3.2007 \(\int \frac{(a+b x) (a^2+2 a b x+b^2 x^2)^{5/2}}{(d+e x)^{10}} \, dx\)

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

[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)

________________________________________________________________________________________

Rubi [A]  time = 0.0646765, antiderivative size = 149, normalized size of antiderivative = 1., 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 = {770, 21, 45, 37} \[ \frac{b^2 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^6}{252 (d+e x)^7 (b d-a e)^3}+\frac{b \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^6}{36 (d+e x)^8 (b d-a e)^2}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^6}{9 (d+e x)^9 (b d-a e)} \]

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 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 )^{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*}

Mathematica [A]  time = 0.11782, size = 295, normalized size = 1.98 \[ -\frac{\sqrt{(a+b x)^2} \left (6 a^2 b^4 e^2 \left (36 d^2 e^2 x^2+9 d^3 e x+d^4+84 d e^3 x^3+126 e^4 x^4\right )+10 a^3 b^3 e^3 \left (9 d^2 e x+d^3+36 d e^2 x^2+84 e^3 x^3\right )+15 a^4 b^2 e^4 \left (d^2+9 d e x+36 e^2 x^2\right )+21 a^5 b e^5 (d+9 e x)+28 a^6 e^6+3 a b^5 e \left (36 d^3 e^2 x^2+84 d^2 e^3 x^3+9 d^4 e x+d^5+126 d e^4 x^4+126 e^5 x^5\right )+b^6 \left (36 d^4 e^2 x^2+84 d^3 e^3 x^3+126 d^2 e^4 x^4+9 d^5 e x+d^6+126 d e^5 x^5+84 e^6 x^6\right )\right )}{252 e^7 (a+b x) (d+e x)^9} \]

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]

-(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 + 1
26*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)))/(252*e^7*(a + b*x)*(d + e*x)^9)

________________________________________________________________________________________

Maple [B]  time = 0.009, size = 392, normalized size = 2.6 \begin{align*} -{\frac{84\,{x}^{6}{b}^{6}{e}^{6}+378\,{x}^{5}a{b}^{5}{e}^{6}+126\,{x}^{5}{b}^{6}d{e}^{5}+756\,{x}^{4}{a}^{2}{b}^{4}{e}^{6}+378\,{x}^{4}a{b}^{5}d{e}^{5}+126\,{x}^{4}{b}^{6}{d}^{2}{e}^{4}+840\,{x}^{3}{a}^{3}{b}^{3}{e}^{6}+504\,{x}^{3}{a}^{2}{b}^{4}d{e}^{5}+252\,{x}^{3}a{b}^{5}{d}^{2}{e}^{4}+84\,{x}^{3}{b}^{6}{d}^{3}{e}^{3}+540\,{x}^{2}{a}^{4}{b}^{2}{e}^{6}+360\,{x}^{2}{a}^{3}{b}^{3}d{e}^{5}+216\,{x}^{2}{a}^{2}{b}^{4}{d}^{2}{e}^{4}+108\,{x}^{2}a{b}^{5}{d}^{3}{e}^{3}+36\,{x}^{2}{b}^{6}{d}^{4}{e}^{2}+189\,x{a}^{5}b{e}^{6}+135\,x{a}^{4}{b}^{2}d{e}^{5}+90\,x{a}^{3}{b}^{3}{d}^{2}{e}^{4}+54\,x{a}^{2}{b}^{4}{d}^{3}{e}^{3}+27\,xa{b}^{5}{d}^{4}{e}^{2}+9\,x{b}^{6}{d}^{5}e+28\,{a}^{6}{e}^{6}+21\,d{e}^{5}{a}^{5}b+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}}{252\,{e}^{7} \left ( ex+d \right ) ^{9} \left ( bx+a \right ) ^{5}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{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)^(5/2)/(e*x+d)^10,x)

[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

________________________________________________________________________________________

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)^(5/2)/(e*x+d)^10,x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [B]  time = 1.60737, size = 915, normalized size = 6.14 \begin{align*} -\frac{84 \, b^{6} e^{6} x^{6} + b^{6} d^{6} + 3 \, a b^{5} d^{5} e + 6 \, a^{2} b^{4} d^{4} e^{2} + 10 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} + 21 \, a^{5} b d e^{5} + 28 \, a^{6} e^{6} + 126 \,{\left (b^{6} d e^{5} + 3 \, a b^{5} e^{6}\right )} x^{5} + 126 \,{\left (b^{6} d^{2} e^{4} + 3 \, a b^{5} d e^{5} + 6 \, a^{2} b^{4} e^{6}\right )} x^{4} + 84 \,{\left (b^{6} d^{3} e^{3} + 3 \, a b^{5} d^{2} e^{4} + 6 \, a^{2} b^{4} d e^{5} + 10 \, a^{3} b^{3} e^{6}\right )} x^{3} + 36 \,{\left (b^{6} d^{4} e^{2} + 3 \, a b^{5} d^{3} e^{3} + 6 \, a^{2} b^{4} d^{2} e^{4} + 10 \, a^{3} b^{3} d e^{5} + 15 \, a^{4} b^{2} e^{6}\right )} x^{2} + 9 \,{\left (b^{6} d^{5} e + 3 \, a b^{5} d^{4} e^{2} + 6 \, a^{2} b^{4} d^{3} e^{3} + 10 \, a^{3} b^{3} d^{2} e^{4} + 15 \, a^{4} b^{2} d e^{5} + 21 \, a^{5} b e^{6}\right )} x}{252 \,{\left (e^{16} x^{9} + 9 \, d e^{15} x^{8} + 36 \, d^{2} e^{14} x^{7} + 84 \, d^{3} e^{13} x^{6} + 126 \, d^{4} e^{12} x^{5} + 126 \, d^{5} e^{11} x^{4} + 84 \, d^{6} e^{10} x^{3} + 36 \, d^{7} e^{9} x^{2} + 9 \, d^{8} e^{8} x + d^{9} e^{7}\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)^(5/2)/(e*x+d)^10,x, algorithm="fricas")

[Out]

-1/252*(84*b^6*e^6*x^6 + b^6*d^6 + 3*a*b^5*d^5*e + 6*a^2*b^4*d^4*e^2 + 10*a^3*b^3*d^3*e^3 + 15*a^4*b^2*d^2*e^4
+ 21*a^5*b*d*e^5 + 28*a^6*e^6 + 126*(b^6*d*e^5 + 3*a*b^5*e^6)*x^5 + 126*(b^6*d^2*e^4 + 3*a*b^5*d*e^5 + 6*a^2*
b^4*e^6)*x^4 + 84*(b^6*d^3*e^3 + 3*a*b^5*d^2*e^4 + 6*a^2*b^4*d*e^5 + 10*a^3*b^3*e^6)*x^3 + 36*(b^6*d^4*e^2 + 3
*a*b^5*d^3*e^3 + 6*a^2*b^4*d^2*e^4 + 10*a^3*b^3*d*e^5 + 15*a^4*b^2*e^6)*x^2 + 9*(b^6*d^5*e + 3*a*b^5*d^4*e^2 +
6*a^2*b^4*d^3*e^3 + 10*a^3*b^3*d^2*e^4 + 15*a^4*b^2*d*e^5 + 21*a^5*b*e^6)*x)/(e^16*x^9 + 9*d*e^15*x^8 + 36*d^
2*e^14*x^7 + 84*d^3*e^13*x^6 + 126*d^4*e^12*x^5 + 126*d^5*e^11*x^4 + 84*d^6*e^10*x^3 + 36*d^7*e^9*x^2 + 9*d^8*
e^8*x + d^9*e^7)

________________________________________________________________________________________

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)**(5/2)/(e*x+d)**10,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B]  time = 1.14802, size = 702, normalized size = 4.71 \begin{align*} -\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{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)^(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