3.18.59 \(\int \frac {(a+b x) (a^2+2 a b x+b^2 x^2)^{3/2}}{(d+e x)^9} \, dx\)

Optimal. Leaf size=252 \[ -\frac {b^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}{e^5 (a+b x) (d+e x)^6}+\frac {4 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3}{7 e^5 (a+b x) (d+e x)^7}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4}{8 e^5 (a+b x) (d+e x)^8}-\frac {b^4 \sqrt {a^2+2 a b x+b^2 x^2}}{4 e^5 (a+b x) (d+e x)^4}+\frac {4 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}{5 e^5 (a+b x) (d+e x)^5} \]

________________________________________________________________________________________

Rubi [A]  time = 0.13, antiderivative size = 252, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {770, 21, 43} \begin {gather*} -\frac {b^4 \sqrt {a^2+2 a b x+b^2 x^2}}{4 e^5 (a+b x) (d+e x)^4}+\frac {4 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}{5 e^5 (a+b x) (d+e x)^5}-\frac {b^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}{e^5 (a+b x) (d+e x)^6}+\frac {4 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3}{7 e^5 (a+b x) (d+e x)^7}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4}{8 e^5 (a+b x) (d+e x)^8} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2))/(d + e*x)^9,x]

[Out]

-((b*d - a*e)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(8*e^5*(a + b*x)*(d + e*x)^8) + (4*b*(b*d - a*e)^3*Sqrt[a^2 + 2
*a*b*x + b^2*x^2])/(7*e^5*(a + b*x)*(d + e*x)^7) - (b^2*(b*d - a*e)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^5*(a +
b*x)*(d + e*x)^6) + (4*b^3*(b*d - a*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*e^5*(a + b*x)*(d + e*x)^5) - (b^4*Sq
rt[a^2 + 2*a*b*x + b^2*x^2])/(4*e^5*(a + b*x)*(d + e*x)^4)

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 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

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]

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)^9} \, 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)^9} \, 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)^9} \, dx}{a b+b^2 x}\\ &=\frac {\left (b \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int \left (\frac {(-b d+a e)^4}{e^4 (d+e x)^9}-\frac {4 b (b d-a e)^3}{e^4 (d+e x)^8}+\frac {6 b^2 (b d-a e)^2}{e^4 (d+e x)^7}-\frac {4 b^3 (b d-a e)}{e^4 (d+e x)^6}+\frac {b^4}{e^4 (d+e x)^5}\right ) \, dx}{a b+b^2 x}\\ &=-\frac {(b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{8 e^5 (a+b x) (d+e x)^8}+\frac {4 b (b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^5 (a+b x) (d+e x)^7}-\frac {b^2 (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{e^5 (a+b x) (d+e x)^6}+\frac {4 b^3 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^5 (a+b x) (d+e x)^5}-\frac {b^4 \sqrt {a^2+2 a b x+b^2 x^2}}{4 e^5 (a+b x) (d+e x)^4}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.06, size = 162, normalized size = 0.64 \begin {gather*} -\frac {\sqrt {(a+b x)^2} \left (35 a^4 e^4+20 a^3 b e^3 (d+8 e x)+10 a^2 b^2 e^2 \left (d^2+8 d e x+28 e^2 x^2\right )+4 a b^3 e \left (d^3+8 d^2 e x+28 d e^2 x^2+56 e^3 x^3\right )+b^4 \left (d^4+8 d^3 e x+28 d^2 e^2 x^2+56 d e^3 x^3+70 e^4 x^4\right )\right )}{280 e^5 (a+b x) (d+e x)^8} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2))/(d + e*x)^9,x]

[Out]

-1/280*(Sqrt[(a + b*x)^2]*(35*a^4*e^4 + 20*a^3*b*e^3*(d + 8*e*x) + 10*a^2*b^2*e^2*(d^2 + 8*d*e*x + 28*e^2*x^2)
+ 4*a*b^3*e*(d^3 + 8*d^2*e*x + 28*d*e^2*x^2 + 56*e^3*x^3) + b^4*(d^4 + 8*d^3*e*x + 28*d^2*e^2*x^2 + 56*d*e^3*
x^3 + 70*e^4*x^4)))/(e^5*(a + b*x)*(d + e*x)^8)

________________________________________________________________________________________

IntegrateAlgebraic [F]  time = 180.03, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2))/(d + e*x)^9,x]

[Out]

$Aborted

________________________________________________________________________________________

fricas [A]  time = 0.43, size = 258, normalized size = 1.02 \begin {gather*} -\frac {70 \, b^{4} e^{4} x^{4} + b^{4} d^{4} + 4 \, a b^{3} d^{3} e + 10 \, a^{2} b^{2} d^{2} e^{2} + 20 \, a^{3} b d e^{3} + 35 \, a^{4} e^{4} + 56 \, {\left (b^{4} d e^{3} + 4 \, a b^{3} e^{4}\right )} x^{3} + 28 \, {\left (b^{4} d^{2} e^{2} + 4 \, a b^{3} d e^{3} + 10 \, a^{2} b^{2} e^{4}\right )} x^{2} + 8 \, {\left (b^{4} d^{3} e + 4 \, a b^{3} d^{2} e^{2} + 10 \, a^{2} b^{2} d e^{3} + 20 \, a^{3} b e^{4}\right )} x}{280 \, {\left (e^{13} x^{8} + 8 \, d e^{12} x^{7} + 28 \, d^{2} e^{11} x^{6} + 56 \, d^{3} e^{10} x^{5} + 70 \, d^{4} e^{9} x^{4} + 56 \, d^{5} e^{8} x^{3} + 28 \, d^{6} e^{7} x^{2} + 8 \, d^{7} e^{6} x + d^{8} e^{5}\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)^(3/2)/(e*x+d)^9,x, algorithm="fricas")

[Out]

-1/280*(70*b^4*e^4*x^4 + b^4*d^4 + 4*a*b^3*d^3*e + 10*a^2*b^2*d^2*e^2 + 20*a^3*b*d*e^3 + 35*a^4*e^4 + 56*(b^4*
d*e^3 + 4*a*b^3*e^4)*x^3 + 28*(b^4*d^2*e^2 + 4*a*b^3*d*e^3 + 10*a^2*b^2*e^4)*x^2 + 8*(b^4*d^3*e + 4*a*b^3*d^2*
e^2 + 10*a^2*b^2*d*e^3 + 20*a^3*b*e^4)*x)/(e^13*x^8 + 8*d*e^12*x^7 + 28*d^2*e^11*x^6 + 56*d^3*e^10*x^5 + 70*d^
4*e^9*x^4 + 56*d^5*e^8*x^3 + 28*d^6*e^7*x^2 + 8*d^7*e^6*x + d^8*e^5)

________________________________________________________________________________________

giac [A]  time = 0.20, size = 264, normalized size = 1.05 \begin {gather*} -\frac {{\left (70 \, b^{4} x^{4} e^{4} \mathrm {sgn}\left (b x + a\right ) + 56 \, b^{4} d x^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) + 28 \, b^{4} d^{2} x^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) + 8 \, b^{4} d^{3} x e \mathrm {sgn}\left (b x + a\right ) + b^{4} d^{4} \mathrm {sgn}\left (b x + a\right ) + 224 \, a b^{3} x^{3} e^{4} \mathrm {sgn}\left (b x + a\right ) + 112 \, a b^{3} d x^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) + 32 \, a b^{3} d^{2} x e^{2} \mathrm {sgn}\left (b x + a\right ) + 4 \, a b^{3} d^{3} e \mathrm {sgn}\left (b x + a\right ) + 280 \, a^{2} b^{2} x^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) + 80 \, a^{2} b^{2} d x e^{3} \mathrm {sgn}\left (b x + a\right ) + 10 \, a^{2} b^{2} d^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) + 160 \, a^{3} b x e^{4} \mathrm {sgn}\left (b x + a\right ) + 20 \, a^{3} b d e^{3} \mathrm {sgn}\left (b x + a\right ) + 35 \, a^{4} e^{4} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-5\right )}}{280 \, {\left (x e + d\right )}^{8}} \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)^(3/2)/(e*x+d)^9,x, algorithm="giac")

[Out]

-1/280*(70*b^4*x^4*e^4*sgn(b*x + a) + 56*b^4*d*x^3*e^3*sgn(b*x + a) + 28*b^4*d^2*x^2*e^2*sgn(b*x + a) + 8*b^4*
d^3*x*e*sgn(b*x + a) + b^4*d^4*sgn(b*x + a) + 224*a*b^3*x^3*e^4*sgn(b*x + a) + 112*a*b^3*d*x^2*e^3*sgn(b*x + a
) + 32*a*b^3*d^2*x*e^2*sgn(b*x + a) + 4*a*b^3*d^3*e*sgn(b*x + a) + 280*a^2*b^2*x^2*e^4*sgn(b*x + a) + 80*a^2*b
^2*d*x*e^3*sgn(b*x + a) + 10*a^2*b^2*d^2*e^2*sgn(b*x + a) + 160*a^3*b*x*e^4*sgn(b*x + a) + 20*a^3*b*d*e^3*sgn(
b*x + a) + 35*a^4*e^4*sgn(b*x + a))*e^(-5)/(x*e + d)^8

________________________________________________________________________________________

maple [A]  time = 0.07, size = 201, normalized size = 0.80 \begin {gather*} -\frac {\left (70 b^{4} e^{4} x^{4}+224 a \,b^{3} e^{4} x^{3}+56 b^{4} d \,e^{3} x^{3}+280 a^{2} b^{2} e^{4} x^{2}+112 a \,b^{3} d \,e^{3} x^{2}+28 b^{4} d^{2} e^{2} x^{2}+160 a^{3} b \,e^{4} x +80 a^{2} b^{2} d \,e^{3} x +32 a \,b^{3} d^{2} e^{2} x +8 b^{4} d^{3} e x +35 a^{4} e^{4}+20 a^{3} b d \,e^{3}+10 a^{2} b^{2} d^{2} e^{2}+4 a \,b^{3} d^{3} e +b^{4} d^{4}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{280 \left (e x +d \right )^{8} \left (b x +a \right )^{3} e^{5}} \end {gather*}

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)^9,x)

[Out]

-1/280/e^5*(70*b^4*e^4*x^4+224*a*b^3*e^4*x^3+56*b^4*d*e^3*x^3+280*a^2*b^2*e^4*x^2+112*a*b^3*d*e^3*x^2+28*b^4*d
^2*e^2*x^2+160*a^3*b*e^4*x+80*a^2*b^2*d*e^3*x+32*a*b^3*d^2*e^2*x+8*b^4*d^3*e*x+35*a^4*e^4+20*a^3*b*d*e^3+10*a^
2*b^2*d^2*e^2+4*a*b^3*d^3*e+b^4*d^4)*((b*x+a)^2)^(3/2)/(e*x+d)^8/(b*x+a)^3

________________________________________________________________________________________

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)^(3/2)/(e*x+d)^9,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 = 2.17, size = 449, normalized size = 1.78 \begin {gather*} \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}{7\,e^5}+\frac {d\,\left (\frac {d\,\left (\frac {b^4\,d}{7\,e^3}-\frac {b^3\,\left (4\,a\,e-b\,d\right )}{7\,e^3}\right )}{e}+\frac {b^2\,\left (6\,a^2\,e^2-4\,a\,b\,d\,e+b^2\,d^2\right )}{7\,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 )}^7}-\frac {\left (\frac {a^4}{8\,e}-\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {a\,b^3}{2\,e}-\frac {b^4\,d}{8\,e^2}\right )}{e}-\frac {3\,a^2\,b^2}{4\,e}\right )}{e}+\frac {a^3\,b}{2\,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 {6\,a^2\,b^2\,e^2-8\,a\,b^3\,d\,e+3\,b^4\,d^2}{6\,e^5}+\frac {d\,\left (\frac {b^4\,d}{6\,e^4}-\frac {b^3\,\left (2\,a\,e-b\,d\right )}{3\,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 )}^6}+\frac {\left (\frac {3\,b^4\,d-4\,a\,b^3\,e}{5\,e^5}+\frac {b^4\,d}{5\,e^5}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^5}-\frac {b^4\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{4\,e^5\,\left (a+b\,x\right )\,{\left (d+e\,x\right )}^4} \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)^(3/2))/(d + e*x)^9,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)/(7*e^5) + (d*((d*((b^4*d)/(7*e^3) - (b^3*(4*a*e -
b*d))/(7*e^3)))/e + (b^2*(6*a^2*e^2 + b^2*d^2 - 4*a*b*d*e))/(7*e^4)))/e)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a
+ b*x)*(d + e*x)^7) - ((a^4/(8*e) - (d*((d*((d*((a*b^3)/(2*e) - (b^4*d)/(8*e^2)))/e - (3*a^2*b^2)/(4*e)))/e +
(a^3*b)/(2*e)))/e)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^8) - (((3*b^4*d^2 + 6*a^2*b^2*e^2 - 8
*a*b^3*d*e)/(6*e^5) + (d*((b^4*d)/(6*e^4) - (b^3*(2*a*e - b*d))/(3*e^4)))/e)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/
((a + b*x)*(d + e*x)^6) + (((3*b^4*d - 4*a*b^3*e)/(5*e^5) + (b^4*d)/(5*e^5))*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/
((a + b*x)*(d + e*x)^5) - (b^4*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/(4*e^5*(a + b*x)*(d + e*x)^4)

________________________________________________________________________________________

sympy [F(-1)]  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)**(3/2)/(e*x+d)**9,x)

[Out]

Timed out

________________________________________________________________________________________