3.1982 \(\int \frac{(a+b x) (a^2+2 a b x+b^2 x^2)^{3/2}}{(d+e x)^7} \, dx\)

Optimal. Leaf size=98 \[ \frac{b \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^4}{30 (d+e x)^5 (b d-a e)^2}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^4}{6 (d+e x)^6 (b d-a e)} \]

[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)

________________________________________________________________________________________

Rubi [A]  time = 0.0520627, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.121, Rules used = {770, 21, 45, 37} \[ \frac{b \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^4}{30 (d+e x)^5 (b d-a e)^2}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^4}{6 (d+e x)^6 (b d-a e)} \]

Antiderivative was successfully verified.

[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 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]

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 45

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 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]

Rubi steps

\begin{align*} \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^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]  time = 0.0616189, size = 162, normalized size = 1.65 \[ -\frac{\sqrt{(a+b x)^2} \left (3 a^2 b^2 e^2 \left (d^2+6 d e x+15 e^2 x^2\right )+4 a^3 b e^3 (d+6 e x)+5 a^4 e^4+2 a b^3 e \left (6 d^2 e x+d^3+15 d e^2 x^2+20 e^3 x^3\right )+b^4 \left (15 d^2 e^2 x^2+6 d^3 e x+d^4+20 d e^3 x^3+15 e^4 x^4\right )\right )}{30 e^5 (a+b x) (d+e x)^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(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)))/(30*e^5*(a + b*x)*(d + e*x)^6)

________________________________________________________________________________________

Maple [B]  time = 0.008, size = 201, normalized size = 2.1 \begin{align*} -{\frac{15\,{x}^{4}{b}^{4}{e}^{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\,xa{b}^{3}{d}^{2}{e}^{2}+6\,x{b}^{4}{d}^{3}e+5\,{a}^{4}{e}^{4}+4\,d{e}^{3}{a}^{3}b+3\,{a}^{2}{b}^{2}{d}^{2}{e}^{2}+2\,a{b}^{3}{d}^{3}e+{b}^{4}{d}^{4}}{30\,{e}^{5} \left ( ex+d \right ) ^{6} \left ( bx+a \right ) ^{3}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \end{align*}

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

[Out]

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

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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)^(3/2)/(e*x+d)^7,x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [B]  time = 1.54349, size = 482, normalized size = 4.92 \begin{align*} -\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 )}} \end{align*}

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)^(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(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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

[Out]

Timed out

________________________________________________________________________________________

Giac [B]  time = 1.14452, size = 356, normalized size = 3.63 \begin{align*} -\frac{{\left (15 \, b^{4} x^{4} e^{4} \mathrm{sgn}\left (b x + a\right ) + 20 \, b^{4} d x^{3} e^{3} \mathrm{sgn}\left (b x + a\right ) + 15 \, b^{4} d^{2} x^{2} e^{2} \mathrm{sgn}\left (b x + a\right ) + 6 \, b^{4} d^{3} x e \mathrm{sgn}\left (b x + a\right ) + b^{4} d^{4} \mathrm{sgn}\left (b x + a\right ) + 40 \, a b^{3} x^{3} e^{4} \mathrm{sgn}\left (b x + a\right ) + 30 \, a b^{3} d x^{2} e^{3} \mathrm{sgn}\left (b x + a\right ) + 12 \, a b^{3} d^{2} x e^{2} \mathrm{sgn}\left (b x + a\right ) + 2 \, a b^{3} d^{3} e \mathrm{sgn}\left (b x + a\right ) + 45 \, a^{2} b^{2} x^{2} e^{4} \mathrm{sgn}\left (b x + a\right ) + 18 \, a^{2} b^{2} d x e^{3} \mathrm{sgn}\left (b x + a\right ) + 3 \, a^{2} b^{2} d^{2} e^{2} \mathrm{sgn}\left (b x + a\right ) + 24 \, a^{3} b x e^{4} \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 )\right )} e^{\left (-5\right )}}{30 \,{\left (x e + d\right )}^{6}} \end{align*}

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

[Out]

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