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

Optimal. Leaf size=212 \[ -\frac {14 b^6 (d+e x)^{13/2} (b d-a e)}{13 e^8}+\frac {42 b^5 (d+e x)^{11/2} (b d-a e)^2}{11 e^8}-\frac {70 b^4 (d+e x)^{9/2} (b d-a e)^3}{9 e^8}+\frac {10 b^3 (d+e x)^{7/2} (b d-a e)^4}{e^8}-\frac {42 b^2 (d+e x)^{5/2} (b d-a e)^5}{5 e^8}+\frac {14 b (d+e x)^{3/2} (b d-a e)^6}{3 e^8}-\frac {2 \sqrt {d+e x} (b d-a e)^7}{e^8}+\frac {2 b^7 (d+e x)^{15/2}}{15 e^8} \]

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Rubi [A]  time = 0.07, antiderivative size = 212, 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)^{13/2} (b d-a e)}{13 e^8}+\frac {42 b^5 (d+e x)^{11/2} (b d-a e)^2}{11 e^8}-\frac {70 b^4 (d+e x)^{9/2} (b d-a e)^3}{9 e^8}+\frac {10 b^3 (d+e x)^{7/2} (b d-a e)^4}{e^8}-\frac {42 b^2 (d+e x)^{5/2} (b d-a e)^5}{5 e^8}+\frac {14 b (d+e x)^{3/2} (b d-a e)^6}{3 e^8}-\frac {2 \sqrt {d+e x} (b d-a e)^7}{e^8}+\frac {2 b^7 (d+e x)^{15/2}}{15 e^8} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(b*d - a*e)^7*Sqrt[d + e*x])/e^8 + (14*b*(b*d - a*e)^6*(d + e*x)^(3/2))/(3*e^8) - (42*b^2*(b*d - a*e)^5*(d
+ e*x)^(5/2))/(5*e^8) + (10*b^3*(b*d - a*e)^4*(d + e*x)^(7/2))/e^8 - (70*b^4*(b*d - a*e)^3*(d + e*x)^(9/2))/(
9*e^8) + (42*b^5*(b*d - a*e)^2*(d + e*x)^(11/2))/(11*e^8) - (14*b^6*(b*d - a*e)*(d + e*x)^(13/2))/(13*e^8) + (
2*b^7*(d + e*x)^(15/2))/(15*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}{\sqrt {d+e x}} \, dx &=\int \frac {(a+b x)^7}{\sqrt {d+e x}} \, dx\\ &=\int \left (\frac {(-b d+a e)^7}{e^7 \sqrt {d+e x}}+\frac {7 b (b d-a e)^6 \sqrt {d+e x}}{e^7}-\frac {21 b^2 (b d-a e)^5 (d+e x)^{3/2}}{e^7}+\frac {35 b^3 (b d-a e)^4 (d+e x)^{5/2}}{e^7}-\frac {35 b^4 (b d-a e)^3 (d+e x)^{7/2}}{e^7}+\frac {21 b^5 (b d-a e)^2 (d+e x)^{9/2}}{e^7}-\frac {7 b^6 (b d-a e) (d+e x)^{11/2}}{e^7}+\frac {b^7 (d+e x)^{13/2}}{e^7}\right ) \, dx\\ &=-\frac {2 (b d-a e)^7 \sqrt {d+e x}}{e^8}+\frac {14 b (b d-a e)^6 (d+e x)^{3/2}}{3 e^8}-\frac {42 b^2 (b d-a e)^5 (d+e x)^{5/2}}{5 e^8}+\frac {10 b^3 (b d-a e)^4 (d+e x)^{7/2}}{e^8}-\frac {70 b^4 (b d-a e)^3 (d+e x)^{9/2}}{9 e^8}+\frac {42 b^5 (b d-a e)^2 (d+e x)^{11/2}}{11 e^8}-\frac {14 b^6 (b d-a e) (d+e x)^{13/2}}{13 e^8}+\frac {2 b^7 (d+e x)^{15/2}}{15 e^8}\\ \end {align*}

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Mathematica [A]  time = 0.10, size = 167, normalized size = 0.79 \begin {gather*} \frac {2 \sqrt {d+e x} \left (-3465 b^6 (d+e x)^6 (b d-a e)+12285 b^5 (d+e x)^5 (b d-a e)^2-25025 b^4 (d+e x)^4 (b d-a e)^3+32175 b^3 (d+e x)^3 (b d-a e)^4-27027 b^2 (d+e x)^2 (b d-a e)^5+15015 b (d+e x) (b d-a e)^6-6435 (b d-a e)^7+429 b^7 (d+e x)^7\right )}{6435 e^8} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[d + e*x]*(-6435*(b*d - a*e)^7 + 15015*b*(b*d - a*e)^6*(d + e*x) - 27027*b^2*(b*d - a*e)^5*(d + e*x)^2
+ 32175*b^3*(b*d - a*e)^4*(d + e*x)^3 - 25025*b^4*(b*d - a*e)^3*(d + e*x)^4 + 12285*b^5*(b*d - a*e)^2*(d + e*x
)^5 - 3465*b^6*(b*d - a*e)*(d + e*x)^6 + 429*b^7*(d + e*x)^7))/(6435*e^8)

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IntegrateAlgebraic [B]  time = 0.16, size = 582, normalized size = 2.75 \begin {gather*} \frac {2 \sqrt {d+e x} \left (6435 a^7 e^7+15015 a^6 b e^6 (d+e x)-45045 a^6 b d e^6+135135 a^5 b^2 d^2 e^5+27027 a^5 b^2 e^5 (d+e x)^2-90090 a^5 b^2 d e^5 (d+e x)-225225 a^4 b^3 d^3 e^4+225225 a^4 b^3 d^2 e^4 (d+e x)+32175 a^4 b^3 e^4 (d+e x)^3-135135 a^4 b^3 d e^4 (d+e x)^2+225225 a^3 b^4 d^4 e^3-300300 a^3 b^4 d^3 e^3 (d+e x)+270270 a^3 b^4 d^2 e^3 (d+e x)^2+25025 a^3 b^4 e^3 (d+e x)^4-128700 a^3 b^4 d e^3 (d+e x)^3-135135 a^2 b^5 d^5 e^2+225225 a^2 b^5 d^4 e^2 (d+e x)-270270 a^2 b^5 d^3 e^2 (d+e x)^2+193050 a^2 b^5 d^2 e^2 (d+e x)^3+12285 a^2 b^5 e^2 (d+e x)^5-75075 a^2 b^5 d e^2 (d+e x)^4+45045 a b^6 d^6 e-90090 a b^6 d^5 e (d+e x)+135135 a b^6 d^4 e (d+e x)^2-128700 a b^6 d^3 e (d+e x)^3+75075 a b^6 d^2 e (d+e x)^4+3465 a b^6 e (d+e x)^6-24570 a b^6 d e (d+e x)^5-6435 b^7 d^7+15015 b^7 d^6 (d+e x)-27027 b^7 d^5 (d+e x)^2+32175 b^7 d^4 (d+e x)^3-25025 b^7 d^3 (d+e x)^4+12285 b^7 d^2 (d+e x)^5+429 b^7 (d+e x)^7-3465 b^7 d (d+e x)^6\right )}{6435 e^8} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[d + e*x]*(-6435*b^7*d^7 + 45045*a*b^6*d^6*e - 135135*a^2*b^5*d^5*e^2 + 225225*a^3*b^4*d^4*e^3 - 225225
*a^4*b^3*d^3*e^4 + 135135*a^5*b^2*d^2*e^5 - 45045*a^6*b*d*e^6 + 6435*a^7*e^7 + 15015*b^7*d^6*(d + e*x) - 90090
*a*b^6*d^5*e*(d + e*x) + 225225*a^2*b^5*d^4*e^2*(d + e*x) - 300300*a^3*b^4*d^3*e^3*(d + e*x) + 225225*a^4*b^3*
d^2*e^4*(d + e*x) - 90090*a^5*b^2*d*e^5*(d + e*x) + 15015*a^6*b*e^6*(d + e*x) - 27027*b^7*d^5*(d + e*x)^2 + 13
5135*a*b^6*d^4*e*(d + e*x)^2 - 270270*a^2*b^5*d^3*e^2*(d + e*x)^2 + 270270*a^3*b^4*d^2*e^3*(d + e*x)^2 - 13513
5*a^4*b^3*d*e^4*(d + e*x)^2 + 27027*a^5*b^2*e^5*(d + e*x)^2 + 32175*b^7*d^4*(d + e*x)^3 - 128700*a*b^6*d^3*e*(
d + e*x)^3 + 193050*a^2*b^5*d^2*e^2*(d + e*x)^3 - 128700*a^3*b^4*d*e^3*(d + e*x)^3 + 32175*a^4*b^3*e^4*(d + e*
x)^3 - 25025*b^7*d^3*(d + e*x)^4 + 75075*a*b^6*d^2*e*(d + e*x)^4 - 75075*a^2*b^5*d*e^2*(d + e*x)^4 + 25025*a^3
*b^4*e^3*(d + e*x)^4 + 12285*b^7*d^2*(d + e*x)^5 - 24570*a*b^6*d*e*(d + e*x)^5 + 12285*a^2*b^5*e^2*(d + e*x)^5
- 3465*b^7*d*(d + e*x)^6 + 3465*a*b^6*e*(d + e*x)^6 + 429*b^7*(d + e*x)^7))/(6435*e^8)

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fricas [B]  time = 0.42, size = 463, normalized size = 2.18 \begin {gather*} \frac {2 \, {\left (429 \, b^{7} e^{7} x^{7} - 2048 \, b^{7} d^{7} + 15360 \, a b^{6} d^{6} e - 49920 \, a^{2} b^{5} d^{5} e^{2} + 91520 \, a^{3} b^{4} d^{4} e^{3} - 102960 \, a^{4} b^{3} d^{3} e^{4} + 72072 \, a^{5} b^{2} d^{2} e^{5} - 30030 \, a^{6} b d e^{6} + 6435 \, a^{7} e^{7} - 231 \, {\left (2 \, b^{7} d e^{6} - 15 \, a b^{6} e^{7}\right )} x^{6} + 63 \, {\left (8 \, b^{7} d^{2} e^{5} - 60 \, a b^{6} d e^{6} + 195 \, a^{2} b^{5} e^{7}\right )} x^{5} - 35 \, {\left (16 \, b^{7} d^{3} e^{4} - 120 \, a b^{6} d^{2} e^{5} + 390 \, a^{2} b^{5} d e^{6} - 715 \, a^{3} b^{4} e^{7}\right )} x^{4} + 5 \, {\left (128 \, b^{7} d^{4} e^{3} - 960 \, a b^{6} d^{3} e^{4} + 3120 \, a^{2} b^{5} d^{2} e^{5} - 5720 \, a^{3} b^{4} d e^{6} + 6435 \, a^{4} b^{3} e^{7}\right )} x^{3} - 3 \, {\left (256 \, b^{7} d^{5} e^{2} - 1920 \, a b^{6} d^{4} e^{3} + 6240 \, a^{2} b^{5} d^{3} e^{4} - 11440 \, a^{3} b^{4} d^{2} e^{5} + 12870 \, a^{4} b^{3} d e^{6} - 9009 \, a^{5} b^{2} e^{7}\right )} x^{2} + {\left (1024 \, b^{7} d^{6} e - 7680 \, a b^{6} d^{5} e^{2} + 24960 \, a^{2} b^{5} d^{4} e^{3} - 45760 \, a^{3} b^{4} d^{3} e^{4} + 51480 \, a^{4} b^{3} d^{2} e^{5} - 36036 \, a^{5} b^{2} d e^{6} + 15015 \, a^{6} b e^{7}\right )} x\right )} \sqrt {e x + d}}{6435 \, e^{8}} \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)^(1/2),x, algorithm="fricas")

[Out]

2/6435*(429*b^7*e^7*x^7 - 2048*b^7*d^7 + 15360*a*b^6*d^6*e - 49920*a^2*b^5*d^5*e^2 + 91520*a^3*b^4*d^4*e^3 - 1
02960*a^4*b^3*d^3*e^4 + 72072*a^5*b^2*d^2*e^5 - 30030*a^6*b*d*e^6 + 6435*a^7*e^7 - 231*(2*b^7*d*e^6 - 15*a*b^6
*e^7)*x^6 + 63*(8*b^7*d^2*e^5 - 60*a*b^6*d*e^6 + 195*a^2*b^5*e^7)*x^5 - 35*(16*b^7*d^3*e^4 - 120*a*b^6*d^2*e^5
+ 390*a^2*b^5*d*e^6 - 715*a^3*b^4*e^7)*x^4 + 5*(128*b^7*d^4*e^3 - 960*a*b^6*d^3*e^4 + 3120*a^2*b^5*d^2*e^5 -
5720*a^3*b^4*d*e^6 + 6435*a^4*b^3*e^7)*x^3 - 3*(256*b^7*d^5*e^2 - 1920*a*b^6*d^4*e^3 + 6240*a^2*b^5*d^3*e^4 -
11440*a^3*b^4*d^2*e^5 + 12870*a^4*b^3*d*e^6 - 9009*a^5*b^2*e^7)*x^2 + (1024*b^7*d^6*e - 7680*a*b^6*d^5*e^2 + 2
4960*a^2*b^5*d^4*e^3 - 45760*a^3*b^4*d^3*e^4 + 51480*a^4*b^3*d^2*e^5 - 36036*a^5*b^2*d*e^6 + 15015*a^6*b*e^7)*
x)*sqrt(e*x + d)/e^8

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giac [B]  time = 0.19, size = 505, normalized size = 2.38 \begin {gather*} \frac {2}{6435} \, {\left (15015 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} a^{6} b e^{\left (-1\right )} + 9009 \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} - 10 \, {\left (x e + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {x e + d} d^{2}\right )} a^{5} b^{2} e^{\left (-2\right )} + 6435 \, {\left (5 \, {\left (x e + d\right )}^{\frac {7}{2}} - 21 \, {\left (x e + d\right )}^{\frac {5}{2}} d + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {x e + d} d^{3}\right )} a^{4} b^{3} e^{\left (-3\right )} + 715 \, {\left (35 \, {\left (x e + d\right )}^{\frac {9}{2}} - 180 \, {\left (x e + d\right )}^{\frac {7}{2}} d + 378 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {x e + d} d^{4}\right )} a^{3} b^{4} e^{\left (-4\right )} + 195 \, {\left (63 \, {\left (x e + d\right )}^{\frac {11}{2}} - 385 \, {\left (x e + d\right )}^{\frac {9}{2}} d + 990 \, {\left (x e + d\right )}^{\frac {7}{2}} d^{2} - 1386 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{3} + 1155 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{4} - 693 \, \sqrt {x e + d} d^{5}\right )} a^{2} b^{5} e^{\left (-5\right )} + 15 \, {\left (231 \, {\left (x e + d\right )}^{\frac {13}{2}} - 1638 \, {\left (x e + d\right )}^{\frac {11}{2}} d + 5005 \, {\left (x e + d\right )}^{\frac {9}{2}} d^{2} - 8580 \, {\left (x e + d\right )}^{\frac {7}{2}} d^{3} + 9009 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{4} - 6006 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{5} + 3003 \, \sqrt {x e + d} d^{6}\right )} a b^{6} e^{\left (-6\right )} + {\left (429 \, {\left (x e + d\right )}^{\frac {15}{2}} - 3465 \, {\left (x e + d\right )}^{\frac {13}{2}} d + 12285 \, {\left (x e + d\right )}^{\frac {11}{2}} d^{2} - 25025 \, {\left (x e + d\right )}^{\frac {9}{2}} d^{3} + 32175 \, {\left (x e + d\right )}^{\frac {7}{2}} d^{4} - 27027 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{5} + 15015 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{6} - 6435 \, \sqrt {x e + d} d^{7}\right )} b^{7} e^{\left (-7\right )} + 6435 \, \sqrt {x e + d} a^{7}\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)^(1/2),x, algorithm="giac")

[Out]

2/6435*(15015*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*a^6*b*e^(-1) + 9009*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2
)*d + 15*sqrt(x*e + d)*d^2)*a^5*b^2*e^(-2) + 6435*(5*(x*e + d)^(7/2) - 21*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/
2)*d^2 - 35*sqrt(x*e + d)*d^3)*a^4*b^3*e^(-3) + 715*(35*(x*e + d)^(9/2) - 180*(x*e + d)^(7/2)*d + 378*(x*e + d
)^(5/2)*d^2 - 420*(x*e + d)^(3/2)*d^3 + 315*sqrt(x*e + d)*d^4)*a^3*b^4*e^(-4) + 195*(63*(x*e + d)^(11/2) - 385
*(x*e + d)^(9/2)*d + 990*(x*e + d)^(7/2)*d^2 - 1386*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4 - 693*sqrt(
x*e + d)*d^5)*a^2*b^5*e^(-5) + 15*(231*(x*e + d)^(13/2) - 1638*(x*e + d)^(11/2)*d + 5005*(x*e + d)^(9/2)*d^2 -
8580*(x*e + d)^(7/2)*d^3 + 9009*(x*e + d)^(5/2)*d^4 - 6006*(x*e + d)^(3/2)*d^5 + 3003*sqrt(x*e + d)*d^6)*a*b^
6*e^(-6) + (429*(x*e + d)^(15/2) - 3465*(x*e + d)^(13/2)*d + 12285*(x*e + d)^(11/2)*d^2 - 25025*(x*e + d)^(9/2
)*d^3 + 32175*(x*e + d)^(7/2)*d^4 - 27027*(x*e + d)^(5/2)*d^5 + 15015*(x*e + d)^(3/2)*d^6 - 6435*sqrt(x*e + d)
*d^7)*b^7*e^(-7) + 6435*sqrt(x*e + d)*a^7)*e^(-1)

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maple [B]  time = 0.06, size = 498, normalized size = 2.35 \begin {gather*} \frac {2 \left (429 b^{7} x^{7} e^{7}+3465 a \,b^{6} e^{7} x^{6}-462 b^{7} d \,e^{6} x^{6}+12285 a^{2} b^{5} e^{7} x^{5}-3780 a \,b^{6} d \,e^{6} x^{5}+504 b^{7} d^{2} e^{5} x^{5}+25025 a^{3} b^{4} e^{7} x^{4}-13650 a^{2} b^{5} d \,e^{6} x^{4}+4200 a \,b^{6} d^{2} e^{5} x^{4}-560 b^{7} d^{3} e^{4} x^{4}+32175 a^{4} b^{3} e^{7} x^{3}-28600 a^{3} b^{4} d \,e^{6} x^{3}+15600 a^{2} b^{5} d^{2} e^{5} x^{3}-4800 a \,b^{6} d^{3} e^{4} x^{3}+640 b^{7} d^{4} e^{3} x^{3}+27027 a^{5} b^{2} e^{7} x^{2}-38610 a^{4} b^{3} d \,e^{6} x^{2}+34320 a^{3} b^{4} d^{2} e^{5} x^{2}-18720 a^{2} b^{5} d^{3} e^{4} x^{2}+5760 a \,b^{6} d^{4} e^{3} x^{2}-768 b^{7} d^{5} e^{2} x^{2}+15015 a^{6} b \,e^{7} x -36036 a^{5} b^{2} d \,e^{6} x +51480 a^{4} b^{3} d^{2} e^{5} x -45760 a^{3} b^{4} d^{3} e^{4} x +24960 a^{2} b^{5} d^{4} e^{3} x -7680 a \,b^{6} d^{5} e^{2} x +1024 b^{7} d^{6} e x +6435 a^{7} e^{7}-30030 a^{6} b d \,e^{6}+72072 a^{5} b^{2} d^{2} e^{5}-102960 a^{4} b^{3} d^{3} e^{4}+91520 a^{3} b^{4} d^{4} e^{3}-49920 a^{2} b^{5} d^{5} e^{2}+15360 a \,b^{6} d^{6} e -2048 b^{7} d^{7}\right ) \sqrt {e x +d}}{6435 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)^(1/2),x)

[Out]

2/6435*(429*b^7*e^7*x^7+3465*a*b^6*e^7*x^6-462*b^7*d*e^6*x^6+12285*a^2*b^5*e^7*x^5-3780*a*b^6*d*e^6*x^5+504*b^
7*d^2*e^5*x^5+25025*a^3*b^4*e^7*x^4-13650*a^2*b^5*d*e^6*x^4+4200*a*b^6*d^2*e^5*x^4-560*b^7*d^3*e^4*x^4+32175*a
^4*b^3*e^7*x^3-28600*a^3*b^4*d*e^6*x^3+15600*a^2*b^5*d^2*e^5*x^3-4800*a*b^6*d^3*e^4*x^3+640*b^7*d^4*e^3*x^3+27
027*a^5*b^2*e^7*x^2-38610*a^4*b^3*d*e^6*x^2+34320*a^3*b^4*d^2*e^5*x^2-18720*a^2*b^5*d^3*e^4*x^2+5760*a*b^6*d^4
*e^3*x^2-768*b^7*d^5*e^2*x^2+15015*a^6*b*e^7*x-36036*a^5*b^2*d*e^6*x+51480*a^4*b^3*d^2*e^5*x-45760*a^3*b^4*d^3
*e^4*x+24960*a^2*b^5*d^4*e^3*x-7680*a*b^6*d^5*e^2*x+1024*b^7*d^6*e*x+6435*a^7*e^7-30030*a^6*b*d*e^6+72072*a^5*
b^2*d^2*e^5-102960*a^4*b^3*d^3*e^4+91520*a^3*b^4*d^4*e^3-49920*a^2*b^5*d^5*e^2+15360*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 = 0.71, size = 456, normalized size = 2.15 \begin {gather*} \frac {2 \, {\left (429 \, {\left (e x + d\right )}^{\frac {15}{2}} b^{7} - 3465 \, {\left (b^{7} d - a b^{6} e\right )} {\left (e x + d\right )}^{\frac {13}{2}} + 12285 \, {\left (b^{7} d^{2} - 2 \, a b^{6} d e + a^{2} b^{5} e^{2}\right )} {\left (e x + d\right )}^{\frac {11}{2}} - 25025 \, {\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 {9}{2}} + 32175 \, {\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 {7}{2}} - 27027 \, {\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 {5}{2}} + 15015 \, {\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 )}^{\frac {3}{2}} - 6435 \, {\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}\right )}}{6435 \, e^{8}} \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)^(1/2),x, algorithm="maxima")

[Out]

2/6435*(429*(e*x + d)^(15/2)*b^7 - 3465*(b^7*d - a*b^6*e)*(e*x + d)^(13/2) + 12285*(b^7*d^2 - 2*a*b^6*d*e + a^
2*b^5*e^2)*(e*x + d)^(11/2) - 25025*(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)^(9/2)
+ 32175*(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)^(7/2) - 27027*
(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)^
(5/2) + 15015*(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)^(3/2) - 6435*(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)*sqrt(e*x + d))/e^8

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mupad [B]  time = 2.07, size = 187, normalized size = 0.88 \begin {gather*} \frac {2\,b^7\,{\left (d+e\,x\right )}^{15/2}}{15\,e^8}-\frac {\left (14\,b^7\,d-14\,a\,b^6\,e\right )\,{\left (d+e\,x\right )}^{13/2}}{13\,e^8}+\frac {2\,{\left (a\,e-b\,d\right )}^7\,\sqrt {d+e\,x}}{e^8}+\frac {42\,b^2\,{\left (a\,e-b\,d\right )}^5\,{\left (d+e\,x\right )}^{5/2}}{5\,e^8}+\frac {10\,b^3\,{\left (a\,e-b\,d\right )}^4\,{\left (d+e\,x\right )}^{7/2}}{e^8}+\frac {70\,b^4\,{\left (a\,e-b\,d\right )}^3\,{\left (d+e\,x\right )}^{9/2}}{9\,e^8}+\frac {42\,b^5\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{11/2}}{11\,e^8}+\frac {14\,b\,{\left (a\,e-b\,d\right )}^6\,{\left (d+e\,x\right )}^{3/2}}{3\,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)^(1/2),x)

[Out]

(2*b^7*(d + e*x)^(15/2))/(15*e^8) - ((14*b^7*d - 14*a*b^6*e)*(d + e*x)^(13/2))/(13*e^8) + (2*(a*e - b*d)^7*(d
+ e*x)^(1/2))/e^8 + (42*b^2*(a*e - b*d)^5*(d + e*x)^(5/2))/(5*e^8) + (10*b^3*(a*e - b*d)^4*(d + e*x)^(7/2))/e^
8 + (70*b^4*(a*e - b*d)^3*(d + e*x)^(9/2))/(9*e^8) + (42*b^5*(a*e - b*d)^2*(d + e*x)^(11/2))/(11*e^8) + (14*b*
(a*e - b*d)^6*(d + e*x)^(3/2))/(3*e^8)

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sympy [A]  time = 140.05, size = 1217, normalized size = 5.74

result too large to display

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)**(1/2),x)

[Out]

Piecewise(((-2*a**7*d/sqrt(d + e*x) - 2*a**7*(-d/sqrt(d + e*x) - sqrt(d + e*x)) - 14*a**6*b*d*(-d/sqrt(d + e*x
) - sqrt(d + e*x))/e - 14*a**6*b*(d**2/sqrt(d + e*x) + 2*d*sqrt(d + e*x) - (d + e*x)**(3/2)/3)/e - 42*a**5*b**
2*d*(d**2/sqrt(d + e*x) + 2*d*sqrt(d + e*x) - (d + e*x)**(3/2)/3)/e**2 - 42*a**5*b**2*(-d**3/sqrt(d + e*x) - 3
*d**2*sqrt(d + e*x) + d*(d + e*x)**(3/2) - (d + e*x)**(5/2)/5)/e**2 - 70*a**4*b**3*d*(-d**3/sqrt(d + e*x) - 3*
d**2*sqrt(d + e*x) + d*(d + e*x)**(3/2) - (d + e*x)**(5/2)/5)/e**3 - 70*a**4*b**3*(d**4/sqrt(d + e*x) + 4*d**3
*sqrt(d + e*x) - 2*d**2*(d + e*x)**(3/2) + 4*d*(d + e*x)**(5/2)/5 - (d + e*x)**(7/2)/7)/e**3 - 70*a**3*b**4*d*
(d**4/sqrt(d + e*x) + 4*d**3*sqrt(d + e*x) - 2*d**2*(d + e*x)**(3/2) + 4*d*(d + e*x)**(5/2)/5 - (d + e*x)**(7/
2)/7)/e**4 - 70*a**3*b**4*(-d**5/sqrt(d + e*x) - 5*d**4*sqrt(d + e*x) + 10*d**3*(d + e*x)**(3/2)/3 - 2*d**2*(d
+ e*x)**(5/2) + 5*d*(d + e*x)**(7/2)/7 - (d + e*x)**(9/2)/9)/e**4 - 42*a**2*b**5*d*(-d**5/sqrt(d + e*x) - 5*d
**4*sqrt(d + e*x) + 10*d**3*(d + e*x)**(3/2)/3 - 2*d**2*(d + e*x)**(5/2) + 5*d*(d + e*x)**(7/2)/7 - (d + e*x)*
*(9/2)/9)/e**5 - 42*a**2*b**5*(d**6/sqrt(d + e*x) + 6*d**5*sqrt(d + e*x) - 5*d**4*(d + e*x)**(3/2) + 4*d**3*(d
+ e*x)**(5/2) - 15*d**2*(d + e*x)**(7/2)/7 + 2*d*(d + e*x)**(9/2)/3 - (d + e*x)**(11/2)/11)/e**5 - 14*a*b**6*
d*(d**6/sqrt(d + e*x) + 6*d**5*sqrt(d + e*x) - 5*d**4*(d + e*x)**(3/2) + 4*d**3*(d + e*x)**(5/2) - 15*d**2*(d
+ e*x)**(7/2)/7 + 2*d*(d + e*x)**(9/2)/3 - (d + e*x)**(11/2)/11)/e**6 - 14*a*b**6*(-d**7/sqrt(d + e*x) - 7*d**
6*sqrt(d + e*x) + 7*d**5*(d + e*x)**(3/2) - 7*d**4*(d + e*x)**(5/2) + 5*d**3*(d + e*x)**(7/2) - 7*d**2*(d + e*
x)**(9/2)/3 + 7*d*(d + e*x)**(11/2)/11 - (d + e*x)**(13/2)/13)/e**6 - 2*b**7*d*(-d**7/sqrt(d + e*x) - 7*d**6*s
qrt(d + e*x) + 7*d**5*(d + e*x)**(3/2) - 7*d**4*(d + e*x)**(5/2) + 5*d**3*(d + e*x)**(7/2) - 7*d**2*(d + e*x)*
*(9/2)/3 + 7*d*(d + e*x)**(11/2)/11 - (d + e*x)**(13/2)/13)/e**7 - 2*b**7*(d**8/sqrt(d + e*x) + 8*d**7*sqrt(d
+ e*x) - 28*d**6*(d + e*x)**(3/2)/3 + 56*d**5*(d + e*x)**(5/2)/5 - 10*d**4*(d + e*x)**(7/2) + 56*d**3*(d + e*x
)**(9/2)/9 - 28*d**2*(d + e*x)**(11/2)/11 + 8*d*(d + e*x)**(13/2)/13 - (d + e*x)**(15/2)/15)/e**7)/e, Ne(e, 0)
), (Piecewise((a**7*x, Eq(b, 0)), ((a**2 + 2*a*b*x + b**2*x**2)**4/(8*b), True))/sqrt(d), True))

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