3.2012 \(\int \frac {(a+b x) (a^2+2 a b x+b^2 x^2)^{5/2}}{(d+e x)^{15}} \, dx\)

Optimal. Leaf size=362 \[ -\frac {5 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4}{4 e^7 (a+b x) (d+e x)^{12}}+\frac {6 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5}{13 e^7 (a+b x) (d+e x)^{13}}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^6}{14 e^7 (a+b x) (d+e x)^{14}}-\frac {b^6 \sqrt {a^2+2 a b x+b^2 x^2}}{8 e^7 (a+b x) (d+e x)^8}+\frac {2 b^5 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}{3 e^7 (a+b x) (d+e x)^9}-\frac {3 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}{2 e^7 (a+b x) (d+e x)^{10}}+\frac {20 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3}{11 e^7 (a+b x) (d+e x)^{11}} \]

[Out]

-1/14*(-a*e+b*d)^6*((b*x+a)^2)^(1/2)/e^7/(b*x+a)/(e*x+d)^14+6/13*b*(-a*e+b*d)^5*((b*x+a)^2)^(1/2)/e^7/(b*x+a)/
(e*x+d)^13-5/4*b^2*(-a*e+b*d)^4*((b*x+a)^2)^(1/2)/e^7/(b*x+a)/(e*x+d)^12+20/11*b^3*(-a*e+b*d)^3*((b*x+a)^2)^(1
/2)/e^7/(b*x+a)/(e*x+d)^11-3/2*b^4*(-a*e+b*d)^2*((b*x+a)^2)^(1/2)/e^7/(b*x+a)/(e*x+d)^10+2/3*b^5*(-a*e+b*d)*((
b*x+a)^2)^(1/2)/e^7/(b*x+a)/(e*x+d)^9-1/8*b^6*((b*x+a)^2)^(1/2)/e^7/(b*x+a)/(e*x+d)^8

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Rubi [A]  time = 0.20, antiderivative size = 362, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {770, 21, 43} \[ -\frac {b^6 \sqrt {a^2+2 a b x+b^2 x^2}}{8 e^7 (a+b x) (d+e x)^8}+\frac {2 b^5 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}{3 e^7 (a+b x) (d+e x)^9}-\frac {3 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}{2 e^7 (a+b x) (d+e x)^{10}}+\frac {20 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3}{11 e^7 (a+b x) (d+e x)^{11}}-\frac {5 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4}{4 e^7 (a+b x) (d+e x)^{12}}+\frac {6 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5}{13 e^7 (a+b x) (d+e x)^{13}}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^6}{14 e^7 (a+b x) (d+e x)^{14}} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^15,x]

[Out]

-((b*d - a*e)^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(14*e^7*(a + b*x)*(d + e*x)^14) + (6*b*(b*d - a*e)^5*Sqrt[a^2 +
2*a*b*x + b^2*x^2])/(13*e^7*(a + b*x)*(d + e*x)^13) - (5*b^2*(b*d - a*e)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(4*
e^7*(a + b*x)*(d + e*x)^12) + (20*b^3*(b*d - a*e)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*e^7*(a + b*x)*(d + e*x)
^11) - (3*b^4*(b*d - a*e)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(2*e^7*(a + b*x)*(d + e*x)^10) + (2*b^5*(b*d - a*e)
*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^7*(a + b*x)*(d + e*x)^9) - (b^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(8*e^7*(a
+ b*x)*(d + e*x)^8)

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)^{15}} \, 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)^{15}} \, 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)^{15}} \, 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)^{15}}-\frac {6 b (b d-a e)^5}{e^6 (d+e x)^{14}}+\frac {15 b^2 (b d-a e)^4}{e^6 (d+e x)^{13}}-\frac {20 b^3 (b d-a e)^3}{e^6 (d+e x)^{12}}+\frac {15 b^4 (b d-a e)^2}{e^6 (d+e x)^{11}}-\frac {6 b^5 (b d-a e)}{e^6 (d+e x)^{10}}+\frac {b^6}{e^6 (d+e x)^9}\right ) \, dx}{a b+b^2 x}\\ &=-\frac {(b d-a e)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{14 e^7 (a+b x) (d+e x)^{14}}+\frac {6 b (b d-a e)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{13 e^7 (a+b x) (d+e x)^{13}}-\frac {5 b^2 (b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{4 e^7 (a+b x) (d+e x)^{12}}+\frac {20 b^3 (b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{11 e^7 (a+b x) (d+e x)^{11}}-\frac {3 b^4 (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{2 e^7 (a+b x) (d+e x)^{10}}+\frac {2 b^5 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^7 (a+b x) (d+e x)^9}-\frac {b^6 \sqrt {a^2+2 a b x+b^2 x^2}}{8 e^7 (a+b x) (d+e x)^8}\\ \end {align*}

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Mathematica [A]  time = 0.11, size = 295, normalized size = 0.81 \[ -\frac {\sqrt {(a+b x)^2} \left (1716 a^6 e^6+792 a^5 b e^5 (d+14 e x)+330 a^4 b^2 e^4 \left (d^2+14 d e x+91 e^2 x^2\right )+120 a^3 b^3 e^3 \left (d^3+14 d^2 e x+91 d e^2 x^2+364 e^3 x^3\right )+36 a^2 b^4 e^2 \left (d^4+14 d^3 e x+91 d^2 e^2 x^2+364 d e^3 x^3+1001 e^4 x^4\right )+8 a b^5 e \left (d^5+14 d^4 e x+91 d^3 e^2 x^2+364 d^2 e^3 x^3+1001 d e^4 x^4+2002 e^5 x^5\right )+b^6 \left (d^6+14 d^5 e x+91 d^4 e^2 x^2+364 d^3 e^3 x^3+1001 d^2 e^4 x^4+2002 d e^5 x^5+3003 e^6 x^6\right )\right )}{24024 e^7 (a+b x) (d+e x)^{14}} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^15,x]

[Out]

-1/24024*(Sqrt[(a + b*x)^2]*(1716*a^6*e^6 + 792*a^5*b*e^5*(d + 14*e*x) + 330*a^4*b^2*e^4*(d^2 + 14*d*e*x + 91*
e^2*x^2) + 120*a^3*b^3*e^3*(d^3 + 14*d^2*e*x + 91*d*e^2*x^2 + 364*e^3*x^3) + 36*a^2*b^4*e^2*(d^4 + 14*d^3*e*x
+ 91*d^2*e^2*x^2 + 364*d*e^3*x^3 + 1001*e^4*x^4) + 8*a*b^5*e*(d^5 + 14*d^4*e*x + 91*d^3*e^2*x^2 + 364*d^2*e^3*
x^3 + 1001*d*e^4*x^4 + 2002*e^5*x^5) + b^6*(d^6 + 14*d^5*e*x + 91*d^4*e^2*x^2 + 364*d^3*e^3*x^3 + 1001*d^2*e^4
*x^4 + 2002*d*e^5*x^5 + 3003*e^6*x^6)))/(e^7*(a + b*x)*(d + e*x)^14)

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fricas [A]  time = 0.87, size = 496, normalized size = 1.37 \[ -\frac {3003 \, b^{6} e^{6} x^{6} + b^{6} d^{6} + 8 \, a b^{5} d^{5} e + 36 \, a^{2} b^{4} d^{4} e^{2} + 120 \, a^{3} b^{3} d^{3} e^{3} + 330 \, a^{4} b^{2} d^{2} e^{4} + 792 \, a^{5} b d e^{5} + 1716 \, a^{6} e^{6} + 2002 \, {\left (b^{6} d e^{5} + 8 \, a b^{5} e^{6}\right )} x^{5} + 1001 \, {\left (b^{6} d^{2} e^{4} + 8 \, a b^{5} d e^{5} + 36 \, a^{2} b^{4} e^{6}\right )} x^{4} + 364 \, {\left (b^{6} d^{3} e^{3} + 8 \, a b^{5} d^{2} e^{4} + 36 \, a^{2} b^{4} d e^{5} + 120 \, a^{3} b^{3} e^{6}\right )} x^{3} + 91 \, {\left (b^{6} d^{4} e^{2} + 8 \, a b^{5} d^{3} e^{3} + 36 \, a^{2} b^{4} d^{2} e^{4} + 120 \, a^{3} b^{3} d e^{5} + 330 \, a^{4} b^{2} e^{6}\right )} x^{2} + 14 \, {\left (b^{6} d^{5} e + 8 \, a b^{5} d^{4} e^{2} + 36 \, a^{2} b^{4} d^{3} e^{3} + 120 \, a^{3} b^{3} d^{2} e^{4} + 330 \, a^{4} b^{2} d e^{5} + 792 \, a^{5} b e^{6}\right )} x}{24024 \, {\left (e^{21} x^{14} + 14 \, d e^{20} x^{13} + 91 \, d^{2} e^{19} x^{12} + 364 \, d^{3} e^{18} x^{11} + 1001 \, d^{4} e^{17} x^{10} + 2002 \, d^{5} e^{16} x^{9} + 3003 \, d^{6} e^{15} x^{8} + 3432 \, d^{7} e^{14} x^{7} + 3003 \, d^{8} e^{13} x^{6} + 2002 \, d^{9} e^{12} x^{5} + 1001 \, d^{10} e^{11} x^{4} + 364 \, d^{11} e^{10} x^{3} + 91 \, d^{12} e^{9} x^{2} + 14 \, d^{13} e^{8} x + d^{14} e^{7}\right )}} \]

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)^15,x, algorithm="fricas")

[Out]

-1/24024*(3003*b^6*e^6*x^6 + b^6*d^6 + 8*a*b^5*d^5*e + 36*a^2*b^4*d^4*e^2 + 120*a^3*b^3*d^3*e^3 + 330*a^4*b^2*
d^2*e^4 + 792*a^5*b*d*e^5 + 1716*a^6*e^6 + 2002*(b^6*d*e^5 + 8*a*b^5*e^6)*x^5 + 1001*(b^6*d^2*e^4 + 8*a*b^5*d*
e^5 + 36*a^2*b^4*e^6)*x^4 + 364*(b^6*d^3*e^3 + 8*a*b^5*d^2*e^4 + 36*a^2*b^4*d*e^5 + 120*a^3*b^3*e^6)*x^3 + 91*
(b^6*d^4*e^2 + 8*a*b^5*d^3*e^3 + 36*a^2*b^4*d^2*e^4 + 120*a^3*b^3*d*e^5 + 330*a^4*b^2*e^6)*x^2 + 14*(b^6*d^5*e
+ 8*a*b^5*d^4*e^2 + 36*a^2*b^4*d^3*e^3 + 120*a^3*b^3*d^2*e^4 + 330*a^4*b^2*d*e^5 + 792*a^5*b*e^6)*x)/(e^21*x^
14 + 14*d*e^20*x^13 + 91*d^2*e^19*x^12 + 364*d^3*e^18*x^11 + 1001*d^4*e^17*x^10 + 2002*d^5*e^16*x^9 + 3003*d^6
*e^15*x^8 + 3432*d^7*e^14*x^7 + 3003*d^8*e^13*x^6 + 2002*d^9*e^12*x^5 + 1001*d^10*e^11*x^4 + 364*d^11*e^10*x^3
+ 91*d^12*e^9*x^2 + 14*d^13*e^8*x + d^14*e^7)

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giac [A]  time = 0.20, size = 520, normalized size = 1.44 \[ -\frac {{\left (3003 \, b^{6} x^{6} e^{6} \mathrm {sgn}\left (b x + a\right ) + 2002 \, b^{6} d x^{5} e^{5} \mathrm {sgn}\left (b x + a\right ) + 1001 \, b^{6} d^{2} x^{4} e^{4} \mathrm {sgn}\left (b x + a\right ) + 364 \, b^{6} d^{3} x^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) + 91 \, b^{6} d^{4} x^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) + 14 \, b^{6} d^{5} x e \mathrm {sgn}\left (b x + a\right ) + b^{6} d^{6} \mathrm {sgn}\left (b x + a\right ) + 16016 \, a b^{5} x^{5} e^{6} \mathrm {sgn}\left (b x + a\right ) + 8008 \, a b^{5} d x^{4} e^{5} \mathrm {sgn}\left (b x + a\right ) + 2912 \, a b^{5} d^{2} x^{3} e^{4} \mathrm {sgn}\left (b x + a\right ) + 728 \, a b^{5} d^{3} x^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) + 112 \, a b^{5} d^{4} x e^{2} \mathrm {sgn}\left (b x + a\right ) + 8 \, a b^{5} d^{5} e \mathrm {sgn}\left (b x + a\right ) + 36036 \, a^{2} b^{4} x^{4} e^{6} \mathrm {sgn}\left (b x + a\right ) + 13104 \, a^{2} b^{4} d x^{3} e^{5} \mathrm {sgn}\left (b x + a\right ) + 3276 \, a^{2} b^{4} d^{2} x^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) + 504 \, a^{2} b^{4} d^{3} x e^{3} \mathrm {sgn}\left (b x + a\right ) + 36 \, a^{2} b^{4} d^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) + 43680 \, a^{3} b^{3} x^{3} e^{6} \mathrm {sgn}\left (b x + a\right ) + 10920 \, a^{3} b^{3} d x^{2} e^{5} \mathrm {sgn}\left (b x + a\right ) + 1680 \, a^{3} b^{3} d^{2} x e^{4} \mathrm {sgn}\left (b x + a\right ) + 120 \, a^{3} b^{3} d^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) + 30030 \, a^{4} b^{2} x^{2} e^{6} \mathrm {sgn}\left (b x + a\right ) + 4620 \, a^{4} b^{2} d x e^{5} \mathrm {sgn}\left (b x + a\right ) + 330 \, a^{4} b^{2} d^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) + 11088 \, a^{5} b x e^{6} \mathrm {sgn}\left (b x + a\right ) + 792 \, a^{5} b d e^{5} \mathrm {sgn}\left (b x + a\right ) + 1716 \, a^{6} e^{6} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-7\right )}}{24024 \, {\left (x e + d\right )}^{14}} \]

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)^15,x, algorithm="giac")

[Out]

-1/24024*(3003*b^6*x^6*e^6*sgn(b*x + a) + 2002*b^6*d*x^5*e^5*sgn(b*x + a) + 1001*b^6*d^2*x^4*e^4*sgn(b*x + a)
+ 364*b^6*d^3*x^3*e^3*sgn(b*x + a) + 91*b^6*d^4*x^2*e^2*sgn(b*x + a) + 14*b^6*d^5*x*e*sgn(b*x + a) + b^6*d^6*s
gn(b*x + a) + 16016*a*b^5*x^5*e^6*sgn(b*x + a) + 8008*a*b^5*d*x^4*e^5*sgn(b*x + a) + 2912*a*b^5*d^2*x^3*e^4*sg
n(b*x + a) + 728*a*b^5*d^3*x^2*e^3*sgn(b*x + a) + 112*a*b^5*d^4*x*e^2*sgn(b*x + a) + 8*a*b^5*d^5*e*sgn(b*x + a
) + 36036*a^2*b^4*x^4*e^6*sgn(b*x + a) + 13104*a^2*b^4*d*x^3*e^5*sgn(b*x + a) + 3276*a^2*b^4*d^2*x^2*e^4*sgn(b
*x + a) + 504*a^2*b^4*d^3*x*e^3*sgn(b*x + a) + 36*a^2*b^4*d^4*e^2*sgn(b*x + a) + 43680*a^3*b^3*x^3*e^6*sgn(b*x
+ a) + 10920*a^3*b^3*d*x^2*e^5*sgn(b*x + a) + 1680*a^3*b^3*d^2*x*e^4*sgn(b*x + a) + 120*a^3*b^3*d^3*e^3*sgn(b
*x + a) + 30030*a^4*b^2*x^2*e^6*sgn(b*x + a) + 4620*a^4*b^2*d*x*e^5*sgn(b*x + a) + 330*a^4*b^2*d^2*e^4*sgn(b*x
+ a) + 11088*a^5*b*x*e^6*sgn(b*x + a) + 792*a^5*b*d*e^5*sgn(b*x + a) + 1716*a^6*e^6*sgn(b*x + a))*e^(-7)/(x*e
+ d)^14

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maple [A]  time = 0.06, size = 392, normalized size = 1.08 \[ -\frac {\left (3003 b^{6} e^{6} x^{6}+16016 a \,b^{5} e^{6} x^{5}+2002 b^{6} d \,e^{5} x^{5}+36036 a^{2} b^{4} e^{6} x^{4}+8008 a \,b^{5} d \,e^{5} x^{4}+1001 b^{6} d^{2} e^{4} x^{4}+43680 a^{3} b^{3} e^{6} x^{3}+13104 a^{2} b^{4} d \,e^{5} x^{3}+2912 a \,b^{5} d^{2} e^{4} x^{3}+364 b^{6} d^{3} e^{3} x^{3}+30030 a^{4} b^{2} e^{6} x^{2}+10920 a^{3} b^{3} d \,e^{5} x^{2}+3276 a^{2} b^{4} d^{2} e^{4} x^{2}+728 a \,b^{5} d^{3} e^{3} x^{2}+91 b^{6} d^{4} e^{2} x^{2}+11088 a^{5} b \,e^{6} x +4620 a^{4} b^{2} d \,e^{5} x +1680 a^{3} b^{3} d^{2} e^{4} x +504 a^{2} b^{4} d^{3} e^{3} x +112 a \,b^{5} d^{4} e^{2} x +14 b^{6} d^{5} e x +1716 a^{6} e^{6}+792 a^{5} b d \,e^{5}+330 a^{4} b^{2} d^{2} e^{4}+120 a^{3} b^{3} d^{3} e^{3}+36 a^{2} b^{4} d^{4} e^{2}+8 a \,b^{5} d^{5} e +b^{6} d^{6}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{24024 \left (e x +d \right )^{14} \left (b x +a \right )^{5} e^{7}} \]

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)^15,x)

[Out]

-1/24024/e^7*(3003*b^6*e^6*x^6+16016*a*b^5*e^6*x^5+2002*b^6*d*e^5*x^5+36036*a^2*b^4*e^6*x^4+8008*a*b^5*d*e^5*x
^4+1001*b^6*d^2*e^4*x^4+43680*a^3*b^3*e^6*x^3+13104*a^2*b^4*d*e^5*x^3+2912*a*b^5*d^2*e^4*x^3+364*b^6*d^3*e^3*x
^3+30030*a^4*b^2*e^6*x^2+10920*a^3*b^3*d*e^5*x^2+3276*a^2*b^4*d^2*e^4*x^2+728*a*b^5*d^3*e^3*x^2+91*b^6*d^4*e^2
*x^2+11088*a^5*b*e^6*x+4620*a^4*b^2*d*e^5*x+1680*a^3*b^3*d^2*e^4*x+504*a^2*b^4*d^3*e^3*x+112*a*b^5*d^4*e^2*x+1
4*b^6*d^5*e*x+1716*a^6*e^6+792*a^5*b*d*e^5+330*a^4*b^2*d^2*e^4+120*a^3*b^3*d^3*e^3+36*a^2*b^4*d^4*e^2+8*a*b^5*
d^5*e+b^6*d^6)*((b*x+a)^2)^(5/2)/(e*x+d)^14/(b*x+a)^5

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

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)^15,x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(a*e-b*d>0)', see `assume?` for
more details)Is a*e-b*d zero or nonzero?

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mupad [B]  time = 2.40, size = 1010, normalized size = 2.79 \[ \frac {\left (\frac {-6\,a^5\,b\,e^5+15\,a^4\,b^2\,d\,e^4-20\,a^3\,b^3\,d^2\,e^3+15\,a^2\,b^4\,d^3\,e^2-6\,a\,b^5\,d^4\,e+b^6\,d^5}{13\,e^7}+\frac {d\,\left (\frac {15\,a^4\,b^2\,e^5-20\,a^3\,b^3\,d\,e^4+15\,a^2\,b^4\,d^2\,e^3-6\,a\,b^5\,d^3\,e^2+b^6\,d^4\,e}{13\,e^7}-\frac {d\,\left (\frac {20\,a^3\,b^3\,e^5-15\,a^2\,b^4\,d\,e^4+6\,a\,b^5\,d^2\,e^3-b^6\,d^3\,e^2}{13\,e^7}-\frac {d\,\left (\frac {d\,\left (\frac {b^6\,d}{13\,e^3}-\frac {b^5\,\left (6\,a\,e-b\,d\right )}{13\,e^3}\right )}{e}+\frac {b^4\,\left (15\,a^2\,e^2-6\,a\,b\,d\,e+b^2\,d^2\right )}{13\,e^4}\right )}{e}\right )}{e}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^{13}}-\frac {\left (\frac {15\,a^2\,b^4\,e^2-24\,a\,b^5\,d\,e+10\,b^6\,d^2}{10\,e^7}+\frac {d\,\left (\frac {b^6\,d}{10\,e^6}-\frac {b^5\,\left (3\,a\,e-2\,b\,d\right )}{5\,e^6}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^{10}}-\frac {\left (\frac {a^6}{14\,e}-\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {3\,a\,b^5}{7\,e}-\frac {b^6\,d}{14\,e^2}\right )}{e}-\frac {15\,a^2\,b^4}{14\,e}\right )}{e}+\frac {10\,a^3\,b^3}{7\,e}\right )}{e}-\frac {15\,a^4\,b^2}{14\,e}\right )}{e}+\frac {3\,a^5\,b}{7\,e}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^{14}}-\frac {\left (\frac {15\,a^4\,b^2\,e^4-40\,a^3\,b^3\,d\,e^3+45\,a^2\,b^4\,d^2\,e^2-24\,a\,b^5\,d^3\,e+5\,b^6\,d^4}{12\,e^7}+\frac {d\,\left (\frac {-20\,a^3\,b^3\,e^4+30\,a^2\,b^4\,d\,e^3-18\,a\,b^5\,d^2\,e^2+4\,b^6\,d^3\,e}{12\,e^7}+\frac {d\,\left (\frac {d\,\left (\frac {b^6\,d}{12\,e^4}-\frac {b^5\,\left (3\,a\,e-b\,d\right )}{6\,e^4}\right )}{e}+\frac {b^4\,\left (5\,a^2\,e^2-4\,a\,b\,d\,e+b^2\,d^2\right )}{4\,e^5}\right )}{e}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^{12}}+\frac {\left (\frac {5\,b^6\,d-6\,a\,b^5\,e}{9\,e^7}+\frac {b^6\,d}{9\,e^7}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^9}+\frac {\left (\frac {-20\,a^3\,b^3\,e^3+45\,a^2\,b^4\,d\,e^2-36\,a\,b^5\,d^2\,e+10\,b^6\,d^3}{11\,e^7}+\frac {d\,\left (\frac {d\,\left (\frac {b^6\,d}{11\,e^5}-\frac {3\,b^5\,\left (2\,a\,e-b\,d\right )}{11\,e^5}\right )}{e}+\frac {3\,b^4\,\left (5\,a^2\,e^2-6\,a\,b\,d\,e+2\,b^2\,d^2\right )}{11\,e^6}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^{11}}-\frac {b^6\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{8\,e^7\,\left (a+b\,x\right )\,{\left (d+e\,x\right )}^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)^15,x)

[Out]

(((b^6*d^5 - 6*a^5*b*e^5 + 15*a^4*b^2*d*e^4 + 15*a^2*b^4*d^3*e^2 - 20*a^3*b^3*d^2*e^3 - 6*a*b^5*d^4*e)/(13*e^7
) + (d*((b^6*d^4*e + 15*a^4*b^2*e^5 - 6*a*b^5*d^3*e^2 - 20*a^3*b^3*d*e^4 + 15*a^2*b^4*d^2*e^3)/(13*e^7) - (d*(
(20*a^3*b^3*e^5 - b^6*d^3*e^2 + 6*a*b^5*d^2*e^3 - 15*a^2*b^4*d*e^4)/(13*e^7) - (d*((d*((b^6*d)/(13*e^3) - (b^5
*(6*a*e - b*d))/(13*e^3)))/e + (b^4*(15*a^2*e^2 + b^2*d^2 - 6*a*b*d*e))/(13*e^4)))/e))/e))/e)*(a^2 + b^2*x^2 +
2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^13) - (((10*b^6*d^2 + 15*a^2*b^4*e^2 - 24*a*b^5*d*e)/(10*e^7) + (d*((b^6
*d)/(10*e^6) - (b^5*(3*a*e - 2*b*d))/(5*e^6)))/e)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^10) -
((a^6/(14*e) - (d*((d*((d*((d*((d*((3*a*b^5)/(7*e) - (b^6*d)/(14*e^2)))/e - (15*a^2*b^4)/(14*e)))/e + (10*a^3*
b^3)/(7*e)))/e - (15*a^4*b^2)/(14*e)))/e + (3*a^5*b)/(7*e)))/e)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a + b*x)*(d
+ e*x)^14) - (((5*b^6*d^4 + 15*a^4*b^2*e^4 - 40*a^3*b^3*d*e^3 + 45*a^2*b^4*d^2*e^2 - 24*a*b^5*d^3*e)/(12*e^7)
+ (d*((4*b^6*d^3*e - 20*a^3*b^3*e^4 - 18*a*b^5*d^2*e^2 + 30*a^2*b^4*d*e^3)/(12*e^7) + (d*((d*((b^6*d)/(12*e^4
) - (b^5*(3*a*e - b*d))/(6*e^4)))/e + (b^4*(5*a^2*e^2 + b^2*d^2 - 4*a*b*d*e))/(4*e^5)))/e))/e)*(a^2 + b^2*x^2
+ 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^12) + (((5*b^6*d - 6*a*b^5*e)/(9*e^7) + (b^6*d)/(9*e^7))*(a^2 + b^2*x^2
+ 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^9) + (((10*b^6*d^3 - 20*a^3*b^3*e^3 + 45*a^2*b^4*d*e^2 - 36*a*b^5*d^2*
e)/(11*e^7) + (d*((d*((b^6*d)/(11*e^5) - (3*b^5*(2*a*e - b*d))/(11*e^5)))/e + (3*b^4*(5*a^2*e^2 + 2*b^2*d^2 -
6*a*b*d*e))/(11*e^6)))/e)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^11) - (b^6*(a^2 + b^2*x^2 + 2*
a*b*x)^(1/2))/(8*e^7*(a + b*x)*(d + e*x)^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)**15,x)

[Out]

Timed out

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