3.2115 \(\int \frac {(a+b x) (a^2+2 a b x+b^2 x^2)^{5/2}}{(d+e x)^{5/2}} \, dx\)

Optimal. Leaf size=370 \[ \frac {30 b^2 \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x} (b d-a e)^4}{e^7 (a+b x)}+\frac {12 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5}{e^7 (a+b x) \sqrt {d+e x}}-\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^6}{3 e^7 (a+b x) (d+e x)^{3/2}}+\frac {2 b^6 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{9/2}}{9 e^7 (a+b x)}-\frac {12 b^5 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)}{7 e^7 (a+b x)}+\frac {6 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^2}{e^7 (a+b x)}-\frac {40 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^3}{3 e^7 (a+b x)} \]

[Out]

-2/3*(-a*e+b*d)^6*((b*x+a)^2)^(1/2)/e^7/(b*x+a)/(e*x+d)^(3/2)-40/3*b^3*(-a*e+b*d)^3*(e*x+d)^(3/2)*((b*x+a)^2)^
(1/2)/e^7/(b*x+a)+6*b^4*(-a*e+b*d)^2*(e*x+d)^(5/2)*((b*x+a)^2)^(1/2)/e^7/(b*x+a)-12/7*b^5*(-a*e+b*d)*(e*x+d)^(
7/2)*((b*x+a)^2)^(1/2)/e^7/(b*x+a)+2/9*b^6*(e*x+d)^(9/2)*((b*x+a)^2)^(1/2)/e^7/(b*x+a)+12*b*(-a*e+b*d)^5*((b*x
+a)^2)^(1/2)/e^7/(b*x+a)/(e*x+d)^(1/2)+30*b^2*(-a*e+b*d)^4*(e*x+d)^(1/2)*((b*x+a)^2)^(1/2)/e^7/(b*x+a)

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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} \[ \frac {2 b^6 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{9/2}}{9 e^7 (a+b x)}-\frac {12 b^5 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)}{7 e^7 (a+b x)}+\frac {6 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^2}{e^7 (a+b x)}-\frac {40 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^3}{3 e^7 (a+b x)}+\frac {30 b^2 \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x} (b d-a e)^4}{e^7 (a+b x)}+\frac {12 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5}{e^7 (a+b x) \sqrt {d+e x}}-\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^6}{3 e^7 (a+b x) (d+e x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(b*d - a*e)^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^7*(a + b*x)*(d + e*x)^(3/2)) + (12*b*(b*d - a*e)^5*Sqrt[
a^2 + 2*a*b*x + b^2*x^2])/(e^7*(a + b*x)*Sqrt[d + e*x]) + (30*b^2*(b*d - a*e)^4*Sqrt[d + e*x]*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)^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^7*(
a + b*x)) + (6*b^4*(b*d - a*e)^2*(d + e*x)^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(a + b*x)) - (12*b^5*(b*d
- a*e)*(d + e*x)^(7/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*e^7*(a + b*x)) + (2*b^6*(d + e*x)^(9/2)*Sqrt[a^2 + 2
*a*b*x + b^2*x^2])/(9*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}}{(d+e x)^{5/2}} \, 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}{(d+e x)^{5/2}} \, 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}{(d+e x)^{5/2}} \, 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 (d+e x)^{5/2}}-\frac {6 b (b d-a e)^5}{e^6 (d+e x)^{3/2}}+\frac {15 b^2 (b d-a e)^4}{e^6 \sqrt {d+e x}}-\frac {20 b^3 (b d-a e)^3 \sqrt {d+e x}}{e^6}+\frac {15 b^4 (b d-a e)^2 (d+e x)^{3/2}}{e^6}-\frac {6 b^5 (b d-a e) (d+e x)^{5/2}}{e^6}+\frac {b^6 (d+e x)^{7/2}}{e^6}\right ) \, dx}{a b+b^2 x}\\ &=-\frac {2 (b d-a e)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^7 (a+b x) (d+e x)^{3/2}}+\frac {12 b (b d-a e)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x) \sqrt {d+e x}}+\frac {30 b^2 (b d-a e)^4 \sqrt {d+e x} \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)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^7 (a+b x)}+\frac {6 b^4 (b d-a e)^2 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x)}-\frac {12 b^5 (b d-a e) (d+e x)^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^7 (a+b x)}+\frac {2 b^6 (d+e x)^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}{9 e^7 (a+b x)}\\ \end {align*}

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Mathematica [A]  time = 0.15, size = 163, normalized size = 0.44 \[ \frac {2 \sqrt {(a+b x)^2} \left (-54 b^5 (d+e x)^5 (b d-a e)+189 b^4 (d+e x)^4 (b d-a e)^2-420 b^3 (d+e x)^3 (b d-a e)^3+945 b^2 (d+e x)^2 (b d-a e)^4+378 b (d+e x) (b d-a e)^5-21 (b d-a e)^6+7 b^6 (d+e x)^6\right )}{63 e^7 (a+b x) (d+e x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[(a + b*x)^2]*(-21*(b*d - a*e)^6 + 378*b*(b*d - a*e)^5*(d + e*x) + 945*b^2*(b*d - a*e)^4*(d + e*x)^2 -
420*b^3*(b*d - a*e)^3*(d + e*x)^3 + 189*b^4*(b*d - a*e)^2*(d + e*x)^4 - 54*b^5*(b*d - a*e)*(d + e*x)^5 + 7*b^6
*(d + e*x)^6))/(63*e^7*(a + b*x)*(d + e*x)^(3/2))

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fricas [A]  time = 0.77, size = 377, normalized size = 1.02 \[ \frac {2 \, {\left (7 \, b^{6} e^{6} x^{6} + 1024 \, b^{6} d^{6} - 4608 \, a b^{5} d^{5} e + 8064 \, a^{2} b^{4} d^{4} e^{2} - 6720 \, a^{3} b^{3} d^{3} e^{3} + 2520 \, a^{4} b^{2} d^{2} e^{4} - 252 \, a^{5} b d e^{5} - 21 \, a^{6} e^{6} - 6 \, {\left (2 \, b^{6} d e^{5} - 9 \, a b^{5} e^{6}\right )} x^{5} + 3 \, {\left (8 \, b^{6} d^{2} e^{4} - 36 \, a b^{5} d e^{5} + 63 \, a^{2} b^{4} e^{6}\right )} x^{4} - 4 \, {\left (16 \, b^{6} d^{3} e^{3} - 72 \, a b^{5} d^{2} e^{4} + 126 \, a^{2} b^{4} d e^{5} - 105 \, a^{3} b^{3} e^{6}\right )} x^{3} + 3 \, {\left (128 \, b^{6} d^{4} e^{2} - 576 \, a b^{5} d^{3} e^{3} + 1008 \, a^{2} b^{4} d^{2} e^{4} - 840 \, a^{3} b^{3} d e^{5} + 315 \, a^{4} b^{2} e^{6}\right )} x^{2} + 6 \, {\left (256 \, b^{6} d^{5} e - 1152 \, a b^{5} d^{4} e^{2} + 2016 \, a^{2} b^{4} d^{3} e^{3} - 1680 \, a^{3} b^{3} d^{2} e^{4} + 630 \, a^{4} b^{2} d e^{5} - 63 \, a^{5} b e^{6}\right )} x\right )} \sqrt {e x + d}}{63 \, {\left (e^{9} x^{2} + 2 \, d e^{8} x + d^{2} e^{7}\right )}} \]

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)^(5/2),x, algorithm="fricas")

[Out]

2/63*(7*b^6*e^6*x^6 + 1024*b^6*d^6 - 4608*a*b^5*d^5*e + 8064*a^2*b^4*d^4*e^2 - 6720*a^3*b^3*d^3*e^3 + 2520*a^4
*b^2*d^2*e^4 - 252*a^5*b*d*e^5 - 21*a^6*e^6 - 6*(2*b^6*d*e^5 - 9*a*b^5*e^6)*x^5 + 3*(8*b^6*d^2*e^4 - 36*a*b^5*
d*e^5 + 63*a^2*b^4*e^6)*x^4 - 4*(16*b^6*d^3*e^3 - 72*a*b^5*d^2*e^4 + 126*a^2*b^4*d*e^5 - 105*a^3*b^3*e^6)*x^3
+ 3*(128*b^6*d^4*e^2 - 576*a*b^5*d^3*e^3 + 1008*a^2*b^4*d^2*e^4 - 840*a^3*b^3*d*e^5 + 315*a^4*b^2*e^6)*x^2 + 6
*(256*b^6*d^5*e - 1152*a*b^5*d^4*e^2 + 2016*a^2*b^4*d^3*e^3 - 1680*a^3*b^3*d^2*e^4 + 630*a^4*b^2*d*e^5 - 63*a^
5*b*e^6)*x)*sqrt(e*x + d)/(e^9*x^2 + 2*d*e^8*x + d^2*e^7)

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giac [B]  time = 0.28, size = 630, normalized size = 1.70 \[ \frac {2}{63} \, {\left (7 \, {\left (x e + d\right )}^{\frac {9}{2}} b^{6} e^{56} \mathrm {sgn}\left (b x + a\right ) - 54 \, {\left (x e + d\right )}^{\frac {7}{2}} b^{6} d e^{56} \mathrm {sgn}\left (b x + a\right ) + 189 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{6} d^{2} e^{56} \mathrm {sgn}\left (b x + a\right ) - 420 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{6} d^{3} e^{56} \mathrm {sgn}\left (b x + a\right ) + 945 \, \sqrt {x e + d} b^{6} d^{4} e^{56} \mathrm {sgn}\left (b x + a\right ) + 54 \, {\left (x e + d\right )}^{\frac {7}{2}} a b^{5} e^{57} \mathrm {sgn}\left (b x + a\right ) - 378 \, {\left (x e + d\right )}^{\frac {5}{2}} a b^{5} d e^{57} \mathrm {sgn}\left (b x + a\right ) + 1260 \, {\left (x e + d\right )}^{\frac {3}{2}} a b^{5} d^{2} e^{57} \mathrm {sgn}\left (b x + a\right ) - 3780 \, \sqrt {x e + d} a b^{5} d^{3} e^{57} \mathrm {sgn}\left (b x + a\right ) + 189 \, {\left (x e + d\right )}^{\frac {5}{2}} a^{2} b^{4} e^{58} \mathrm {sgn}\left (b x + a\right ) - 1260 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{2} b^{4} d e^{58} \mathrm {sgn}\left (b x + a\right ) + 5670 \, \sqrt {x e + d} a^{2} b^{4} d^{2} e^{58} \mathrm {sgn}\left (b x + a\right ) + 420 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{3} b^{3} e^{59} \mathrm {sgn}\left (b x + a\right ) - 3780 \, \sqrt {x e + d} a^{3} b^{3} d e^{59} \mathrm {sgn}\left (b x + a\right ) + 945 \, \sqrt {x e + d} a^{4} b^{2} e^{60} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-63\right )} + \frac {2 \, {\left (18 \, {\left (x e + d\right )} b^{6} d^{5} \mathrm {sgn}\left (b x + a\right ) - b^{6} d^{6} \mathrm {sgn}\left (b x + a\right ) - 90 \, {\left (x e + d\right )} a b^{5} d^{4} e \mathrm {sgn}\left (b x + a\right ) + 6 \, a b^{5} d^{5} e \mathrm {sgn}\left (b x + a\right ) + 180 \, {\left (x e + d\right )} a^{2} b^{4} d^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) - 15 \, a^{2} b^{4} d^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) - 180 \, {\left (x e + d\right )} a^{3} b^{3} d^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) + 20 \, a^{3} b^{3} d^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) + 90 \, {\left (x e + d\right )} a^{4} b^{2} d e^{4} \mathrm {sgn}\left (b x + a\right ) - 15 \, a^{4} b^{2} d^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) - 18 \, {\left (x e + d\right )} a^{5} b e^{5} \mathrm {sgn}\left (b x + a\right ) + 6 \, a^{5} b d e^{5} \mathrm {sgn}\left (b x + a\right ) - a^{6} e^{6} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-7\right )}}{3 \, {\left (x e + d\right )}^{\frac {3}{2}}} \]

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)^(5/2),x, algorithm="giac")

[Out]

2/63*(7*(x*e + d)^(9/2)*b^6*e^56*sgn(b*x + a) - 54*(x*e + d)^(7/2)*b^6*d*e^56*sgn(b*x + a) + 189*(x*e + d)^(5/
2)*b^6*d^2*e^56*sgn(b*x + a) - 420*(x*e + d)^(3/2)*b^6*d^3*e^56*sgn(b*x + a) + 945*sqrt(x*e + d)*b^6*d^4*e^56*
sgn(b*x + a) + 54*(x*e + d)^(7/2)*a*b^5*e^57*sgn(b*x + a) - 378*(x*e + d)^(5/2)*a*b^5*d*e^57*sgn(b*x + a) + 12
60*(x*e + d)^(3/2)*a*b^5*d^2*e^57*sgn(b*x + a) - 3780*sqrt(x*e + d)*a*b^5*d^3*e^57*sgn(b*x + a) + 189*(x*e + d
)^(5/2)*a^2*b^4*e^58*sgn(b*x + a) - 1260*(x*e + d)^(3/2)*a^2*b^4*d*e^58*sgn(b*x + a) + 5670*sqrt(x*e + d)*a^2*
b^4*d^2*e^58*sgn(b*x + a) + 420*(x*e + d)^(3/2)*a^3*b^3*e^59*sgn(b*x + a) - 3780*sqrt(x*e + d)*a^3*b^3*d*e^59*
sgn(b*x + a) + 945*sqrt(x*e + d)*a^4*b^2*e^60*sgn(b*x + a))*e^(-63) + 2/3*(18*(x*e + d)*b^6*d^5*sgn(b*x + a) -
b^6*d^6*sgn(b*x + a) - 90*(x*e + d)*a*b^5*d^4*e*sgn(b*x + a) + 6*a*b^5*d^5*e*sgn(b*x + a) + 180*(x*e + d)*a^2
*b^4*d^3*e^2*sgn(b*x + a) - 15*a^2*b^4*d^4*e^2*sgn(b*x + a) - 180*(x*e + d)*a^3*b^3*d^2*e^3*sgn(b*x + a) + 20*
a^3*b^3*d^3*e^3*sgn(b*x + a) + 90*(x*e + d)*a^4*b^2*d*e^4*sgn(b*x + a) - 15*a^4*b^2*d^2*e^4*sgn(b*x + a) - 18*
(x*e + d)*a^5*b*e^5*sgn(b*x + a) + 6*a^5*b*d*e^5*sgn(b*x + a) - a^6*e^6*sgn(b*x + a))*e^(-7)/(x*e + d)^(3/2)

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maple [A]  time = 0.05, size = 393, normalized size = 1.06 \[ -\frac {2 \left (-7 b^{6} e^{6} x^{6}-54 a \,b^{5} e^{6} x^{5}+12 b^{6} d \,e^{5} x^{5}-189 a^{2} b^{4} e^{6} x^{4}+108 a \,b^{5} d \,e^{5} x^{4}-24 b^{6} d^{2} e^{4} x^{4}-420 a^{3} b^{3} e^{6} x^{3}+504 a^{2} b^{4} d \,e^{5} x^{3}-288 a \,b^{5} d^{2} e^{4} x^{3}+64 b^{6} d^{3} e^{3} x^{3}-945 a^{4} b^{2} e^{6} x^{2}+2520 a^{3} b^{3} d \,e^{5} x^{2}-3024 a^{2} b^{4} d^{2} e^{4} x^{2}+1728 a \,b^{5} d^{3} e^{3} x^{2}-384 b^{6} d^{4} e^{2} x^{2}+378 a^{5} b \,e^{6} x -3780 a^{4} b^{2} d \,e^{5} x +10080 a^{3} b^{3} d^{2} e^{4} x -12096 a^{2} b^{4} d^{3} e^{3} x +6912 a \,b^{5} d^{4} e^{2} x -1536 b^{6} d^{5} e x +21 a^{6} e^{6}+252 a^{5} b d \,e^{5}-2520 a^{4} b^{2} d^{2} e^{4}+6720 a^{3} b^{3} d^{3} e^{3}-8064 a^{2} b^{4} d^{4} e^{2}+4608 a \,b^{5} d^{5} e -1024 b^{6} d^{6}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{63 \left (e x +d \right )^{\frac {3}{2}} \left (b x +a \right )^{5} e^{7}} \]

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)^(5/2),x)

[Out]

-2/63/(e*x+d)^(3/2)*(-7*b^6*e^6*x^6-54*a*b^5*e^6*x^5+12*b^6*d*e^5*x^5-189*a^2*b^4*e^6*x^4+108*a*b^5*d*e^5*x^4-
24*b^6*d^2*e^4*x^4-420*a^3*b^3*e^6*x^3+504*a^2*b^4*d*e^5*x^3-288*a*b^5*d^2*e^4*x^3+64*b^6*d^3*e^3*x^3-945*a^4*
b^2*e^6*x^2+2520*a^3*b^3*d*e^5*x^2-3024*a^2*b^4*d^2*e^4*x^2+1728*a*b^5*d^3*e^3*x^2-384*b^6*d^4*e^2*x^2+378*a^5
*b*e^6*x-3780*a^4*b^2*d*e^5*x+10080*a^3*b^3*d^2*e^4*x-12096*a^2*b^4*d^3*e^3*x+6912*a*b^5*d^4*e^2*x-1536*b^6*d^
5*e*x+21*a^6*e^6+252*a^5*b*d*e^5-2520*a^4*b^2*d^2*e^4+6720*a^3*b^3*d^3*e^3-8064*a^2*b^4*d^4*e^2+4608*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 = 0.88, size = 625, normalized size = 1.69 \[ \frac {2 \, {\left (3 \, b^{5} e^{5} x^{5} - 256 \, b^{5} d^{5} + 896 \, a b^{4} d^{4} e - 1120 \, a^{2} b^{3} d^{3} e^{2} + 560 \, a^{3} b^{2} d^{2} e^{3} - 70 \, a^{4} b d e^{4} - 7 \, a^{5} e^{5} - 3 \, {\left (2 \, b^{5} d e^{4} - 7 \, a b^{4} e^{5}\right )} x^{4} + 2 \, {\left (8 \, b^{5} d^{2} e^{3} - 28 \, a b^{4} d e^{4} + 35 \, a^{2} b^{3} e^{5}\right )} x^{3} - 6 \, {\left (16 \, b^{5} d^{3} e^{2} - 56 \, a b^{4} d^{2} e^{3} + 70 \, a^{2} b^{3} d e^{4} - 35 \, a^{3} b^{2} e^{5}\right )} x^{2} - 3 \, {\left (128 \, b^{5} d^{4} e - 448 \, a b^{4} d^{3} e^{2} + 560 \, a^{2} b^{3} d^{2} e^{3} - 280 \, a^{3} b^{2} d e^{4} + 35 \, a^{4} b e^{5}\right )} x\right )} a}{21 \, {\left (e^{7} x + d e^{6}\right )} \sqrt {e x + d}} + \frac {2 \, {\left (7 \, b^{5} e^{6} x^{6} + 1024 \, b^{5} d^{6} - 3840 \, a b^{4} d^{5} e + 5376 \, a^{2} b^{3} d^{4} e^{2} - 3360 \, a^{3} b^{2} d^{3} e^{3} + 840 \, a^{4} b d^{2} e^{4} - 42 \, a^{5} d e^{5} - 3 \, {\left (4 \, b^{5} d e^{5} - 15 \, a b^{4} e^{6}\right )} x^{5} + 6 \, {\left (4 \, b^{5} d^{2} e^{4} - 15 \, a b^{4} d e^{5} + 21 \, a^{2} b^{3} e^{6}\right )} x^{4} - 2 \, {\left (32 \, b^{5} d^{3} e^{3} - 120 \, a b^{4} d^{2} e^{4} + 168 \, a^{2} b^{3} d e^{5} - 105 \, a^{3} b^{2} e^{6}\right )} x^{3} + 3 \, {\left (128 \, b^{5} d^{4} e^{2} - 480 \, a b^{4} d^{3} e^{3} + 672 \, a^{2} b^{3} d^{2} e^{4} - 420 \, a^{3} b^{2} d e^{5} + 105 \, a^{4} b e^{6}\right )} x^{2} + 3 \, {\left (512 \, b^{5} d^{5} e - 1920 \, a b^{4} d^{4} e^{2} + 2688 \, a^{2} b^{3} d^{3} e^{3} - 1680 \, a^{3} b^{2} d^{2} e^{4} + 420 \, a^{4} b d e^{5} - 21 \, a^{5} e^{6}\right )} x\right )} b}{63 \, {\left (e^{8} x + d e^{7}\right )} \sqrt {e x + d}} \]

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)^(5/2),x, algorithm="maxima")

[Out]

2/21*(3*b^5*e^5*x^5 - 256*b^5*d^5 + 896*a*b^4*d^4*e - 1120*a^2*b^3*d^3*e^2 + 560*a^3*b^2*d^2*e^3 - 70*a^4*b*d*
e^4 - 7*a^5*e^5 - 3*(2*b^5*d*e^4 - 7*a*b^4*e^5)*x^4 + 2*(8*b^5*d^2*e^3 - 28*a*b^4*d*e^4 + 35*a^2*b^3*e^5)*x^3
- 6*(16*b^5*d^3*e^2 - 56*a*b^4*d^2*e^3 + 70*a^2*b^3*d*e^4 - 35*a^3*b^2*e^5)*x^2 - 3*(128*b^5*d^4*e - 448*a*b^4
*d^3*e^2 + 560*a^2*b^3*d^2*e^3 - 280*a^3*b^2*d*e^4 + 35*a^4*b*e^5)*x)*a/((e^7*x + d*e^6)*sqrt(e*x + d)) + 2/63
*(7*b^5*e^6*x^6 + 1024*b^5*d^6 - 3840*a*b^4*d^5*e + 5376*a^2*b^3*d^4*e^2 - 3360*a^3*b^2*d^3*e^3 + 840*a^4*b*d^
2*e^4 - 42*a^5*d*e^5 - 3*(4*b^5*d*e^5 - 15*a*b^4*e^6)*x^5 + 6*(4*b^5*d^2*e^4 - 15*a*b^4*d*e^5 + 21*a^2*b^3*e^6
)*x^4 - 2*(32*b^5*d^3*e^3 - 120*a*b^4*d^2*e^4 + 168*a^2*b^3*d*e^5 - 105*a^3*b^2*e^6)*x^3 + 3*(128*b^5*d^4*e^2
- 480*a*b^4*d^3*e^3 + 672*a^2*b^3*d^2*e^4 - 420*a^3*b^2*d*e^5 + 105*a^4*b*e^6)*x^2 + 3*(512*b^5*d^5*e - 1920*a
*b^4*d^4*e^2 + 2688*a^2*b^3*d^3*e^3 - 1680*a^3*b^2*d^2*e^4 + 420*a^4*b*d*e^5 - 21*a^5*e^6)*x)*b/((e^8*x + d*e^
7)*sqrt(e*x + d))

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mupad [B]  time = 3.15, size = 432, normalized size = 1.17 \[ \frac {\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}\,\left (\frac {2\,b^5\,x^6}{9\,e^2}-\frac {42\,a^6\,e^6+504\,a^5\,b\,d\,e^5-5040\,a^4\,b^2\,d^2\,e^4+13440\,a^3\,b^3\,d^3\,e^3-16128\,a^2\,b^4\,d^4\,e^2+9216\,a\,b^5\,d^5\,e-2048\,b^6\,d^6}{63\,b\,e^8}-\frac {x\,\left (756\,a^5\,b\,e^6-7560\,a^4\,b^2\,d\,e^5+20160\,a^3\,b^3\,d^2\,e^4-24192\,a^2\,b^4\,d^3\,e^3+13824\,a\,b^5\,d^4\,e^2-3072\,b^6\,d^5\,e\right )}{63\,b\,e^8}+\frac {8\,b^2\,x^3\,\left (105\,a^3\,e^3-126\,a^2\,b\,d\,e^2+72\,a\,b^2\,d^2\,e-16\,b^3\,d^3\right )}{63\,e^5}+\frac {4\,b^4\,x^5\,\left (9\,a\,e-2\,b\,d\right )}{21\,e^3}+\frac {2\,b^3\,x^4\,\left (63\,a^2\,e^2-36\,a\,b\,d\,e+8\,b^2\,d^2\right )}{21\,e^4}+\frac {x^2\,\left (1890\,a^4\,b^2\,e^6-5040\,a^3\,b^3\,d\,e^5+6048\,a^2\,b^4\,d^2\,e^4-3456\,a\,b^5\,d^3\,e^3+768\,b^6\,d^4\,e^2\right )}{63\,b\,e^8}\right )}{x^2\,\sqrt {d+e\,x}+\frac {a\,d\,\sqrt {d+e\,x}}{b\,e}+\frac {x\,\left (63\,a\,e^8+63\,b\,d\,e^7\right )\,\sqrt {d+e\,x}}{63\,b\,e^8}} \]

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)^(5/2),x)

[Out]

((a^2 + b^2*x^2 + 2*a*b*x)^(1/2)*((2*b^5*x^6)/(9*e^2) - (42*a^6*e^6 - 2048*b^6*d^6 - 16128*a^2*b^4*d^4*e^2 + 1
3440*a^3*b^3*d^3*e^3 - 5040*a^4*b^2*d^2*e^4 + 9216*a*b^5*d^5*e + 504*a^5*b*d*e^5)/(63*b*e^8) - (x*(756*a^5*b*e
^6 - 3072*b^6*d^5*e + 13824*a*b^5*d^4*e^2 - 7560*a^4*b^2*d*e^5 - 24192*a^2*b^4*d^3*e^3 + 20160*a^3*b^3*d^2*e^4
))/(63*b*e^8) + (8*b^2*x^3*(105*a^3*e^3 - 16*b^3*d^3 + 72*a*b^2*d^2*e - 126*a^2*b*d*e^2))/(63*e^5) + (4*b^4*x^
5*(9*a*e - 2*b*d))/(21*e^3) + (2*b^3*x^4*(63*a^2*e^2 + 8*b^2*d^2 - 36*a*b*d*e))/(21*e^4) + (x^2*(1890*a^4*b^2*
e^6 + 768*b^6*d^4*e^2 - 3456*a*b^5*d^3*e^3 - 5040*a^3*b^3*d*e^5 + 6048*a^2*b^4*d^2*e^4))/(63*b*e^8)))/(x^2*(d
+ e*x)^(1/2) + (a*d*(d + e*x)^(1/2))/(b*e) + (x*(63*a*e^8 + 63*b*d*e^7)*(d + e*x)^(1/2))/(63*b*e^8))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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)**(5/2),x)

[Out]

Timed out

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