x)^3 - 12012*a^3*b^4*d*e^3*(d + e*x)^3 + 3003*a^4*b^3*e^4*(d + e*x)^3 - 2145*b^7*d^3*(d + e*x)^4 + 6435*a*
b^6*d^2*e*(d + e*x)^4 - 6435*a^2*b^5*d*e^2*(d + e*x)^4 + 2145*a^3*b^4*e^3*(d + e*x)^4 + 1001*b^7*d^2*(d + e*x)
^5 - 2002*a*b^6*d*e*(d + e*x)^5 + 1001*a^2*b^5*e^2*(d + e*x)^5 - 273*b^7*d*(d + e*x)^6 + 273*a*b^6*e*(d + e*x)
^6 + 33*b^7*(d + e*x)^7))/(429*e^8*Sqrt[d + e*x])

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fricas [B]  time = 0.43, size = 472, normalized size = 2.29 \begin {gather*} \frac {2 \, {\left (33 \, b^{7} e^{7} x^{7} + 2048 \, b^{7} d^{7} - 13312 \, a b^{6} d^{6} e + 36608 \, a^{2} b^{5} d^{5} e^{2} - 54912 \, a^{3} b^{4} d^{4} e^{3} + 48048 \, a^{4} b^{3} d^{3} e^{4} - 24024 \, a^{5} b^{2} d^{2} e^{5} + 6006 \, a^{6} b d e^{6} - 429 \, a^{7} e^{7} - 21 \, {\left (2 \, b^{7} d e^{6} - 13 \, a b^{6} e^{7}\right )} x^{6} + 7 \, {\left (8 \, b^{7} d^{2} e^{5} - 52 \, a b^{6} d e^{6} + 143 \, a^{2} b^{5} e^{7}\right )} x^{5} - 5 \, {\left (16 \, b^{7} d^{3} e^{4} - 104 \, a b^{6} d^{2} e^{5} + 286 \, a^{2} b^{5} d e^{6} - 429 \, a^{3} b^{4} e^{7}\right )} x^{4} + {\left (128 \, b^{7} d^{4} e^{3} - 832 \, a b^{6} d^{3} e^{4} + 2288 \, a^{2} b^{5} d^{2} e^{5} - 3432 \, a^{3} b^{4} d e^{6} + 3003 \, a^{4} b^{3} e^{7}\right )} x^{3} - {\left (256 \, b^{7} d^{5} e^{2} - 1664 \, a b^{6} d^{4} e^{3} + 4576 \, a^{2} b^{5} d^{3} e^{4} - 6864 \, a^{3} b^{4} d^{2} e^{5} + 6006 \, a^{4} b^{3} d e^{6} - 3003 \, a^{5} b^{2} e^{7}\right )} x^{2} + {\left (1024 \, b^{7} d^{6} e - 6656 \, a b^{6} d^{5} e^{2} + 18304 \, a^{2} b^{5} d^{4} e^{3} - 27456 \, a^{3} b^{4} d^{3} e^{4} + 24024 \, a^{4} b^{3} d^{2} e^{5} - 12012 \, a^{5} b^{2} d e^{6} + 3003 \, a^{6} b e^{7}\right )} x\right )} \sqrt {e x + d}}{429 \, {\left (e^{9} x + d 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)^(3/2),x, algorithm="fricas")

[Out]

2/429*(33*b^7*e^7*x^7 + 2048*b^7*d^7 - 13312*a*b^6*d^6*e + 36608*a^2*b^5*d^5*e^2 - 54912*a^3*b^4*d^4*e^3 + 480
48*a^4*b^3*d^3*e^4 - 24024*a^5*b^2*d^2*e^5 + 6006*a^6*b*d*e^6 - 429*a^7*e^7 - 21*(2*b^7*d*e^6 - 13*a*b^6*e^7)*
x^6 + 7*(8*b^7*d^2*e^5 - 52*a*b^6*d*e^6 + 143*a^2*b^5*e^7)*x^5 - 5*(16*b^7*d^3*e^4 - 104*a*b^6*d^2*e^5 + 286*a
^2*b^5*d*e^6 - 429*a^3*b^4*e^7)*x^4 + (128*b^7*d^4*e^3 - 832*a*b^6*d^3*e^4 + 2288*a^2*b^5*d^2*e^5 - 3432*a^3*b
^4*d*e^6 + 3003*a^4*b^3*e^7)*x^3 - (256*b^7*d^5*e^2 - 1664*a*b^6*d^4*e^3 + 4576*a^2*b^5*d^3*e^4 - 6864*a^3*b^4
*d^2*e^5 + 6006*a^4*b^3*d*e^6 - 3003*a^5*b^2*e^7)*x^2 + (1024*b^7*d^6*e - 6656*a*b^6*d^5*e^2 + 18304*a^2*b^5*d
^4*e^3 - 27456*a^3*b^4*d^3*e^4 + 24024*a^4*b^3*d^2*e^5 - 12012*a^5*b^2*d*e^6 + 3003*a^6*b*e^7)*x)*sqrt(e*x + d
)/(e^9*x + d*e^8)

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giac [B]  time = 0.23, size = 625, normalized size = 3.03 \begin {gather*} \frac {2}{429} \, {\left (33 \, {\left (x e + d\right )}^{\frac {13}{2}} b^{7} e^{96} - 273 \, {\left (x e + d\right )}^{\frac {11}{2}} b^{7} d e^{96} + 1001 \, {\left (x e + d\right )}^{\frac {9}{2}} b^{7} d^{2} e^{96} - 2145 \, {\left (x e + d\right )}^{\frac {7}{2}} b^{7} d^{3} e^{96} + 3003 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{7} d^{4} e^{96} - 3003 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{7} d^{5} e^{96} + 3003 \, \sqrt {x e + d} b^{7} d^{6} e^{96} + 273 \, {\left (x e + d\right )}^{\frac {11}{2}} a b^{6} e^{97} - 2002 \, {\left (x e + d\right )}^{\frac {9}{2}} a b^{6} d e^{97} + 6435 \, {\left (x e + d\right )}^{\frac {7}{2}} a b^{6} d^{2} e^{97} - 12012 \, {\left (x e + d\right )}^{\frac {5}{2}} a b^{6} d^{3} e^{97} + 15015 \, {\left (x e + d\right )}^{\frac {3}{2}} a b^{6} d^{4} e^{97} - 18018 \, \sqrt {x e + d} a b^{6} d^{5} e^{97} + 1001 \, {\left (x e + d\right )}^{\frac {9}{2}} a^{2} b^{5} e^{98} - 6435 \, {\left (x e + d\right )}^{\frac {7}{2}} a^{2} b^{5} d e^{98} + 18018 \, {\left (x e + d\right )}^{\frac {5}{2}} a^{2} b^{5} d^{2} e^{98} - 30030 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{2} b^{5} d^{3} e^{98} + 45045 \, \sqrt {x e + d} a^{2} b^{5} d^{4} e^{98} + 2145 \, {\left (x e + d\right )}^{\frac {7}{2}} a^{3} b^{4} e^{99} - 12012 \, {\left (x e + d\right )}^{\frac {5}{2}} a^{3} b^{4} d e^{99} + 30030 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{3} b^{4} d^{2} e^{99} - 60060 \, \sqrt {x e + d} a^{3} b^{4} d^{3} e^{99} + 3003 \, {\left (x e + d\right )}^{\frac {5}{2}} a^{4} b^{3} e^{100} - 15015 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{4} b^{3} d e^{100} + 45045 \, \sqrt {x e + d} a^{4} b^{3} d^{2} e^{100} + 3003 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{5} b^{2} e^{101} - 18018 \, \sqrt {x e + d} a^{5} b^{2} d e^{101} + 3003 \, \sqrt {x e + d} a^{6} b e^{102}\right )} e^{\left (-104\right )} + \frac {2 \, {\left (b^{7} d^{7} - 7 \, a b^{6} d^{6} e + 21 \, a^{2} b^{5} d^{5} e^{2} - 35 \, a^{3} b^{4} d^{4} e^{3} + 35 \, a^{4} b^{3} d^{3} e^{4} - 21 \, a^{5} b^{2} d^{2} e^{5} + 7 \, a^{6} b d e^{6} - a^{7} e^{7}\right )} e^{\left (-8\right )}}{\sqrt {x e + d}} \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)^(3/2),x, algorithm="giac")

[Out]

2/429*(33*(x*e + d)^(13/2)*b^7*e^96 - 273*(x*e + d)^(11/2)*b^7*d*e^96 + 1001*(x*e + d)^(9/2)*b^7*d^2*e^96 - 21
45*(x*e + d)^(7/2)*b^7*d^3*e^96 + 3003*(x*e + d)^(5/2)*b^7*d^4*e^96 - 3003*(x*e + d)^(3/2)*b^7*d^5*e^96 + 3003
*sqrt(x*e + d)*b^7*d^6*e^96 + 273*(x*e + d)^(11/2)*a*b^6*e^97 - 2002*(x*e + d)^(9/2)*a*b^6*d*e^97 + 6435*(x*e
+ d)^(7/2)*a*b^6*d^2*e^97 - 12012*(x*e + d)^(5/2)*a*b^6*d^3*e^97 + 15015*(x*e + d)^(3/2)*a*b^6*d^4*e^97 - 1801
8*sqrt(x*e + d)*a*b^6*d^5*e^97 + 1001*(x*e + d)^(9/2)*a^2*b^5*e^98 - 6435*(x*e + d)^(7/2)*a^2*b^5*d*e^98 + 180
18*(x*e + d)^(5/2)*a^2*b^5*d^2*e^98 - 30030*(x*e + d)^(3/2)*a^2*b^5*d^3*e^98 + 45045*sqrt(x*e + d)*a^2*b^5*d^4
*e^98 + 2145*(x*e + d)^(7/2)*a^3*b^4*e^99 - 12012*(x*e + d)^(5/2)*a^3*b^4*d*e^99 + 30030*(x*e + d)^(3/2)*a^3*b
^4*d^2*e^99 - 60060*sqrt(x*e + d)*a^3*b^4*d^3*e^99 + 3003*(x*e + d)^(5/2)*a^4*b^3*e^100 - 15015*(x*e + d)^(3/2
)*a^4*b^3*d*e^100 + 45045*sqrt(x*e + d)*a^4*b^3*d^2*e^100 + 3003*(x*e + d)^(3/2)*a^5*b^2*e^101 - 18018*sqrt(x*
e + d)*a^5*b^2*d*e^101 + 3003*sqrt(x*e + d)*a^6*b*e^102)*e^(-104) + 2*(b^7*d^7 - 7*a*b^6*d^6*e + 21*a^2*b^5*d^
5*e^2 - 35*a^3*b^4*d^4*e^3 + 35*a^4*b^3*d^3*e^4 - 21*a^5*b^2*d^2*e^5 + 7*a^6*b*d*e^6 - a^7*e^7)*e^(-8)/sqrt(x*
e + d)

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maple [B]  time = 0.05, size = 498, normalized size = 2.42 \begin {gather*} -\frac {2 \left (-33 b^{7} x^{7} e^{7}-273 a \,b^{6} e^{7} x^{6}+42 b^{7} d \,e^{6} x^{6}-1001 a^{2} b^{5} e^{7} x^{5}+364 a \,b^{6} d \,e^{6} x^{5}-56 b^{7} d^{2} e^{5} x^{5}-2145 a^{3} b^{4} e^{7} x^{4}+1430 a^{2} b^{5} d \,e^{6} x^{4}-520 a \,b^{6} d^{2} e^{5} x^{4}+80 b^{7} d^{3} e^{4} x^{4}-3003 a^{4} b^{3} e^{7} x^{3}+3432 a^{3} b^{4} d \,e^{6} x^{3}-2288 a^{2} b^{5} d^{2} e^{5} x^{3}+832 a \,b^{6} d^{3} e^{4} x^{3}-128 b^{7} d^{4} e^{3} x^{3}-3003 a^{5} b^{2} e^{7} x^{2}+6006 a^{4} b^{3} d \,e^{6} x^{2}-6864 a^{3} b^{4} d^{2} e^{5} x^{2}+4576 a^{2} b^{5} d^{3} e^{4} x^{2}-1664 a \,b^{6} d^{4} e^{3} x^{2}+256 b^{7} d^{5} e^{2} x^{2}-3003 a^{6} b \,e^{7} x +12012 a^{5} b^{2} d \,e^{6} x -24024 a^{4} b^{3} d^{2} e^{5} x +27456 a^{3} b^{4} d^{3} e^{4} x -18304 a^{2} b^{5} d^{4} e^{3} x +6656 a \,b^{6} d^{5} e^{2} x -1024 b^{7} d^{6} e x +429 a^{7} e^{7}-6006 a^{6} b d \,e^{6}+24024 a^{5} b^{2} d^{2} e^{5}-48048 a^{4} b^{3} d^{3} e^{4}+54912 a^{3} b^{4} d^{4} e^{3}-36608 a^{2} b^{5} d^{5} e^{2}+13312 a \,b^{6} d^{6} e -2048 b^{7} d^{7}\right )}{429 \sqrt {e x +d}\, 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)^(3/2),x)

[Out]

-2/429*(-33*b^7*e^7*x^7-273*a*b^6*e^7*x^6+42*b^7*d*e^6*x^6-1001*a^2*b^5*e^7*x^5+364*a*b^6*d*e^6*x^5-56*b^7*d^2
*e^5*x^5-2145*a^3*b^4*e^7*x^4+1430*a^2*b^5*d*e^6*x^4-520*a*b^6*d^2*e^5*x^4+80*b^7*d^3*e^4*x^4-3003*a^4*b^3*e^7
*x^3+3432*a^3*b^4*d*e^6*x^3-2288*a^2*b^5*d^2*e^5*x^3+832*a*b^6*d^3*e^4*x^3-128*b^7*d^4*e^3*x^3-3003*a^5*b^2*e^
7*x^2+6006*a^4*b^3*d*e^6*x^2-6864*a^3*b^4*d^2*e^5*x^2+4576*a^2*b^5*d^3*e^4*x^2-1664*a*b^6*d^4*e^3*x^2+256*b^7*
d^5*e^2*x^2-3003*a^6*b*e^7*x+12012*a^5*b^2*d*e^6*x-24024*a^4*b^3*d^2*e^5*x+27456*a^3*b^4*d^3*e^4*x-18304*a^2*b
^5*d^4*e^3*x+6656*a*b^6*d^5*e^2*x-1024*b^7*d^6*e*x+429*a^7*e^7-6006*a^6*b*d*e^6+24024*a^5*b^2*d^2*e^5-48048*a^
4*b^3*d^3*e^4+54912*a^3*b^4*d^4*e^3-36608*a^2*b^5*d^5*e^2+13312*a*b^6*d^6*e-2048*b^7*d^7)/(e*x+d)^(1/2)/e^8

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maxima [B]  time = 0.50, size = 464, normalized size = 2.25 \begin {gather*} \frac {2 \, {\left (\frac {33 \, {\left (e x + d\right )}^{\frac {13}{2}} b^{7} - 273 \, {\left (b^{7} d - a b^{6} e\right )} {\left (e x + d\right )}^{\frac {11}{2}} + 1001 \, {\left (b^{7} d^{2} - 2 \, a b^{6} d e + a^{2} b^{5} e^{2}\right )} {\left (e x + d\right )}^{\frac {9}{2}} - 2145 \, {\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 {7}{2}} + 3003 \, {\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 )} {\left (e x + d\right )}^{\frac {5}{2}} - 3003 \, {\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 )}^{\frac {3}{2}} + 3003 \, {\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 )} \sqrt {e x + d}}{e^{7}} + \frac {429 \, {\left (b^{7} d^{7} - 7 \, a b^{6} d^{6} e + 21 \, a^{2} b^{5} d^{5} e^{2} - 35 \, a^{3} b^{4} d^{4} e^{3} + 35 \, a^{4} b^{3} d^{3} e^{4} - 21 \, a^{5} b^{2} d^{2} e^{5} + 7 \, a^{6} b d e^{6} - a^{7} e^{7}\right )}}{\sqrt {e x + d} e^{7}}\right )}}{429 \, 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)^(3/2),x, algorithm="maxima")

[Out]

2/429*((33*(e*x + d)^(13/2)*b^7 - 273*(b^7*d - a*b^6*e)*(e*x + d)^(11/2) + 1001*(b^7*d^2 - 2*a*b^6*d*e + a^2*b
^5*e^2)*(e*x + d)^(9/2) - 2145*(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)^(7/2) + 300
3*(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)*(e*x + d)^(5/2) - 3003*(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)^(3/2) +
3003*(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)*sqrt(e*x + d))/e^7 + 429*(b^7*d^7 - 7*a*b^6*d^6*e + 21*a^2*b^5*d^5*e^2 - 35*a^3*b^4*d^4*e^3 + 3
5*a^4*b^3*d^3*e^4 - 21*a^5*b^2*d^2*e^5 + 7*a^6*b*d*e^6 - a^7*e^7)/(sqrt(e*x + d)*e^7))/e

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mupad [B]  time = 2.07, size = 270, normalized size = 1.31 \begin {gather*} \frac {2\,b^7\,{\left (d+e\,x\right )}^{13/2}}{13\,e^8}-\frac {\left (14\,b^7\,d-14\,a\,b^6\,e\right )\,{\left (d+e\,x\right )}^{11/2}}{11\,e^8}-\frac {2\,a^7\,e^7-14\,a^6\,b\,d\,e^6+42\,a^5\,b^2\,d^2\,e^5-70\,a^4\,b^3\,d^3\,e^4+70\,a^3\,b^4\,d^4\,e^3-42\,a^2\,b^5\,d^5\,e^2+14\,a\,b^6\,d^6\,e-2\,b^7\,d^7}{e^8\,\sqrt {d+e\,x}}+\frac {14\,b^2\,{\left (a\,e-b\,d\right )}^5\,{\left (d+e\,x\right )}^{3/2}}{e^8}+\frac {14\,b^3\,{\left (a\,e-b\,d\right )}^4\,{\left (d+e\,x\right )}^{5/2}}{e^8}+\frac {10\,b^4\,{\left (a\,e-b\,d\right )}^3\,{\left (d+e\,x\right )}^{7/2}}{e^8}+\frac {14\,b^5\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{9/2}}{3\,e^8}+\frac {14\,b\,{\left (a\,e-b\,d\right )}^6\,\sqrt {d+e\,x}}{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)^(3/2),x)

[Out]

(2*b^7*(d + e*x)^(13/2))/(13*e^8) - ((14*b^7*d - 14*a*b^6*e)*(d + e*x)^(11/2))/(11*e^8) - (2*a^7*e^7 - 2*b^7*d
^7 - 42*a^2*b^5*d^5*e^2 + 70*a^3*b^4*d^4*e^3 - 70*a^4*b^3*d^3*e^4 + 42*a^5*b^2*d^2*e^5 + 14*a*b^6*d^6*e - 14*a
^6*b*d*e^6)/(e^8*(d + e*x)^(1/2)) + (14*b^2*(a*e - b*d)^5*(d + e*x)^(3/2))/e^8 + (14*b^3*(a*e - b*d)^4*(d + e*
x)^(5/2))/e^8 + (10*b^4*(a*e - b*d)^3*(d + e*x)^(7/2))/e^8 + (14*b^5*(a*e - b*d)^2*(d + e*x)^(9/2))/(3*e^8) +
(14*b*(a*e - b*d)^6*(d + e*x)^(1/2))/e^8

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sympy [B]  time = 99.36, size = 439, normalized size = 2.13 \begin {gather*} \frac {2 b^{7} \left (d + e x\right )^{\frac {13}{2}}}{13 e^{8}} + \frac {\left (d + e x\right )^{\frac {11}{2}} \left (14 a b^{6} e - 14 b^{7} d\right )}{11 e^{8}} + \frac {\left (d + e x\right )^{\frac {9}{2}} \left (42 a^{2} b^{5} e^{2} - 84 a b^{6} d e + 42 b^{7} d^{2}\right )}{9 e^{8}} + \frac {\left (d + e x\right )^{\frac {7}{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 )}{7 e^{8}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \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 )}{5 e^{8}} + \frac {\left (d + e x\right )^{\frac {3}{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 )}{3 e^{8}} + \frac {\sqrt {d + e x} \left (14 a^{6} b e^{6} - 84 a^{5} b^{2} d e^{5} + 210 a^{4} b^{3} d^{2} e^{4} - 280 a^{3} b^{4} d^{3} e^{3} + 210 a^{2} b^{5} d^{4} e^{2} - 84 a b^{6} d^{5} e + 14 b^{7} d^{6}\right )}{e^{8}} - \frac {2 \left (a e - b d\right )^{7}}{e^{8} \sqrt {d + e x}} \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)**(3/2),x)

[Out]

2*b**7*(d + e*x)**(13/2)/(13*e**8) + (d + e*x)**(11/2)*(14*a*b**6*e - 14*b**7*d)/(11*e**8) + (d + e*x)**(9/2)*
(42*a**2*b**5*e**2 - 84*a*b**6*d*e + 42*b**7*d**2)/(9*e**8) + (d + e*x)**(7/2)*(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)/(7*e**8) + (d + e*x)**(5/2)*(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)/(5*e**8) + (d + e*x)**(3/2)*(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)/(3*e**8) + sqrt(d + e*x)*(14*a**6*b*e**6 - 84*a**5*b**2*d*e**5 + 210*a**4*b**3*d**2*e**4 - 280*a**3*b**
4*d**3*e**3 + 210*a**2*b**5*d**4*e**2 - 84*a*b**6*d**5*e + 14*b**7*d**6)/e**8 - 2*(a*e - b*d)**7/(e**8*sqrt(d
+ e*x))

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