3.1581 \(\int \frac {(a^2+2 a b x+b^2 x^2)^{5/2}}{(d+e x)^7} \, dx\)

Optimal. Leaf size=48 \[ \frac {(a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{6 (d+e x)^6 (b d-a e)} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {646, 37} \[ \frac {(a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{6 (d+e x)^6 (b d-a e)} \]

Antiderivative was successfully verified.

[In]

Int[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/(d + e*x)^7,x]

[Out]

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

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 646

Int[((d_.) + (e_.)*(x_))^(m_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[(a + b*x + c*x^2)^Fra
cPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*FracPart[p])), Int[(d + e*x)^m*(b/2 + c*x)^(2*p), x], x] /; FreeQ[{a, b,
c, d, e, m, p}, x] && EqQ[b^2 - 4*a*c, 0] &&  !IntegerQ[p] && NeQ[2*c*d - b*e, 0]

Rubi steps

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

________________________________________________________________________________________

Mathematica [B]  time = 0.08, size = 218, normalized size = 4.54 \[ -\frac {\sqrt {(a+b x)^2} \left (a^5 e^5+a^4 b e^4 (d+6 e x)+a^3 b^2 e^3 \left (d^2+6 d e x+15 e^2 x^2\right )+a^2 b^3 e^2 \left (d^3+6 d^2 e x+15 d e^2 x^2+20 e^3 x^3\right )+a b^4 e \left (d^4+6 d^3 e x+15 d^2 e^2 x^2+20 d e^3 x^3+15 e^4 x^4\right )+b^5 \left (d^5+6 d^4 e x+15 d^3 e^2 x^2+20 d^2 e^3 x^3+15 d e^4 x^4+6 e^5 x^5\right )\right )}{6 e^6 (a+b x) (d+e x)^6} \]

Antiderivative was successfully verified.

[In]

Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/(d + e*x)^7,x]

[Out]

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

________________________________________________________________________________________

fricas [B]  time = 0.74, size = 300, normalized size = 6.25 \[ -\frac {6 \, b^{5} e^{5} x^{5} + b^{5} d^{5} + a b^{4} d^{4} e + a^{2} b^{3} d^{3} e^{2} + a^{3} b^{2} d^{2} e^{3} + a^{4} b d e^{4} + a^{5} e^{5} + 15 \, {\left (b^{5} d e^{4} + a b^{4} e^{5}\right )} x^{4} + 20 \, {\left (b^{5} d^{2} e^{3} + a b^{4} d e^{4} + a^{2} b^{3} e^{5}\right )} x^{3} + 15 \, {\left (b^{5} d^{3} e^{2} + a b^{4} d^{2} e^{3} + a^{2} b^{3} d e^{4} + a^{3} b^{2} e^{5}\right )} x^{2} + 6 \, {\left (b^{5} d^{4} e + a b^{4} d^{3} e^{2} + a^{2} b^{3} d^{2} e^{3} + a^{3} b^{2} d e^{4} + a^{4} b e^{5}\right )} x}{6 \, {\left (e^{12} x^{6} + 6 \, d e^{11} x^{5} + 15 \, d^{2} e^{10} x^{4} + 20 \, d^{3} e^{9} x^{3} + 15 \, d^{4} e^{8} x^{2} + 6 \, d^{5} e^{7} x + d^{6} e^{6}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^7,x, algorithm="fricas")

[Out]

-1/6*(6*b^5*e^5*x^5 + b^5*d^5 + a*b^4*d^4*e + a^2*b^3*d^3*e^2 + a^3*b^2*d^2*e^3 + a^4*b*d*e^4 + a^5*e^5 + 15*(
b^5*d*e^4 + a*b^4*e^5)*x^4 + 20*(b^5*d^2*e^3 + a*b^4*d*e^4 + a^2*b^3*e^5)*x^3 + 15*(b^5*d^3*e^2 + a*b^4*d^2*e^
3 + a^2*b^3*d*e^4 + a^3*b^2*e^5)*x^2 + 6*(b^5*d^4*e + a*b^4*d^3*e^2 + a^2*b^3*d^2*e^3 + a^3*b^2*d*e^4 + a^4*b*
e^5)*x)/(e^12*x^6 + 6*d*e^11*x^5 + 15*d^2*e^10*x^4 + 20*d^3*e^9*x^3 + 15*d^4*e^8*x^2 + 6*d^5*e^7*x + d^6*e^6)

________________________________________________________________________________________

giac [B]  time = 0.21, size = 376, normalized size = 7.83 \[ -\frac {{\left (6 \, b^{5} x^{5} e^{5} \mathrm {sgn}\left (b x + a\right ) + 15 \, b^{5} d x^{4} e^{4} \mathrm {sgn}\left (b x + a\right ) + 20 \, b^{5} d^{2} x^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) + 15 \, b^{5} d^{3} x^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) + 6 \, b^{5} d^{4} x e \mathrm {sgn}\left (b x + a\right ) + b^{5} d^{5} \mathrm {sgn}\left (b x + a\right ) + 15 \, a b^{4} x^{4} e^{5} \mathrm {sgn}\left (b x + a\right ) + 20 \, a b^{4} d x^{3} e^{4} \mathrm {sgn}\left (b x + a\right ) + 15 \, a b^{4} d^{2} x^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) + 6 \, a b^{4} d^{3} x e^{2} \mathrm {sgn}\left (b x + a\right ) + a b^{4} d^{4} e \mathrm {sgn}\left (b x + a\right ) + 20 \, a^{2} b^{3} x^{3} e^{5} \mathrm {sgn}\left (b x + a\right ) + 15 \, a^{2} b^{3} d x^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) + 6 \, a^{2} b^{3} d^{2} x e^{3} \mathrm {sgn}\left (b x + a\right ) + a^{2} b^{3} d^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) + 15 \, a^{3} b^{2} x^{2} e^{5} \mathrm {sgn}\left (b x + a\right ) + 6 \, a^{3} b^{2} d x e^{4} \mathrm {sgn}\left (b x + a\right ) + a^{3} b^{2} d^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) + 6 \, a^{4} b x e^{5} \mathrm {sgn}\left (b x + a\right ) + a^{4} b d e^{4} \mathrm {sgn}\left (b x + a\right ) + a^{5} e^{5} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-6\right )}}{6 \, {\left (x e + d\right )}^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^7,x, algorithm="giac")

[Out]

-1/6*(6*b^5*x^5*e^5*sgn(b*x + a) + 15*b^5*d*x^4*e^4*sgn(b*x + a) + 20*b^5*d^2*x^3*e^3*sgn(b*x + a) + 15*b^5*d^
3*x^2*e^2*sgn(b*x + a) + 6*b^5*d^4*x*e*sgn(b*x + a) + b^5*d^5*sgn(b*x + a) + 15*a*b^4*x^4*e^5*sgn(b*x + a) + 2
0*a*b^4*d*x^3*e^4*sgn(b*x + a) + 15*a*b^4*d^2*x^2*e^3*sgn(b*x + a) + 6*a*b^4*d^3*x*e^2*sgn(b*x + a) + a*b^4*d^
4*e*sgn(b*x + a) + 20*a^2*b^3*x^3*e^5*sgn(b*x + a) + 15*a^2*b^3*d*x^2*e^4*sgn(b*x + a) + 6*a^2*b^3*d^2*x*e^3*s
gn(b*x + a) + a^2*b^3*d^3*e^2*sgn(b*x + a) + 15*a^3*b^2*x^2*e^5*sgn(b*x + a) + 6*a^3*b^2*d*x*e^4*sgn(b*x + a)
+ a^3*b^2*d^2*e^3*sgn(b*x + a) + 6*a^4*b*x*e^5*sgn(b*x + a) + a^4*b*d*e^4*sgn(b*x + a) + a^5*e^5*sgn(b*x + a))
*e^(-6)/(x*e + d)^6

________________________________________________________________________________________

maple [B]  time = 0.05, size = 283, normalized size = 5.90 \[ -\frac {\left (6 b^{5} e^{5} x^{5}+15 a \,b^{4} e^{5} x^{4}+15 b^{5} d \,e^{4} x^{4}+20 a^{2} b^{3} e^{5} x^{3}+20 a \,b^{4} d \,e^{4} x^{3}+20 b^{5} d^{2} e^{3} x^{3}+15 a^{3} b^{2} e^{5} x^{2}+15 a^{2} b^{3} d \,e^{4} x^{2}+15 a \,b^{4} d^{2} e^{3} x^{2}+15 b^{5} d^{3} e^{2} x^{2}+6 a^{4} b \,e^{5} x +6 a^{3} b^{2} d \,e^{4} x +6 a^{2} b^{3} d^{2} e^{3} x +6 a \,b^{4} d^{3} e^{2} x +6 b^{5} d^{4} e x +a^{5} e^{5}+a^{4} b d \,e^{4}+a^{3} b^{2} d^{2} e^{3}+a^{2} b^{3} d^{3} e^{2}+a \,b^{4} d^{4} e +b^{5} d^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{6 \left (e x +d \right )^{6} \left (b x +a \right )^{5} e^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^7,x)

[Out]

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

________________________________________________________________________________________

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^2*x^2+2*a*b*x+a^2)^(5/2)/(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 details)Is a*e-b*d zero or nonzero?

________________________________________________________________________________________

mupad [B]  time = 0.74, size = 687, normalized size = 14.31 \[ \frac {\left (\frac {4\,b^5\,d-5\,a\,b^4\,e}{2\,e^6}+\frac {b^5\,d}{2\,e^6}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^2}-\frac {\left (\frac {5\,a^4\,b\,e^4-10\,a^3\,b^2\,d\,e^3+10\,a^2\,b^3\,d^2\,e^2-5\,a\,b^4\,d^3\,e+b^5\,d^4}{5\,e^6}+\frac {d\,\left (\frac {-10\,a^3\,b^2\,e^4+10\,a^2\,b^3\,d\,e^3-5\,a\,b^4\,d^2\,e^2+b^5\,d^3\,e}{5\,e^6}+\frac {d\,\left (\frac {d\,\left (\frac {b^5\,d}{5\,e^3}-\frac {b^4\,\left (5\,a\,e-b\,d\right )}{5\,e^3}\right )}{e}+\frac {b^3\,\left (10\,a^2\,e^2-5\,a\,b\,d\,e+b^2\,d^2\right )}{5\,e^4}\right )}{e}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^5}-\frac {\left (\frac {10\,a^2\,b^3\,e^2-15\,a\,b^4\,d\,e+6\,b^5\,d^2}{3\,e^6}+\frac {d\,\left (\frac {b^5\,d}{3\,e^5}-\frac {b^4\,\left (5\,a\,e-3\,b\,d\right )}{3\,e^5}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^3}-\frac {\left (\frac {a^5}{6\,e}-\frac {d\,\left (\frac {5\,a^4\,b}{6\,e}-\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {5\,a\,b^4}{6\,e}-\frac {b^5\,d}{6\,e^2}\right )}{e}-\frac {5\,a^2\,b^3}{3\,e}\right )}{e}+\frac {5\,a^3\,b^2}{3\,e}\right )}{e}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^6}+\frac {\left (\frac {-10\,a^3\,b^2\,e^3+20\,a^2\,b^3\,d\,e^2-15\,a\,b^4\,d^2\,e+4\,b^5\,d^3}{4\,e^6}+\frac {d\,\left (\frac {d\,\left (\frac {b^5\,d}{4\,e^4}-\frac {b^4\,\left (5\,a\,e-2\,b\,d\right )}{4\,e^4}\right )}{e}+\frac {b^3\,\left (10\,a^2\,e^2-10\,a\,b\,d\,e+3\,b^2\,d^2\right )}{4\,e^5}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^4}-\frac {b^5\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{e^6\,\left (a+b\,x\right )\,\left (d+e\,x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a^2 + b^2*x^2 + 2*a*b*x)^(5/2)/(d + e*x)^7,x)

[Out]

(((4*b^5*d - 5*a*b^4*e)/(2*e^6) + (b^5*d)/(2*e^6))*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^2) -
(((b^5*d^4 + 5*a^4*b*e^4 - 10*a^3*b^2*d*e^3 + 10*a^2*b^3*d^2*e^2 - 5*a*b^4*d^3*e)/(5*e^6) + (d*((b^5*d^3*e - 1
0*a^3*b^2*e^4 - 5*a*b^4*d^2*e^2 + 10*a^2*b^3*d*e^3)/(5*e^6) + (d*((d*((b^5*d)/(5*e^3) - (b^4*(5*a*e - b*d))/(5
*e^3)))/e + (b^3*(10*a^2*e^2 + b^2*d^2 - 5*a*b*d*e))/(5*e^4)))/e))/e)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a + b
*x)*(d + e*x)^5) - (((6*b^5*d^2 + 10*a^2*b^3*e^2 - 15*a*b^4*d*e)/(3*e^6) + (d*((b^5*d)/(3*e^5) - (b^4*(5*a*e -
3*b*d))/(3*e^5)))/e)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^3) - ((a^5/(6*e) - (d*((5*a^4*b)/(
6*e) - (d*((d*((d*((5*a*b^4)/(6*e) - (b^5*d)/(6*e^2)))/e - (5*a^2*b^3)/(3*e)))/e + (5*a^3*b^2)/(3*e)))/e))/e)*
(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^6) + (((4*b^5*d^3 - 10*a^3*b^2*e^3 + 20*a^2*b^3*d*e^2 -
15*a*b^4*d^2*e)/(4*e^6) + (d*((d*((b^5*d)/(4*e^4) - (b^4*(5*a*e - 2*b*d))/(4*e^4)))/e + (b^3*(10*a^2*e^2 + 3*b
^2*d^2 - 10*a*b*d*e))/(4*e^5)))/e)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^4) - (b^5*(a^2 + b^2*
x^2 + 2*a*b*x)^(1/2))/(e^6*(a + b*x)*(d + e*x))

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (\left (a + b x\right )^{2}\right )^{\frac {5}{2}}}{\left (d + e x\right )^{7}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b**2*x**2+2*a*b*x+a**2)**(5/2)/(e*x+d)**7,x)

[Out]

Integral(((a + b*x)**2)**(5/2)/(d + e*x)**7, x)

________________________________________________________________________________________