3.19.88 \(\int \frac {(a+b x) (a^2+2 a b x+b^2 x^2)^{5/2}}{\sqrt {d+e x}} \, dx\)

Optimal. Leaf size=370 \[ \frac {6 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^4}{e^7 (a+b x)}-\frac {4 b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^5}{e^7 (a+b x)}+\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x} (b d-a e)^6}{e^7 (a+b x)}+\frac {2 b^6 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{13/2}}{13 e^7 (a+b x)}-\frac {12 b^5 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{11/2} (b d-a e)}{11 e^7 (a+b x)}+\frac {10 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e)^2}{3 e^7 (a+b x)}-\frac {40 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)^3}{7 e^7 (a+b x)} \]

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Rubi [A]  time = 0.14, antiderivative size = 370, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.086, Rules used = {770, 21, 43} \begin {gather*} \frac {2 b^6 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{13/2}}{13 e^7 (a+b x)}-\frac {12 b^5 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{11/2} (b d-a e)}{11 e^7 (a+b x)}+\frac {10 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e)^2}{3 e^7 (a+b x)}-\frac {40 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)^3}{7 e^7 (a+b x)}+\frac {6 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^4}{e^7 (a+b x)}-\frac {4 b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^5}{e^7 (a+b x)}+\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x} (b d-a e)^6}{e^7 (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/Sqrt[d + e*x],x]

[Out]

(2*(b*d - a*e)^6*Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(a + b*x)) - (4*b*(b*d - a*e)^5*(d + e*x)^(
3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(a + b*x)) + (6*b^2*(b*d - a*e)^4*(d + e*x)^(5/2)*Sqrt[a^2 + 2*a*b*x
+ b^2*x^2])/(e^7*(a + b*x)) - (40*b^3*(b*d - a*e)^3*(d + e*x)^(7/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*e^7*(a +
b*x)) + (10*b^4*(b*d - a*e)^2*(d + e*x)^(9/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^7*(a + b*x)) - (12*b^5*(b*d
- a*e)*(d + e*x)^(11/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*e^7*(a + b*x)) + (2*b^6*(d + e*x)^(13/2)*Sqrt[a^2
+ 2*a*b*x + b^2*x^2])/(13*e^7*(a + b*x))

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

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])

Rule 770

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dis
t[(a + b*x + c*x^2)^FracPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*FracPart[p])), Int[(d + e*x)^m*(f + g*x)*(b/2 + c
*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && EqQ[b^2 - 4*a*c, 0]

Rubi steps

\begin {align*} \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{\sqrt {d+e x}} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {(a+b x) \left (a b+b^2 x\right )^5}{\sqrt {d+e x}} \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac {\left (b \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int \frac {(a+b x)^6}{\sqrt {d+e x}} \, dx}{a b+b^2 x}\\ &=\frac {\left (b \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int \left (\frac {(-b d+a e)^6}{e^6 \sqrt {d+e x}}-\frac {6 b (b d-a e)^5 \sqrt {d+e x}}{e^6}+\frac {15 b^2 (b d-a e)^4 (d+e x)^{3/2}}{e^6}-\frac {20 b^3 (b d-a e)^3 (d+e x)^{5/2}}{e^6}+\frac {15 b^4 (b d-a e)^2 (d+e x)^{7/2}}{e^6}-\frac {6 b^5 (b d-a e) (d+e x)^{9/2}}{e^6}+\frac {b^6 (d+e x)^{11/2}}{e^6}\right ) \, dx}{a b+b^2 x}\\ &=\frac {2 (b d-a e)^6 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x)}-\frac {4 b (b d-a e)^5 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x)}+\frac {6 b^2 (b d-a e)^4 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x)}-\frac {40 b^3 (b d-a e)^3 (d+e x)^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^7 (a+b x)}+\frac {10 b^4 (b d-a e)^2 (d+e x)^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^7 (a+b x)}-\frac {12 b^5 (b d-a e) (d+e x)^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}}{11 e^7 (a+b x)}+\frac {2 b^6 (d+e x)^{13/2} \sqrt {a^2+2 a b x+b^2 x^2}}{13 e^7 (a+b x)}\\ \end {align*}

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Mathematica [A]  time = 0.10, size = 163, normalized size = 0.44 \begin {gather*} \frac {2 \sqrt {(a+b x)^2} \sqrt {d+e x} \left (-1638 b^5 (d+e x)^5 (b d-a e)+5005 b^4 (d+e x)^4 (b d-a e)^2-8580 b^3 (d+e x)^3 (b d-a e)^3+9009 b^2 (d+e x)^2 (b d-a e)^4-6006 b (d+e x) (b d-a e)^5+3003 (b d-a e)^6+231 b^6 (d+e x)^6\right )}{3003 e^7 (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/Sqrt[d + e*x],x]

[Out]

(2*Sqrt[(a + b*x)^2]*Sqrt[d + e*x]*(3003*(b*d - a*e)^6 - 6006*b*(b*d - a*e)^5*(d + e*x) + 9009*b^2*(b*d - a*e)
^4*(d + e*x)^2 - 8580*b^3*(b*d - a*e)^3*(d + e*x)^3 + 5005*b^4*(b*d - a*e)^2*(d + e*x)^4 - 1638*b^5*(b*d - a*e
)*(d + e*x)^5 + 231*b^6*(d + e*x)^6))/(3003*e^7*(a + b*x))

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IntegrateAlgebraic [A]  time = 35.21, size = 466, normalized size = 1.26 \begin {gather*} \frac {2 \sqrt {d+e x} \sqrt {\frac {(a e+b e x)^2}{e^2}} \left (3003 a^6 e^6+6006 a^5 b e^5 (d+e x)-18018 a^5 b d e^5+45045 a^4 b^2 d^2 e^4+9009 a^4 b^2 e^4 (d+e x)^2-30030 a^4 b^2 d e^4 (d+e x)-60060 a^3 b^3 d^3 e^3+60060 a^3 b^3 d^2 e^3 (d+e x)+8580 a^3 b^3 e^3 (d+e x)^3-36036 a^3 b^3 d e^3 (d+e x)^2+45045 a^2 b^4 d^4 e^2-60060 a^2 b^4 d^3 e^2 (d+e x)+54054 a^2 b^4 d^2 e^2 (d+e x)^2+5005 a^2 b^4 e^2 (d+e x)^4-25740 a^2 b^4 d e^2 (d+e x)^3-18018 a b^5 d^5 e+30030 a b^5 d^4 e (d+e x)-36036 a b^5 d^3 e (d+e x)^2+25740 a b^5 d^2 e (d+e x)^3+1638 a b^5 e (d+e x)^5-10010 a b^5 d e (d+e x)^4+3003 b^6 d^6-6006 b^6 d^5 (d+e x)+9009 b^6 d^4 (d+e x)^2-8580 b^6 d^3 (d+e x)^3+5005 b^6 d^2 (d+e x)^4+231 b^6 (d+e x)^6-1638 b^6 d (d+e x)^5\right )}{3003 e^6 (a e+b e x)} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/Sqrt[d + e*x],x]

[Out]

(2*Sqrt[d + e*x]*Sqrt[(a*e + b*e*x)^2/e^2]*(3003*b^6*d^6 - 18018*a*b^5*d^5*e + 45045*a^2*b^4*d^4*e^2 - 60060*a
^3*b^3*d^3*e^3 + 45045*a^4*b^2*d^2*e^4 - 18018*a^5*b*d*e^5 + 3003*a^6*e^6 - 6006*b^6*d^5*(d + e*x) + 30030*a*b
^5*d^4*e*(d + e*x) - 60060*a^2*b^4*d^3*e^2*(d + e*x) + 60060*a^3*b^3*d^2*e^3*(d + e*x) - 30030*a^4*b^2*d*e^4*(
d + e*x) + 6006*a^5*b*e^5*(d + e*x) + 9009*b^6*d^4*(d + e*x)^2 - 36036*a*b^5*d^3*e*(d + e*x)^2 + 54054*a^2*b^4
*d^2*e^2*(d + e*x)^2 - 36036*a^3*b^3*d*e^3*(d + e*x)^2 + 9009*a^4*b^2*e^4*(d + e*x)^2 - 8580*b^6*d^3*(d + e*x)
^3 + 25740*a*b^5*d^2*e*(d + e*x)^3 - 25740*a^2*b^4*d*e^2*(d + e*x)^3 + 8580*a^3*b^3*e^3*(d + e*x)^3 + 5005*b^6
*d^2*(d + e*x)^4 - 10010*a*b^5*d*e*(d + e*x)^4 + 5005*a^2*b^4*e^2*(d + e*x)^4 - 1638*b^6*d*(d + e*x)^5 + 1638*
a*b^5*e*(d + e*x)^5 + 231*b^6*(d + e*x)^6))/(3003*e^6*(a*e + b*e*x))

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fricas [A]  time = 0.42, size = 356, normalized size = 0.96 \begin {gather*} \frac {2 \, {\left (231 \, b^{6} e^{6} x^{6} + 1024 \, b^{6} d^{6} - 6656 \, a b^{5} d^{5} e + 18304 \, a^{2} b^{4} d^{4} e^{2} - 27456 \, a^{3} b^{3} d^{3} e^{3} + 24024 \, a^{4} b^{2} d^{2} e^{4} - 12012 \, a^{5} b d e^{5} + 3003 \, a^{6} e^{6} - 126 \, {\left (2 \, b^{6} d e^{5} - 13 \, a b^{5} e^{6}\right )} x^{5} + 35 \, {\left (8 \, b^{6} d^{2} e^{4} - 52 \, a b^{5} d e^{5} + 143 \, a^{2} b^{4} e^{6}\right )} x^{4} - 20 \, {\left (16 \, b^{6} d^{3} e^{3} - 104 \, a b^{5} d^{2} e^{4} + 286 \, a^{2} b^{4} d e^{5} - 429 \, a^{3} b^{3} e^{6}\right )} x^{3} + 3 \, {\left (128 \, b^{6} d^{4} e^{2} - 832 \, a b^{5} d^{3} e^{3} + 2288 \, a^{2} b^{4} d^{2} e^{4} - 3432 \, a^{3} b^{3} d e^{5} + 3003 \, a^{4} b^{2} e^{6}\right )} x^{2} - 2 \, {\left (256 \, b^{6} d^{5} e - 1664 \, a b^{5} d^{4} e^{2} + 4576 \, a^{2} b^{4} d^{3} e^{3} - 6864 \, a^{3} b^{3} d^{2} e^{4} + 6006 \, a^{4} b^{2} d e^{5} - 3003 \, a^{5} b e^{6}\right )} x\right )} \sqrt {e x + d}}{3003 \, e^{7}} \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)^(5/2)/(e*x+d)^(1/2),x, algorithm="fricas")

[Out]

2/3003*(231*b^6*e^6*x^6 + 1024*b^6*d^6 - 6656*a*b^5*d^5*e + 18304*a^2*b^4*d^4*e^2 - 27456*a^3*b^3*d^3*e^3 + 24
024*a^4*b^2*d^2*e^4 - 12012*a^5*b*d*e^5 + 3003*a^6*e^6 - 126*(2*b^6*d*e^5 - 13*a*b^5*e^6)*x^5 + 35*(8*b^6*d^2*
e^4 - 52*a*b^5*d*e^5 + 143*a^2*b^4*e^6)*x^4 - 20*(16*b^6*d^3*e^3 - 104*a*b^5*d^2*e^4 + 286*a^2*b^4*d*e^5 - 429
*a^3*b^3*e^6)*x^3 + 3*(128*b^6*d^4*e^2 - 832*a*b^5*d^3*e^3 + 2288*a^2*b^4*d^2*e^4 - 3432*a^3*b^3*d*e^5 + 3003*
a^4*b^2*e^6)*x^2 - 2*(256*b^6*d^5*e - 1664*a*b^5*d^4*e^2 + 4576*a^2*b^4*d^3*e^3 - 6864*a^3*b^3*d^2*e^4 + 6006*
a^4*b^2*d*e^5 - 3003*a^5*b*e^6)*x)*sqrt(e*x + d)/e^7

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giac [A]  time = 0.23, size = 437, normalized size = 1.18 \begin {gather*} \frac {2}{3003} \, {\left (6006 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} a^{5} b e^{\left (-1\right )} \mathrm {sgn}\left (b x + a\right ) + 3003 \, {\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^{4} b^{2} e^{\left (-2\right )} \mathrm {sgn}\left (b x + a\right ) + 1716 \, {\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^{3} b^{3} e^{\left (-3\right )} \mathrm {sgn}\left (b x + a\right ) + 143 \, {\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^{2} b^{4} e^{\left (-4\right )} \mathrm {sgn}\left (b x + a\right ) + 26 \, {\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 b^{5} e^{\left (-5\right )} \mathrm {sgn}\left (b x + a\right ) + {\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 )} b^{6} e^{\left (-6\right )} \mathrm {sgn}\left (b x + a\right ) + 3003 \, \sqrt {x e + d} a^{6} \mathrm {sgn}\left (b x + a\right )\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)^(5/2)/(e*x+d)^(1/2),x, algorithm="giac")

[Out]

2/3003*(6006*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*a^5*b*e^(-1)*sgn(b*x + a) + 3003*(3*(x*e + d)^(5/2) - 10*(x
*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*a^4*b^2*e^(-2)*sgn(b*x + a) + 1716*(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^3*b^3*e^(-3)*sgn(b*x + a) + 143*(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^2*b^4*e
^(-4)*sgn(b*x + a) + 26*(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*b^5*e^(-5)*sgn(b*x + a) + (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)*b^6*e^(-6)*sgn(b*x + a) + 3003*sqrt(x*e + d)*a^6*sgn(b*x
+ a))*e^(-1)

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maple [A]  time = 0.05, size = 393, normalized size = 1.06 \begin {gather*} \frac {2 \sqrt {e x +d}\, \left (231 b^{6} e^{6} x^{6}+1638 a \,b^{5} e^{6} x^{5}-252 b^{6} d \,e^{5} x^{5}+5005 a^{2} b^{4} e^{6} x^{4}-1820 a \,b^{5} d \,e^{5} x^{4}+280 b^{6} d^{2} e^{4} x^{4}+8580 a^{3} b^{3} e^{6} x^{3}-5720 a^{2} b^{4} d \,e^{5} x^{3}+2080 a \,b^{5} d^{2} e^{4} x^{3}-320 b^{6} d^{3} e^{3} x^{3}+9009 a^{4} b^{2} e^{6} x^{2}-10296 a^{3} b^{3} d \,e^{5} x^{2}+6864 a^{2} b^{4} d^{2} e^{4} x^{2}-2496 a \,b^{5} d^{3} e^{3} x^{2}+384 b^{6} d^{4} e^{2} x^{2}+6006 a^{5} b \,e^{6} x -12012 a^{4} b^{2} d \,e^{5} x +13728 a^{3} b^{3} d^{2} e^{4} x -9152 a^{2} b^{4} d^{3} e^{3} x +3328 a \,b^{5} d^{4} e^{2} x -512 b^{6} d^{5} e x +3003 a^{6} e^{6}-12012 a^{5} b d \,e^{5}+24024 a^{4} b^{2} d^{2} e^{4}-27456 a^{3} b^{3} d^{3} e^{3}+18304 a^{2} b^{4} d^{4} e^{2}-6656 a \,b^{5} d^{5} e +1024 b^{6} d^{6}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{3003 \left (b x +a \right )^{5} e^{7}} \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)^(5/2)/(e*x+d)^(1/2),x)

[Out]

2/3003*(e*x+d)^(1/2)*(231*b^6*e^6*x^6+1638*a*b^5*e^6*x^5-252*b^6*d*e^5*x^5+5005*a^2*b^4*e^6*x^4-1820*a*b^5*d*e
^5*x^4+280*b^6*d^2*e^4*x^4+8580*a^3*b^3*e^6*x^3-5720*a^2*b^4*d*e^5*x^3+2080*a*b^5*d^2*e^4*x^3-320*b^6*d^3*e^3*
x^3+9009*a^4*b^2*e^6*x^2-10296*a^3*b^3*d*e^5*x^2+6864*a^2*b^4*d^2*e^4*x^2-2496*a*b^5*d^3*e^3*x^2+384*b^6*d^4*e
^2*x^2+6006*a^5*b*e^6*x-12012*a^4*b^2*d*e^5*x+13728*a^3*b^3*d^2*e^4*x-9152*a^2*b^4*d^3*e^3*x+3328*a*b^5*d^4*e^
2*x-512*b^6*d^5*e*x+3003*a^6*e^6-12012*a^5*b*d*e^5+24024*a^4*b^2*d^2*e^4-27456*a^3*b^3*d^3*e^3+18304*a^2*b^4*d
^4*e^2-6656*a*b^5*d^5*e+1024*b^6*d^6)*((b*x+a)^2)^(5/2)/e^7/(b*x+a)^5

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maxima [B]  time = 0.70, size = 758, normalized size = 2.05 \begin {gather*} \frac {2 \, {\left (63 \, b^{5} e^{6} x^{6} - 256 \, b^{5} d^{6} + 1408 \, a b^{4} d^{5} e - 3168 \, a^{2} b^{3} d^{4} e^{2} + 3696 \, a^{3} b^{2} d^{3} e^{3} - 2310 \, a^{4} b d^{2} e^{4} + 693 \, a^{5} d e^{5} - 7 \, {\left (b^{5} d e^{5} - 55 \, a b^{4} e^{6}\right )} x^{5} + 5 \, {\left (2 \, b^{5} d^{2} e^{4} - 11 \, a b^{4} d e^{5} + 198 \, a^{2} b^{3} e^{6}\right )} x^{4} - 2 \, {\left (8 \, b^{5} d^{3} e^{3} - 44 \, a b^{4} d^{2} e^{4} + 99 \, a^{2} b^{3} d e^{5} - 693 \, a^{3} b^{2} e^{6}\right )} x^{3} + {\left (32 \, b^{5} d^{4} e^{2} - 176 \, a b^{4} d^{3} e^{3} + 396 \, a^{2} b^{3} d^{2} e^{4} - 462 \, a^{3} b^{2} d e^{5} + 1155 \, a^{4} b e^{6}\right )} x^{2} - {\left (128 \, b^{5} d^{5} e - 704 \, a b^{4} d^{4} e^{2} + 1584 \, a^{2} b^{3} d^{3} e^{3} - 1848 \, a^{3} b^{2} d^{2} e^{4} + 1155 \, a^{4} b d e^{5} - 693 \, a^{5} e^{6}\right )} x\right )} a}{693 \, \sqrt {e x + d} e^{6}} + \frac {2 \, {\left (693 \, b^{5} e^{7} x^{7} + 3072 \, b^{5} d^{7} - 16640 \, a b^{4} d^{6} e + 36608 \, a^{2} b^{3} d^{5} e^{2} - 41184 \, a^{3} b^{2} d^{4} e^{3} + 24024 \, a^{4} b d^{3} e^{4} - 6006 \, a^{5} d^{2} e^{5} - 63 \, {\left (b^{5} d e^{6} - 65 \, a b^{4} e^{7}\right )} x^{6} + 7 \, {\left (12 \, b^{5} d^{2} e^{5} - 65 \, a b^{4} d e^{6} + 1430 \, a^{2} b^{3} e^{7}\right )} x^{5} - 10 \, {\left (12 \, b^{5} d^{3} e^{4} - 65 \, a b^{4} d^{2} e^{5} + 143 \, a^{2} b^{3} d e^{6} - 1287 \, a^{3} b^{2} e^{7}\right )} x^{4} + {\left (192 \, b^{5} d^{4} e^{3} - 1040 \, a b^{4} d^{3} e^{4} + 2288 \, a^{2} b^{3} d^{2} e^{5} - 2574 \, a^{3} b^{2} d e^{6} + 9009 \, a^{4} b e^{7}\right )} x^{3} - {\left (384 \, b^{5} d^{5} e^{2} - 2080 \, a b^{4} d^{4} e^{3} + 4576 \, a^{2} b^{3} d^{3} e^{4} - 5148 \, a^{3} b^{2} d^{2} e^{5} + 3003 \, a^{4} b d e^{6} - 3003 \, a^{5} e^{7}\right )} x^{2} + {\left (1536 \, b^{5} d^{6} e - 8320 \, a b^{4} d^{5} e^{2} + 18304 \, a^{2} b^{3} d^{4} e^{3} - 20592 \, a^{3} b^{2} d^{3} e^{4} + 12012 \, a^{4} b d^{2} e^{5} - 3003 \, a^{5} d e^{6}\right )} x\right )} b}{9009 \, \sqrt {e x + d} e^{7}} \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)^(5/2)/(e*x+d)^(1/2),x, algorithm="maxima")

[Out]

2/693*(63*b^5*e^6*x^6 - 256*b^5*d^6 + 1408*a*b^4*d^5*e - 3168*a^2*b^3*d^4*e^2 + 3696*a^3*b^2*d^3*e^3 - 2310*a^
4*b*d^2*e^4 + 693*a^5*d*e^5 - 7*(b^5*d*e^5 - 55*a*b^4*e^6)*x^5 + 5*(2*b^5*d^2*e^4 - 11*a*b^4*d*e^5 + 198*a^2*b
^3*e^6)*x^4 - 2*(8*b^5*d^3*e^3 - 44*a*b^4*d^2*e^4 + 99*a^2*b^3*d*e^5 - 693*a^3*b^2*e^6)*x^3 + (32*b^5*d^4*e^2
- 176*a*b^4*d^3*e^3 + 396*a^2*b^3*d^2*e^4 - 462*a^3*b^2*d*e^5 + 1155*a^4*b*e^6)*x^2 - (128*b^5*d^5*e - 704*a*b
^4*d^4*e^2 + 1584*a^2*b^3*d^3*e^3 - 1848*a^3*b^2*d^2*e^4 + 1155*a^4*b*d*e^5 - 693*a^5*e^6)*x)*a/(sqrt(e*x + d)
*e^6) + 2/9009*(693*b^5*e^7*x^7 + 3072*b^5*d^7 - 16640*a*b^4*d^6*e + 36608*a^2*b^3*d^5*e^2 - 41184*a^3*b^2*d^4
*e^3 + 24024*a^4*b*d^3*e^4 - 6006*a^5*d^2*e^5 - 63*(b^5*d*e^6 - 65*a*b^4*e^7)*x^6 + 7*(12*b^5*d^2*e^5 - 65*a*b
^4*d*e^6 + 1430*a^2*b^3*e^7)*x^5 - 10*(12*b^5*d^3*e^4 - 65*a*b^4*d^2*e^5 + 143*a^2*b^3*d*e^6 - 1287*a^3*b^2*e^
7)*x^4 + (192*b^5*d^4*e^3 - 1040*a*b^4*d^3*e^4 + 2288*a^2*b^3*d^2*e^5 - 2574*a^3*b^2*d*e^6 + 9009*a^4*b*e^7)*x
^3 - (384*b^5*d^5*e^2 - 2080*a*b^4*d^4*e^3 + 4576*a^2*b^3*d^3*e^4 - 5148*a^3*b^2*d^2*e^5 + 3003*a^4*b*d*e^6 -
3003*a^5*e^7)*x^2 + (1536*b^5*d^6*e - 8320*a*b^4*d^5*e^2 + 18304*a^2*b^3*d^4*e^3 - 20592*a^3*b^2*d^3*e^4 + 120
12*a^4*b*d^2*e^5 - 3003*a^5*d*e^6)*x)*b/(sqrt(e*x + d)*e^7)

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mupad [B]  time = 2.78, size = 491, normalized size = 1.33 \begin {gather*} \frac {\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}\,\left (\frac {2\,b^5\,x^7}{13}+\frac {6006\,a^6\,d\,e^6-24024\,a^5\,b\,d^2\,e^5+48048\,a^4\,b^2\,d^3\,e^4-54912\,a^3\,b^3\,d^4\,e^3+36608\,a^2\,b^4\,d^5\,e^2-13312\,a\,b^5\,d^6\,e+2048\,b^6\,d^7}{3003\,b\,e^7}+\frac {10\,b^2\,x^4\,\left (1716\,a^3\,e^3-143\,a^2\,b\,d\,e^2+52\,a\,b^2\,d^2\,e-8\,b^3\,d^3\right )}{3003\,e^3}+\frac {2\,b^4\,x^6\,\left (78\,a\,e-b\,d\right )}{143\,e}+\frac {2\,b^3\,x^5\,\left (715\,a^2\,e^2-26\,a\,b\,d\,e+4\,b^2\,d^2\right )}{429\,e^2}+\frac {x\,\left (6006\,a^6\,e^7-12012\,a^5\,b\,d\,e^6+24024\,a^4\,b^2\,d^2\,e^5-27456\,a^3\,b^3\,d^3\,e^4+18304\,a^2\,b^4\,d^4\,e^3-6656\,a\,b^5\,d^5\,e^2+1024\,b^6\,d^6\,e\right )}{3003\,b\,e^7}+\frac {x^3\,\left (18018\,a^4\,b^2\,e^7-3432\,a^3\,b^3\,d\,e^6+2288\,a^2\,b^4\,d^2\,e^5-832\,a\,b^5\,d^3\,e^4+128\,b^6\,d^4\,e^3\right )}{3003\,b\,e^7}+\frac {x^2\,\left (12012\,a^5\,b\,e^7-6006\,a^4\,b^2\,d\,e^6+6864\,a^3\,b^3\,d^2\,e^5-4576\,a^2\,b^4\,d^3\,e^4+1664\,a\,b^5\,d^4\,e^3-256\,b^6\,d^5\,e^2\right )}{3003\,b\,e^7}\right )}{x\,\sqrt {d+e\,x}+\frac {a\,\sqrt {d+e\,x}}{b}} \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)^(5/2))/(d + e*x)^(1/2),x)

[Out]

((a^2 + b^2*x^2 + 2*a*b*x)^(1/2)*((2*b^5*x^7)/13 + (2048*b^6*d^7 + 6006*a^6*d*e^6 - 24024*a^5*b*d^2*e^5 + 3660
8*a^2*b^4*d^5*e^2 - 54912*a^3*b^3*d^4*e^3 + 48048*a^4*b^2*d^3*e^4 - 13312*a*b^5*d^6*e)/(3003*b*e^7) + (10*b^2*
x^4*(1716*a^3*e^3 - 8*b^3*d^3 + 52*a*b^2*d^2*e - 143*a^2*b*d*e^2))/(3003*e^3) + (2*b^4*x^6*(78*a*e - b*d))/(14
3*e) + (2*b^3*x^5*(715*a^2*e^2 + 4*b^2*d^2 - 26*a*b*d*e))/(429*e^2) + (x*(6006*a^6*e^7 + 1024*b^6*d^6*e - 6656
*a*b^5*d^5*e^2 + 18304*a^2*b^4*d^4*e^3 - 27456*a^3*b^3*d^3*e^4 + 24024*a^4*b^2*d^2*e^5 - 12012*a^5*b*d*e^6))/(
3003*b*e^7) + (x^3*(18018*a^4*b^2*e^7 + 128*b^6*d^4*e^3 - 832*a*b^5*d^3*e^4 - 3432*a^3*b^3*d*e^6 + 2288*a^2*b^
4*d^2*e^5))/(3003*b*e^7) + (x^2*(12012*a^5*b*e^7 - 256*b^6*d^5*e^2 + 1664*a*b^5*d^4*e^3 - 6006*a^4*b^2*d*e^6 -
4576*a^2*b^4*d^3*e^4 + 6864*a^3*b^3*d^2*e^5))/(3003*b*e^7)))/(x*(d + e*x)^(1/2) + (a*(d + e*x)^(1/2))/b)

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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)**(5/2)/(e*x+d)**(1/2),x)

[Out]

Timed out

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