3.21.7 \(\int \frac {(a+b x) (a^2+2 a b x+b^2 x^2)^{5/2}}{(d+e x)^{10}} \, dx\) [2007]

Optimal. Leaf size=149 \[ \frac {(a+b x)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{9 (b d-a e) (d+e x)^9}+\frac {b (a+b x)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{36 (b d-a e)^2 (d+e x)^8}+\frac {b^2 (a+b x)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{252 (b d-a e)^3 (d+e x)^7} \]

[Out]

1/9*(b*x+a)^6*((b*x+a)^2)^(1/2)/(-a*e+b*d)/(e*x+d)^9+1/36*b*(b*x+a)^6*((b*x+a)^2)^(1/2)/(-a*e+b*d)^2/(e*x+d)^8
+1/252*b^2*(b*x+a)^6*((b*x+a)^2)^(1/2)/(-a*e+b*d)^3/(e*x+d)^7

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Rubi [A]
time = 0.04, antiderivative size = 149, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {784, 21, 47, 37} \begin {gather*} \frac {b^2 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^6}{252 (d+e x)^7 (b d-a e)^3}+\frac {b \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^6}{36 (d+e x)^8 (b d-a e)^2}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^6}{9 (d+e x)^9 (b d-a e)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^10,x]

[Out]

((a + b*x)^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*(b*d - a*e)*(d + e*x)^9) + (b*(a + b*x)^6*Sqrt[a^2 + 2*a*b*x +
b^2*x^2])/(36*(b*d - a*e)^2*(d + e*x)^8) + (b^2*(a + b*x)^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(252*(b*d - a*e)^3*
(d + e*x)^7)

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 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n +
1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rule 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n + 1
)/((b*c - a*d)*(m + 1))), x] - Dist[d*(Simplify[m + n + 2]/((b*c - a*d)*(m + 1))), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

Rule 784

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)^{10}} \, 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)^{10}} \, 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)^{10}} \, dx}{a b+b^2 x}\\ &=\frac {(a+b x)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{9 (b d-a e) (d+e x)^9}+\frac {\left (2 b^2 \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int \frac {(a+b x)^6}{(d+e x)^9} \, dx}{9 (b d-a e) \left (a b+b^2 x\right )}\\ &=\frac {(a+b x)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{9 (b d-a e) (d+e x)^9}+\frac {b (a+b x)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{36 (b d-a e)^2 (d+e x)^8}+\frac {\left (b^3 \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int \frac {(a+b x)^6}{(d+e x)^8} \, dx}{36 (b d-a e)^2 \left (a b+b^2 x\right )}\\ &=\frac {(a+b x)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{9 (b d-a e) (d+e x)^9}+\frac {b (a+b x)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{36 (b d-a e)^2 (d+e x)^8}+\frac {b^2 (a+b x)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{252 (b d-a e)^3 (d+e x)^7}\\ \end {align*}

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Mathematica [A]
time = 0.07, size = 295, normalized size = 1.98 \begin {gather*} -\frac {\sqrt {(a+b x)^2} \left (28 a^6 e^6+21 a^5 b e^5 (d+9 e x)+15 a^4 b^2 e^4 \left (d^2+9 d e x+36 e^2 x^2\right )+10 a^3 b^3 e^3 \left (d^3+9 d^2 e x+36 d e^2 x^2+84 e^3 x^3\right )+6 a^2 b^4 e^2 \left (d^4+9 d^3 e x+36 d^2 e^2 x^2+84 d e^3 x^3+126 e^4 x^4\right )+3 a b^5 e \left (d^5+9 d^4 e x+36 d^3 e^2 x^2+84 d^2 e^3 x^3+126 d e^4 x^4+126 e^5 x^5\right )+b^6 \left (d^6+9 d^5 e x+36 d^4 e^2 x^2+84 d^3 e^3 x^3+126 d^2 e^4 x^4+126 d e^5 x^5+84 e^6 x^6\right )\right )}{252 e^7 (a+b x) (d+e x)^9} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^10,x]

[Out]

-1/252*(Sqrt[(a + b*x)^2]*(28*a^6*e^6 + 21*a^5*b*e^5*(d + 9*e*x) + 15*a^4*b^2*e^4*(d^2 + 9*d*e*x + 36*e^2*x^2)
+ 10*a^3*b^3*e^3*(d^3 + 9*d^2*e*x + 36*d*e^2*x^2 + 84*e^3*x^3) + 6*a^2*b^4*e^2*(d^4 + 9*d^3*e*x + 36*d^2*e^2*
x^2 + 84*d*e^3*x^3 + 126*e^4*x^4) + 3*a*b^5*e*(d^5 + 9*d^4*e*x + 36*d^3*e^2*x^2 + 84*d^2*e^3*x^3 + 126*d*e^4*x
^4 + 126*e^5*x^5) + b^6*(d^6 + 9*d^5*e*x + 36*d^4*e^2*x^2 + 84*d^3*e^3*x^3 + 126*d^2*e^4*x^4 + 126*d*e^5*x^5 +
84*e^6*x^6)))/(e^7*(a + b*x)*(d + e*x)^9)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(391\) vs. \(2(110)=220\).
time = 0.08, size = 392, normalized size = 2.63

method	result	size
risch	\(\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (-\frac {b^{6} x^{6}}{3 e}-\frac {b^{5} \left (3 a e +b d \right ) x^{5}}{2 e^{2}}-\frac {b^{4} \left (6 a^{2} e^{2}+3 a b d e +b^{2} d^{2}\right ) x^{4}}{2 e^{3}}-\frac {b^{3} \left (10 a^{3} e^{3}+6 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e +b^{3} d^{3}\right ) x^{3}}{3 e^{4}}-\frac {b^{2} \left (15 a^{4} e^{4}+10 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}+3 a \,b^{3} d^{3} e +b^{4} d^{4}\right ) x^{2}}{7 e^{5}}-\frac {b \left (21 a^{5} e^{5}+15 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}+6 a^{2} b^{3} d^{3} e^{2}+3 a \,b^{4} d^{4} e +b^{5} d^{5}\right ) x}{28 e^{6}}-\frac {28 e^{6} a^{6}+21 d \,e^{5} a^{5} b +15 d^{2} e^{4} a^{4} b^{2}+10 d^{3} e^{3} a^{3} b^{3}+6 d^{4} e^{2} a^{2} b^{4}+3 d^{5} e a \,b^{5}+d^{6} b^{6}}{252 e^{7}}\right )}{\left (b x +a \right ) \left (e x +d \right )^{9}}\)	\(351\)
gosper	\(-\frac {\left (84 b^{6} e^{6} x^{6}+378 a \,b^{5} e^{6} x^{5}+126 b^{6} d \,e^{5} x^{5}+756 a^{2} b^{4} e^{6} x^{4}+378 a \,b^{5} d \,e^{5} x^{4}+126 b^{6} d^{2} e^{4} x^{4}+840 a^{3} b^{3} e^{6} x^{3}+504 a^{2} b^{4} d \,e^{5} x^{3}+252 a \,b^{5} d^{2} e^{4} x^{3}+84 b^{6} d^{3} e^{3} x^{3}+540 a^{4} b^{2} e^{6} x^{2}+360 a^{3} b^{3} d \,e^{5} x^{2}+216 a^{2} b^{4} d^{2} e^{4} x^{2}+108 a \,b^{5} d^{3} e^{3} x^{2}+36 b^{6} d^{4} e^{2} x^{2}+189 a^{5} b \,e^{6} x +135 a^{4} b^{2} d \,e^{5} x +90 a^{3} b^{3} d^{2} e^{4} x +54 a^{2} b^{4} d^{3} e^{3} x +27 a \,b^{5} d^{4} e^{2} x +9 b^{6} d^{5} e x +28 e^{6} a^{6}+21 d \,e^{5} a^{5} b +15 d^{2} e^{4} a^{4} b^{2}+10 d^{3} e^{3} a^{3} b^{3}+6 d^{4} e^{2} a^{2} b^{4}+3 d^{5} e a \,b^{5}+d^{6} b^{6}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{252 e^{7} \left (e x +d \right )^{9} \left (b x +a \right )^{5}}\)	\(392\)
default	\(-\frac {\left (84 b^{6} e^{6} x^{6}+378 a \,b^{5} e^{6} x^{5}+126 b^{6} d \,e^{5} x^{5}+756 a^{2} b^{4} e^{6} x^{4}+378 a \,b^{5} d \,e^{5} x^{4}+126 b^{6} d^{2} e^{4} x^{4}+840 a^{3} b^{3} e^{6} x^{3}+504 a^{2} b^{4} d \,e^{5} x^{3}+252 a \,b^{5} d^{2} e^{4} x^{3}+84 b^{6} d^{3} e^{3} x^{3}+540 a^{4} b^{2} e^{6} x^{2}+360 a^{3} b^{3} d \,e^{5} x^{2}+216 a^{2} b^{4} d^{2} e^{4} x^{2}+108 a \,b^{5} d^{3} e^{3} x^{2}+36 b^{6} d^{4} e^{2} x^{2}+189 a^{5} b \,e^{6} x +135 a^{4} b^{2} d \,e^{5} x +90 a^{3} b^{3} d^{2} e^{4} x +54 a^{2} b^{4} d^{3} e^{3} x +27 a \,b^{5} d^{4} e^{2} x +9 b^{6} d^{5} e x +28 e^{6} a^{6}+21 d \,e^{5} a^{5} b +15 d^{2} e^{4} a^{4} b^{2}+10 d^{3} e^{3} a^{3} b^{3}+6 d^{4} e^{2} a^{2} b^{4}+3 d^{5} e a \,b^{5}+d^{6} b^{6}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{252 e^{7} \left (e x +d \right )^{9} \left (b x +a \right )^{5}}\)	\(392\)

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)^10,x,method=_RETURNVERBOSE)

[Out]

-1/252/e^7*(84*b^6*e^6*x^6+378*a*b^5*e^6*x^5+126*b^6*d*e^5*x^5+756*a^2*b^4*e^6*x^4+378*a*b^5*d*e^5*x^4+126*b^6
*d^2*e^4*x^4+840*a^3*b^3*e^6*x^3+504*a^2*b^4*d*e^5*x^3+252*a*b^5*d^2*e^4*x^3+84*b^6*d^3*e^3*x^3+540*a^4*b^2*e^
6*x^2+360*a^3*b^3*d*e^5*x^2+216*a^2*b^4*d^2*e^4*x^2+108*a*b^5*d^3*e^3*x^2+36*b^6*d^4*e^2*x^2+189*a^5*b*e^6*x+1
35*a^4*b^2*d*e^5*x+90*a^3*b^3*d^2*e^4*x+54*a^2*b^4*d^3*e^3*x+27*a*b^5*d^4*e^2*x+9*b^6*d^5*e*x+28*a^6*e^6+21*a^
5*b*d*e^5+15*a^4*b^2*d^2*e^4+10*a^3*b^3*d^3*e^3+6*a^2*b^4*d^4*e^2+3*a*b^5*d^5*e+b^6*d^6)*((b*x+a)^2)^(5/2)/(e*
x+d)^9/(b*x+a)^5

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \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)^10,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(b*d-%e*a>0)', see `assume?` fo
r more detai

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 407 vs. \(2 (116) = 232\).
time = 2.58, size = 407, normalized size = 2.73 \begin {gather*} -\frac {b^{6} d^{6} + {\left (84 \, b^{6} x^{6} + 378 \, a b^{5} x^{5} + 756 \, a^{2} b^{4} x^{4} + 840 \, a^{3} b^{3} x^{3} + 540 \, a^{4} b^{2} x^{2} + 189 \, a^{5} b x + 28 \, a^{6}\right )} e^{6} + 3 \, {\left (42 \, b^{6} d x^{5} + 126 \, a b^{5} d x^{4} + 168 \, a^{2} b^{4} d x^{3} + 120 \, a^{3} b^{3} d x^{2} + 45 \, a^{4} b^{2} d x + 7 \, a^{5} b d\right )} e^{5} + 3 \, {\left (42 \, b^{6} d^{2} x^{4} + 84 \, a b^{5} d^{2} x^{3} + 72 \, a^{2} b^{4} d^{2} x^{2} + 30 \, a^{3} b^{3} d^{2} x + 5 \, a^{4} b^{2} d^{2}\right )} e^{4} + 2 \, {\left (42 \, b^{6} d^{3} x^{3} + 54 \, a b^{5} d^{3} x^{2} + 27 \, a^{2} b^{4} d^{3} x + 5 \, a^{3} b^{3} d^{3}\right )} e^{3} + 3 \, {\left (12 \, b^{6} d^{4} x^{2} + 9 \, a b^{5} d^{4} x + 2 \, a^{2} b^{4} d^{4}\right )} e^{2} + 3 \, {\left (3 \, b^{6} d^{5} x + a b^{5} d^{5}\right )} e}{252 \, {\left (x^{9} e^{16} + 9 \, d x^{8} e^{15} + 36 \, d^{2} x^{7} e^{14} + 84 \, d^{3} x^{6} e^{13} + 126 \, d^{4} x^{5} e^{12} + 126 \, d^{5} x^{4} e^{11} + 84 \, d^{6} x^{3} e^{10} + 36 \, d^{7} x^{2} e^{9} + 9 \, d^{8} x e^{8} + d^{9} e^{7}\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)^10,x, algorithm="fricas")

[Out]

-1/252*(b^6*d^6 + (84*b^6*x^6 + 378*a*b^5*x^5 + 756*a^2*b^4*x^4 + 840*a^3*b^3*x^3 + 540*a^4*b^2*x^2 + 189*a^5*
b*x + 28*a^6)*e^6 + 3*(42*b^6*d*x^5 + 126*a*b^5*d*x^4 + 168*a^2*b^4*d*x^3 + 120*a^3*b^3*d*x^2 + 45*a^4*b^2*d*x
+ 7*a^5*b*d)*e^5 + 3*(42*b^6*d^2*x^4 + 84*a*b^5*d^2*x^3 + 72*a^2*b^4*d^2*x^2 + 30*a^3*b^3*d^2*x + 5*a^4*b^2*d
^2)*e^4 + 2*(42*b^6*d^3*x^3 + 54*a*b^5*d^3*x^2 + 27*a^2*b^4*d^3*x + 5*a^3*b^3*d^3)*e^3 + 3*(12*b^6*d^4*x^2 + 9
*a*b^5*d^4*x + 2*a^2*b^4*d^4)*e^2 + 3*(3*b^6*d^5*x + a*b^5*d^5)*e)/(x^9*e^16 + 9*d*x^8*e^15 + 36*d^2*x^7*e^14
+ 84*d^3*x^6*e^13 + 126*d^4*x^5*e^12 + 126*d^5*x^4*e^11 + 84*d^6*x^3*e^10 + 36*d^7*x^2*e^9 + 9*d^8*x*e^8 + d^9
*e^7)

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Sympy [F(-1)] Timed out
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)**10,x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 520 vs. \(2 (116) = 232\).
time = 1.04, size = 520, normalized size = 3.49 \begin {gather*} -\frac {{\left (84 \, b^{6} x^{6} e^{6} \mathrm {sgn}\left (b x + a\right ) + 126 \, b^{6} d x^{5} e^{5} \mathrm {sgn}\left (b x + a\right ) + 126 \, b^{6} d^{2} x^{4} e^{4} \mathrm {sgn}\left (b x + a\right ) + 84 \, b^{6} d^{3} x^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) + 36 \, b^{6} d^{4} x^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) + 9 \, b^{6} d^{5} x e \mathrm {sgn}\left (b x + a\right ) + b^{6} d^{6} \mathrm {sgn}\left (b x + a\right ) + 378 \, a b^{5} x^{5} e^{6} \mathrm {sgn}\left (b x + a\right ) + 378 \, a b^{5} d x^{4} e^{5} \mathrm {sgn}\left (b x + a\right ) + 252 \, a b^{5} d^{2} x^{3} e^{4} \mathrm {sgn}\left (b x + a\right ) + 108 \, a b^{5} d^{3} x^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) + 27 \, a b^{5} d^{4} x e^{2} \mathrm {sgn}\left (b x + a\right ) + 3 \, a b^{5} d^{5} e \mathrm {sgn}\left (b x + a\right ) + 756 \, a^{2} b^{4} x^{4} e^{6} \mathrm {sgn}\left (b x + a\right ) + 504 \, a^{2} b^{4} d x^{3} e^{5} \mathrm {sgn}\left (b x + a\right ) + 216 \, a^{2} b^{4} d^{2} x^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) + 54 \, a^{2} b^{4} d^{3} x e^{3} \mathrm {sgn}\left (b x + a\right ) + 6 \, a^{2} b^{4} d^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) + 840 \, a^{3} b^{3} x^{3} e^{6} \mathrm {sgn}\left (b x + a\right ) + 360 \, a^{3} b^{3} d x^{2} e^{5} \mathrm {sgn}\left (b x + a\right ) + 90 \, a^{3} b^{3} d^{2} x e^{4} \mathrm {sgn}\left (b x + a\right ) + 10 \, a^{3} b^{3} d^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) + 540 \, a^{4} b^{2} x^{2} e^{6} \mathrm {sgn}\left (b x + a\right ) + 135 \, a^{4} b^{2} d x e^{5} \mathrm {sgn}\left (b x + a\right ) + 15 \, a^{4} b^{2} d^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) + 189 \, a^{5} b x e^{6} \mathrm {sgn}\left (b x + a\right ) + 21 \, a^{5} b d e^{5} \mathrm {sgn}\left (b x + a\right ) + 28 \, a^{6} e^{6} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-7\right )}}{252 \, {\left (x e + d\right )}^{9}} \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)^10,x, algorithm="giac")

[Out]

-1/252*(84*b^6*x^6*e^6*sgn(b*x + a) + 126*b^6*d*x^5*e^5*sgn(b*x + a) + 126*b^6*d^2*x^4*e^4*sgn(b*x + a) + 84*b
^6*d^3*x^3*e^3*sgn(b*x + a) + 36*b^6*d^4*x^2*e^2*sgn(b*x + a) + 9*b^6*d^5*x*e*sgn(b*x + a) + b^6*d^6*sgn(b*x +
a) + 378*a*b^5*x^5*e^6*sgn(b*x + a) + 378*a*b^5*d*x^4*e^5*sgn(b*x + a) + 252*a*b^5*d^2*x^3*e^4*sgn(b*x + a) +
108*a*b^5*d^3*x^2*e^3*sgn(b*x + a) + 27*a*b^5*d^4*x*e^2*sgn(b*x + a) + 3*a*b^5*d^5*e*sgn(b*x + a) + 756*a^2*b
^4*x^4*e^6*sgn(b*x + a) + 504*a^2*b^4*d*x^3*e^5*sgn(b*x + a) + 216*a^2*b^4*d^2*x^2*e^4*sgn(b*x + a) + 54*a^2*b
^4*d^3*x*e^3*sgn(b*x + a) + 6*a^2*b^4*d^4*e^2*sgn(b*x + a) + 840*a^3*b^3*x^3*e^6*sgn(b*x + a) + 360*a^3*b^3*d*
x^2*e^5*sgn(b*x + a) + 90*a^3*b^3*d^2*x*e^4*sgn(b*x + a) + 10*a^3*b^3*d^3*e^3*sgn(b*x + a) + 540*a^4*b^2*x^2*e
^6*sgn(b*x + a) + 135*a^4*b^2*d*x*e^5*sgn(b*x + a) + 15*a^4*b^2*d^2*e^4*sgn(b*x + a) + 189*a^5*b*x*e^6*sgn(b*x
+ a) + 21*a^5*b*d*e^5*sgn(b*x + a) + 28*a^6*e^6*sgn(b*x + a))*e^(-7)/(x*e + d)^9

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Mupad [B]
time = 2.26, size = 1010, normalized size = 6.78 \begin {gather*} \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}{8\,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}{8\,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}{8\,e^7}-\frac {d\,\left (\frac {d\,\left (\frac {b^6\,d}{8\,e^3}-\frac {b^5\,\left (6\,a\,e-b\,d\right )}{8\,e^3}\right )}{e}+\frac {b^4\,\left (15\,a^2\,e^2-6\,a\,b\,d\,e+b^2\,d^2\right )}{8\,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 )}^8}-\frac {\left (\frac {15\,a^2\,b^4\,e^2-24\,a\,b^5\,d\,e+10\,b^6\,d^2}{5\,e^7}+\frac {d\,\left (\frac {b^6\,d}{5\,e^6}-\frac {2\,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 )}^5}-\frac {\left (\frac {a^6}{9\,e}-\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {2\,a\,b^5}{3\,e}-\frac {b^6\,d}{9\,e^2}\right )}{e}-\frac {5\,a^2\,b^4}{3\,e}\right )}{e}+\frac {20\,a^3\,b^3}{9\,e}\right )}{e}-\frac {5\,a^4\,b^2}{3\,e}\right )}{e}+\frac {2\,a^5\,b}{3\,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 )}^9}-\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}{7\,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}{7\,e^7}+\frac {d\,\left (\frac {d\,\left (\frac {b^6\,d}{7\,e^4}-\frac {2\,b^5\,\left (3\,a\,e-b\,d\right )}{7\,e^4}\right )}{e}+\frac {3\,b^4\,\left (5\,a^2\,e^2-4\,a\,b\,d\,e+b^2\,d^2\right )}{7\,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 )}^7}+\frac {\left (\frac {5\,b^6\,d-6\,a\,b^5\,e}{4\,e^7}+\frac {b^6\,d}{4\,e^7}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^4}+\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}{6\,e^7}+\frac {d\,\left (\frac {d\,\left (\frac {b^6\,d}{6\,e^5}-\frac {b^5\,\left (2\,a\,e-b\,d\right )}{2\,e^5}\right )}{e}+\frac {b^4\,\left (5\,a^2\,e^2-6\,a\,b\,d\,e+2\,b^2\,d^2\right )}{2\,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 )}^6}-\frac {b^6\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{3\,e^7\,\left (a+b\,x\right )\,{\left (d+e\,x\right )}^3} \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)^10,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)/(8*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)/(8*e^7) - (d*((2
0*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)/(8*e^7) - (d*((d*((b^6*d)/(8*e^3) - (b^5*(6*
a*e - b*d))/(8*e^3)))/e + (b^4*(15*a^2*e^2 + b^2*d^2 - 6*a*b*d*e))/(8*e^4)))/e))/e))/e)*(a^2 + b^2*x^2 + 2*a*b
*x)^(1/2))/((a + b*x)*(d + e*x)^8) - (((10*b^6*d^2 + 15*a^2*b^4*e^2 - 24*a*b^5*d*e)/(5*e^7) + (d*((b^6*d)/(5*e
^6) - (2*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)^5) - ((a^6/(9
*e) - (d*((d*((d*((d*((d*((2*a*b^5)/(3*e) - (b^6*d)/(9*e^2)))/e - (5*a^2*b^4)/(3*e)))/e + (20*a^3*b^3)/(9*e)))
/e - (5*a^4*b^2)/(3*e)))/e + (2*a^5*b)/(3*e)))/e)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^9) - (
((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)/(7*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)/(7*e^7) + (d*((d*((b^6*d)/(7*e^4) - (2*b^5*(3*a*e
- b*d))/(7*e^4)))/e + (3*b^4*(5*a^2*e^2 + b^2*d^2 - 4*a*b*d*e))/(7*e^5)))/e))/e)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/
2))/((a + b*x)*(d + e*x)^7) + (((5*b^6*d - 6*a*b^5*e)/(4*e^7) + (b^6*d)/(4*e^7))*(a^2 + b^2*x^2 + 2*a*b*x)^(1/
2))/((a + b*x)*(d + e*x)^4) + (((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)/(6*e^7) + (d
*((d*((b^6*d)/(6*e^5) - (b^5*(2*a*e - b*d))/(2*e^5)))/e + (b^4*(5*a^2*e^2 + 2*b^2*d^2 - 6*a*b*d*e))/(2*e^6)))/
e)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^6) - (b^6*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/(3*e^7*(a
+ b*x)*(d + e*x)^3)

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