\(\int \frac {(a+b x) (a^2+2 a b x+b^2 x^2)^{3/2}}{(d+e x)^7} \, dx\) [1982]

   Optimal result
   Rubi [A] (verified)
   Mathematica [A] (verified)
   Maple [B] (verified)
   Fricas [B] (verification not implemented)
   Sympy [F]
   Maxima [F(-2)]
   Giac [B] (verification not implemented)
   Mupad [B] (verification not implemented)

Optimal result

Integrand size = 33, antiderivative size = 98 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^7} \, dx=\frac {(a+b x)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{6 (b d-a e) (d+e x)^6}+\frac {b (a+b x)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{30 (b d-a e)^2 (d+e x)^5} \]

[Out]

1/6*(b*x+a)^4*((b*x+a)^2)^(1/2)/(-a*e+b*d)/(e*x+d)^6+1/30*b*(b*x+a)^4*((b*x+a)^2)^(1/2)/(-a*e+b*d)^2/(e*x+d)^5

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {784, 21, 47, 37} \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^7} \, dx=\frac {b \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^4}{30 (d+e x)^5 (b d-a e)^2}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^4}{6 (d+e x)^6 (b d-a e)} \]

[In]

Int[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2))/(d + e*x)^7,x]

[Out]

((a + b*x)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(6*(b*d - a*e)*(d + e*x)^6) + (b*(a + b*x)^4*Sqrt[a^2 + 2*a*b*x +
b^2*x^2])/(30*(b*d - a*e)^2*(d + e*x)^5)

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*} \text {integral}& = \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {(a+b x) \left (a b+b^2 x\right )^3}{(d+e x)^7} \, dx}{b^2 \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)^4}{(d+e x)^7} \, dx}{a b+b^2 x} \\ & = \frac {(a+b x)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{6 (b d-a e) (d+e x)^6}+\frac {\left (b^2 \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int \frac {(a+b x)^4}{(d+e x)^6} \, dx}{6 (b d-a e) \left (a b+b^2 x\right )} \\ & = \frac {(a+b x)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{6 (b d-a e) (d+e x)^6}+\frac {b (a+b x)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{30 (b d-a e)^2 (d+e x)^5} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.04 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.65 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^7} \, dx=-\frac {\sqrt {(a+b x)^2} \left (5 a^4 e^4+4 a^3 b e^3 (d+6 e x)+3 a^2 b^2 e^2 \left (d^2+6 d e x+15 e^2 x^2\right )+2 a b^3 e \left (d^3+6 d^2 e x+15 d e^2 x^2+20 e^3 x^3\right )+b^4 \left (d^4+6 d^3 e x+15 d^2 e^2 x^2+20 d e^3 x^3+15 e^4 x^4\right )\right )}{30 e^5 (a+b x) (d+e x)^6} \]

[In]

Integrate[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2))/(d + e*x)^7,x]

[Out]

-1/30*(Sqrt[(a + b*x)^2]*(5*a^4*e^4 + 4*a^3*b*e^3*(d + 6*e*x) + 3*a^2*b^2*e^2*(d^2 + 6*d*e*x + 15*e^2*x^2) + 2
*a*b^3*e*(d^3 + 6*d^2*e*x + 15*d*e^2*x^2 + 20*e^3*x^3) + b^4*(d^4 + 6*d^3*e*x + 15*d^2*e^2*x^2 + 20*d*e^3*x^3
+ 15*e^4*x^4)))/(e^5*(a + b*x)*(d + e*x)^6)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(186\) vs. \(2(72)=144\).

Time = 0.94 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.91

method result size
risch \(\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (-\frac {b^{4} x^{4}}{2 e}-\frac {2 b^{3} \left (2 a e +b d \right ) x^{3}}{3 e^{2}}-\frac {b^{2} \left (3 e^{2} a^{2}+2 a b d e +b^{2} d^{2}\right ) x^{2}}{2 e^{3}}-\frac {b \left (4 a^{3} e^{3}+3 a^{2} b d \,e^{2}+2 a \,b^{2} d^{2} e +b^{3} d^{3}\right ) x}{5 e^{4}}-\frac {5 e^{4} a^{4}+4 b d \,e^{3} a^{3}+3 b^{2} d^{2} e^{2} a^{2}+2 b^{3} d^{3} e a +b^{4} d^{4}}{30 e^{5}}\right )}{\left (b x +a \right ) \left (e x +d \right )^{6}}\) \(187\)
gosper \(-\frac {\left (15 e^{4} x^{4} b^{4}+40 x^{3} a \,b^{3} e^{4}+20 x^{3} b^{4} d \,e^{3}+45 x^{2} a^{2} b^{2} e^{4}+30 x^{2} a \,b^{3} d \,e^{3}+15 x^{2} b^{4} d^{2} e^{2}+24 x \,a^{3} b \,e^{4}+18 x \,a^{2} b^{2} d \,e^{3}+12 x a \,b^{3} d^{2} e^{2}+6 x \,b^{4} d^{3} e +5 e^{4} a^{4}+4 b d \,e^{3} a^{3}+3 b^{2} d^{2} e^{2} a^{2}+2 b^{3} d^{3} e a +b^{4} d^{4}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{30 e^{5} \left (e x +d \right )^{6} \left (b x +a \right )^{3}}\) \(201\)
default \(-\frac {\left (15 e^{4} x^{4} b^{4}+40 x^{3} a \,b^{3} e^{4}+20 x^{3} b^{4} d \,e^{3}+45 x^{2} a^{2} b^{2} e^{4}+30 x^{2} a \,b^{3} d \,e^{3}+15 x^{2} b^{4} d^{2} e^{2}+24 x \,a^{3} b \,e^{4}+18 x \,a^{2} b^{2} d \,e^{3}+12 x a \,b^{3} d^{2} e^{2}+6 x \,b^{4} d^{3} e +5 e^{4} a^{4}+4 b d \,e^{3} a^{3}+3 b^{2} d^{2} e^{2} a^{2}+2 b^{3} d^{3} e a +b^{4} d^{4}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{30 e^{5} \left (e x +d \right )^{6} \left (b x +a \right )^{3}}\) \(201\)

[In]

int((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^(3/2)/(e*x+d)^7,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 236 vs. \(2 (72) = 144\).

Time = 0.30 (sec) , antiderivative size = 236, normalized size of antiderivative = 2.41 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^7} \, dx=-\frac {15 \, b^{4} e^{4} x^{4} + b^{4} d^{4} + 2 \, a b^{3} d^{3} e + 3 \, a^{2} b^{2} d^{2} e^{2} + 4 \, a^{3} b d e^{3} + 5 \, a^{4} e^{4} + 20 \, {\left (b^{4} d e^{3} + 2 \, a b^{3} e^{4}\right )} x^{3} + 15 \, {\left (b^{4} d^{2} e^{2} + 2 \, a b^{3} d e^{3} + 3 \, a^{2} b^{2} e^{4}\right )} x^{2} + 6 \, {\left (b^{4} d^{3} e + 2 \, a b^{3} d^{2} e^{2} + 3 \, a^{2} b^{2} d e^{3} + 4 \, a^{3} b e^{4}\right )} x}{30 \, {\left (e^{11} x^{6} + 6 \, d e^{10} x^{5} + 15 \, d^{2} e^{9} x^{4} + 20 \, d^{3} e^{8} x^{3} + 15 \, d^{4} e^{7} x^{2} + 6 \, d^{5} e^{6} x + d^{6} e^{5}\right )}} \]

[In]

integrate((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^(3/2)/(e*x+d)^7,x, algorithm="fricas")

[Out]

-1/30*(15*b^4*e^4*x^4 + b^4*d^4 + 2*a*b^3*d^3*e + 3*a^2*b^2*d^2*e^2 + 4*a^3*b*d*e^3 + 5*a^4*e^4 + 20*(b^4*d*e^
3 + 2*a*b^3*e^4)*x^3 + 15*(b^4*d^2*e^2 + 2*a*b^3*d*e^3 + 3*a^2*b^2*e^4)*x^2 + 6*(b^4*d^3*e + 2*a*b^3*d^2*e^2 +
 3*a^2*b^2*d*e^3 + 4*a^3*b*e^4)*x)/(e^11*x^6 + 6*d*e^10*x^5 + 15*d^2*e^9*x^4 + 20*d^3*e^8*x^3 + 15*d^4*e^7*x^2
 + 6*d^5*e^6*x + d^6*e^5)

Sympy [F]

\[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^7} \, dx=\int \frac {\left (a + b x\right ) \left (\left (a + b x\right )^{2}\right )^{\frac {3}{2}}}{\left (d + e x\right )^{7}}\, dx \]

[In]

integrate((b*x+a)*(b**2*x**2+2*a*b*x+a**2)**(3/2)/(e*x+d)**7,x)

[Out]

Integral((a + b*x)*((a + b*x)**2)**(3/2)/(d + e*x)**7, x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^7} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^(3/2)/(e*x+d)^7,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 detail

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 314 vs. \(2 (72) = 144\).

Time = 0.27 (sec) , antiderivative size = 314, normalized size of antiderivative = 3.20 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^7} \, dx=\frac {b^{6} \mathrm {sgn}\left (b x + a\right )}{30 \, {\left (b^{2} d^{2} e^{5} - 2 \, a b d e^{6} + a^{2} e^{7}\right )}} - \frac {15 \, b^{4} e^{4} x^{4} \mathrm {sgn}\left (b x + a\right ) + 20 \, b^{4} d e^{3} x^{3} \mathrm {sgn}\left (b x + a\right ) + 40 \, a b^{3} e^{4} x^{3} \mathrm {sgn}\left (b x + a\right ) + 15 \, b^{4} d^{2} e^{2} x^{2} \mathrm {sgn}\left (b x + a\right ) + 30 \, a b^{3} d e^{3} x^{2} \mathrm {sgn}\left (b x + a\right ) + 45 \, a^{2} b^{2} e^{4} x^{2} \mathrm {sgn}\left (b x + a\right ) + 6 \, b^{4} d^{3} e x \mathrm {sgn}\left (b x + a\right ) + 12 \, a b^{3} d^{2} e^{2} x \mathrm {sgn}\left (b x + a\right ) + 18 \, a^{2} b^{2} d e^{3} x \mathrm {sgn}\left (b x + a\right ) + 24 \, a^{3} b e^{4} x \mathrm {sgn}\left (b x + a\right ) + b^{4} d^{4} \mathrm {sgn}\left (b x + a\right ) + 2 \, a b^{3} d^{3} e \mathrm {sgn}\left (b x + a\right ) + 3 \, a^{2} b^{2} d^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) + 4 \, a^{3} b d e^{3} \mathrm {sgn}\left (b x + a\right ) + 5 \, a^{4} e^{4} \mathrm {sgn}\left (b x + a\right )}{30 \, {\left (e x + d\right )}^{6} e^{5}} \]

[In]

integrate((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^(3/2)/(e*x+d)^7,x, algorithm="giac")

[Out]

1/30*b^6*sgn(b*x + a)/(b^2*d^2*e^5 - 2*a*b*d*e^6 + a^2*e^7) - 1/30*(15*b^4*e^4*x^4*sgn(b*x + a) + 20*b^4*d*e^3
*x^3*sgn(b*x + a) + 40*a*b^3*e^4*x^3*sgn(b*x + a) + 15*b^4*d^2*e^2*x^2*sgn(b*x + a) + 30*a*b^3*d*e^3*x^2*sgn(b
*x + a) + 45*a^2*b^2*e^4*x^2*sgn(b*x + a) + 6*b^4*d^3*e*x*sgn(b*x + a) + 12*a*b^3*d^2*e^2*x*sgn(b*x + a) + 18*
a^2*b^2*d*e^3*x*sgn(b*x + a) + 24*a^3*b*e^4*x*sgn(b*x + a) + b^4*d^4*sgn(b*x + a) + 2*a*b^3*d^3*e*sgn(b*x + a)
 + 3*a^2*b^2*d^2*e^2*sgn(b*x + a) + 4*a^3*b*d*e^3*sgn(b*x + a) + 5*a^4*e^4*sgn(b*x + a))/((e*x + d)^6*e^5)

Mupad [B] (verification not implemented)

Time = 10.80 (sec) , antiderivative size = 449, normalized size of antiderivative = 4.58 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^7} \, dx=\frac {\left (\frac {-4\,a^3\,b\,e^3+6\,a^2\,b^2\,d\,e^2-4\,a\,b^3\,d^2\,e+b^4\,d^3}{5\,e^5}+\frac {d\,\left (\frac {d\,\left (\frac {b^4\,d}{5\,e^3}-\frac {b^3\,\left (4\,a\,e-b\,d\right )}{5\,e^3}\right )}{e}+\frac {b^2\,\left (6\,a^2\,e^2-4\,a\,b\,d\,e+b^2\,d^2\right )}{5\,e^4}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^5}-\frac {\left (\frac {a^4}{6\,e}-\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {2\,a\,b^3}{3\,e}-\frac {b^4\,d}{6\,e^2}\right )}{e}-\frac {a^2\,b^2}{e}\right )}{e}+\frac {2\,a^3\,b}{3\,e}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^6}-\frac {\left (\frac {6\,a^2\,b^2\,e^2-8\,a\,b^3\,d\,e+3\,b^4\,d^2}{4\,e^5}+\frac {d\,\left (\frac {b^4\,d}{4\,e^4}-\frac {b^3\,\left (2\,a\,e-b\,d\right )}{2\,e^4}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^4}+\frac {\left (\frac {3\,b^4\,d-4\,a\,b^3\,e}{3\,e^5}+\frac {b^4\,d}{3\,e^5}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^3}-\frac {b^4\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{2\,e^5\,\left (a+b\,x\right )\,{\left (d+e\,x\right )}^2} \]

[In]

int(((a + b*x)*(a^2 + b^2*x^2 + 2*a*b*x)^(3/2))/(d + e*x)^7,x)

[Out]

(((b^4*d^3 - 4*a^3*b*e^3 + 6*a^2*b^2*d*e^2 - 4*a*b^3*d^2*e)/(5*e^5) + (d*((d*((b^4*d)/(5*e^3) - (b^3*(4*a*e -
b*d))/(5*e^3)))/e + (b^2*(6*a^2*e^2 + b^2*d^2 - 4*a*b*d*e))/(5*e^4)))/e)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a
+ b*x)*(d + e*x)^5) - ((a^4/(6*e) - (d*((d*((d*((2*a*b^3)/(3*e) - (b^4*d)/(6*e^2)))/e - (a^2*b^2)/e))/e + (2*a
^3*b)/(3*e)))/e)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^6) - (((3*b^4*d^2 + 6*a^2*b^2*e^2 - 8*a
*b^3*d*e)/(4*e^5) + (d*((b^4*d)/(4*e^4) - (b^3*(2*a*e - b*d))/(2*e^4)))/e)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((
a + b*x)*(d + e*x)^4) + (((3*b^4*d - 4*a*b^3*e)/(3*e^5) + (b^4*d)/(3*e^5))*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((
a + b*x)*(d + e*x)^3) - (b^4*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/(2*e^5*(a + b*x)*(d + e*x)^2)