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

Optimal. Leaf size=258 \[ -\frac {4 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}{e^5 (a+b x) (d+e x)^{3/2}}+\frac {8 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3}{5 e^5 (a+b x) (d+e x)^{5/2}}-\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4}{7 e^5 (a+b x) (d+e x)^{7/2}}+\frac {2 b^4 \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x}}{e^5 (a+b x)}+\frac {8 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}{e^5 (a+b x) \sqrt {d+e x}} \]

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Rubi [A]  time = 0.10, antiderivative size = 258, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.086, Rules used = {770, 21, 43} \begin {gather*} \frac {2 b^4 \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x}}{e^5 (a+b x)}+\frac {8 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}{e^5 (a+b x) \sqrt {d+e x}}-\frac {4 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)^{3/2}}+\frac {8 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3}{5 e^5 (a+b x) (d+e x)^{5/2}}-\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4}{7 e^5 (a+b x) (d+e x)^{7/2}} \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)^(9/2),x]

[Out]

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

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

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Mathematica [A]  time = 0.09, size = 173, normalized size = 0.67 \begin {gather*} -\frac {2 \sqrt {(a+b x)^2} \left (5 a^4 e^4+4 a^3 b e^3 (2 d+7 e x)+2 a^2 b^2 e^2 \left (8 d^2+28 d e x+35 e^2 x^2\right )+4 a b^3 e \left (16 d^3+56 d^2 e x+70 d e^2 x^2+35 e^3 x^3\right )-\left (b^4 \left (128 d^4+448 d^3 e x+560 d^2 e^2 x^2+280 d e^3 x^3+35 e^4 x^4\right )\right )\right )}{35 e^5 (a+b x) (d+e x)^{7/2}} \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)^(9/2),x]

[Out]

(-2*Sqrt[(a + b*x)^2]*(5*a^4*e^4 + 4*a^3*b*e^3*(2*d + 7*e*x) + 2*a^2*b^2*e^2*(8*d^2 + 28*d*e*x + 35*e^2*x^2) +
4*a*b^3*e*(16*d^3 + 56*d^2*e*x + 70*d*e^2*x^2 + 35*e^3*x^3) - b^4*(128*d^4 + 448*d^3*e*x + 560*d^2*e^2*x^2 +
280*d*e^3*x^3 + 35*e^4*x^4)))/(35*e^5*(a + b*x)*(d + e*x)^(7/2))

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IntegrateAlgebraic [A]  time = 26.60, size = 241, normalized size = 0.93 \begin {gather*} \frac {2 \sqrt {\frac {(a e+b e x)^2}{e^2}} \left (-5 a^4 e^4-28 a^3 b e^3 (d+e x)+20 a^3 b d e^3-30 a^2 b^2 d^2 e^2-70 a^2 b^2 e^2 (d+e x)^2+84 a^2 b^2 d e^2 (d+e x)+20 a b^3 d^3 e-84 a b^3 d^2 e (d+e x)-140 a b^3 e (d+e x)^3+140 a b^3 d e (d+e x)^2-5 b^4 d^4+28 b^4 d^3 (d+e x)-70 b^4 d^2 (d+e x)^2+35 b^4 (d+e x)^4+140 b^4 d (d+e x)^3\right )}{35 e^4 (d+e x)^{7/2} (a e+b e x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[(a*e + b*e*x)^2/e^2]*(-5*b^4*d^4 + 20*a*b^3*d^3*e - 30*a^2*b^2*d^2*e^2 + 20*a^3*b*d*e^3 - 5*a^4*e^4 +
28*b^4*d^3*(d + e*x) - 84*a*b^3*d^2*e*(d + e*x) + 84*a^2*b^2*d*e^2*(d + e*x) - 28*a^3*b*e^3*(d + e*x) - 70*b^4
*d^2*(d + e*x)^2 + 140*a*b^3*d*e*(d + e*x)^2 - 70*a^2*b^2*e^2*(d + e*x)^2 + 140*b^4*d*(d + e*x)^3 - 140*a*b^3*
e*(d + e*x)^3 + 35*b^4*(d + e*x)^4))/(35*e^4*(d + e*x)^(7/2)*(a*e + b*e*x))

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fricas [A]  time = 0.44, size = 225, normalized size = 0.87 \begin {gather*} \frac {2 \, {\left (35 \, b^{4} e^{4} x^{4} + 128 \, b^{4} d^{4} - 64 \, a b^{3} d^{3} e - 16 \, a^{2} b^{2} d^{2} e^{2} - 8 \, a^{3} b d e^{3} - 5 \, a^{4} e^{4} + 140 \, {\left (2 \, b^{4} d e^{3} - a b^{3} e^{4}\right )} x^{3} + 70 \, {\left (8 \, b^{4} d^{2} e^{2} - 4 \, a b^{3} d e^{3} - a^{2} b^{2} e^{4}\right )} x^{2} + 28 \, {\left (16 \, b^{4} d^{3} e - 8 \, a b^{3} d^{2} e^{2} - 2 \, a^{2} b^{2} d e^{3} - a^{3} b e^{4}\right )} x\right )} \sqrt {e x + d}}{35 \, {\left (e^{9} x^{4} + 4 \, d e^{8} x^{3} + 6 \, d^{2} e^{7} x^{2} + 4 \, d^{3} e^{6} x + d^{4} 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)^(9/2),x, algorithm="fricas")

[Out]

2/35*(35*b^4*e^4*x^4 + 128*b^4*d^4 - 64*a*b^3*d^3*e - 16*a^2*b^2*d^2*e^2 - 8*a^3*b*d*e^3 - 5*a^4*e^4 + 140*(2*
b^4*d*e^3 - a*b^3*e^4)*x^3 + 70*(8*b^4*d^2*e^2 - 4*a*b^3*d*e^3 - a^2*b^2*e^4)*x^2 + 28*(16*b^4*d^3*e - 8*a*b^3
*d^2*e^2 - 2*a^2*b^2*d*e^3 - a^3*b*e^4)*x)*sqrt(e*x + d)/(e^9*x^4 + 4*d*e^8*x^3 + 6*d^2*e^7*x^2 + 4*d^3*e^6*x
+ d^4*e^5)

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giac [A]  time = 0.23, size = 310, normalized size = 1.20 \begin {gather*} 2 \, \sqrt {x e + d} b^{4} e^{\left (-5\right )} \mathrm {sgn}\left (b x + a\right ) + \frac {2 \, {\left (140 \, {\left (x e + d\right )}^{3} b^{4} d \mathrm {sgn}\left (b x + a\right ) - 70 \, {\left (x e + d\right )}^{2} b^{4} d^{2} \mathrm {sgn}\left (b x + a\right ) + 28 \, {\left (x e + d\right )} b^{4} d^{3} \mathrm {sgn}\left (b x + a\right ) - 5 \, b^{4} d^{4} \mathrm {sgn}\left (b x + a\right ) - 140 \, {\left (x e + d\right )}^{3} a b^{3} e \mathrm {sgn}\left (b x + a\right ) + 140 \, {\left (x e + d\right )}^{2} a b^{3} d e \mathrm {sgn}\left (b x + a\right ) - 84 \, {\left (x e + d\right )} a b^{3} d^{2} e \mathrm {sgn}\left (b x + a\right ) + 20 \, a b^{3} d^{3} e \mathrm {sgn}\left (b x + a\right ) - 70 \, {\left (x e + d\right )}^{2} a^{2} b^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) + 84 \, {\left (x e + d\right )} a^{2} b^{2} d e^{2} \mathrm {sgn}\left (b x + a\right ) - 30 \, a^{2} b^{2} d^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) - 28 \, {\left (x e + d\right )} a^{3} b e^{3} \mathrm {sgn}\left (b x + a\right ) + 20 \, 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 )}}{35 \, {\left (x e + d\right )}^{\frac {7}{2}}} \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)^(9/2),x, algorithm="giac")

[Out]

2*sqrt(x*e + d)*b^4*e^(-5)*sgn(b*x + a) + 2/35*(140*(x*e + d)^3*b^4*d*sgn(b*x + a) - 70*(x*e + d)^2*b^4*d^2*sg
n(b*x + a) + 28*(x*e + d)*b^4*d^3*sgn(b*x + a) - 5*b^4*d^4*sgn(b*x + a) - 140*(x*e + d)^3*a*b^3*e*sgn(b*x + a)
+ 140*(x*e + d)^2*a*b^3*d*e*sgn(b*x + a) - 84*(x*e + d)*a*b^3*d^2*e*sgn(b*x + a) + 20*a*b^3*d^3*e*sgn(b*x + a
) - 70*(x*e + d)^2*a^2*b^2*e^2*sgn(b*x + a) + 84*(x*e + d)*a^2*b^2*d*e^2*sgn(b*x + a) - 30*a^2*b^2*d^2*e^2*sgn
(b*x + a) - 28*(x*e + d)*a^3*b*e^3*sgn(b*x + a) + 20*a^3*b*d*e^3*sgn(b*x + a) - 5*a^4*e^4*sgn(b*x + a))*e^(-5)
/(x*e + d)^(7/2)

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maple [A]  time = 0.05, size = 202, normalized size = 0.78 \begin {gather*} -\frac {2 \left (-35 b^{4} e^{4} x^{4}+140 a \,b^{3} e^{4} x^{3}-280 b^{4} d \,e^{3} x^{3}+70 a^{2} b^{2} e^{4} x^{2}+280 a \,b^{3} d \,e^{3} x^{2}-560 b^{4} d^{2} e^{2} x^{2}+28 a^{3} b \,e^{4} x +56 a^{2} b^{2} d \,e^{3} x +224 a \,b^{3} d^{2} e^{2} x -448 b^{4} d^{3} e x +5 a^{4} e^{4}+8 a^{3} b d \,e^{3}+16 a^{2} b^{2} d^{2} e^{2}+64 a \,b^{3} d^{3} e -128 b^{4} d^{4}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{35 \left (e x +d \right )^{\frac {7}{2}} \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)^(9/2),x)

[Out]

-2/35/(e*x+d)^(7/2)*(-35*b^4*e^4*x^4+140*a*b^3*e^4*x^3-280*b^4*d*e^3*x^3+70*a^2*b^2*e^4*x^2+280*a*b^3*d*e^3*x^
2-560*b^4*d^2*e^2*x^2+28*a^3*b*e^4*x+56*a^2*b^2*d*e^3*x+224*a*b^3*d^2*e^2*x-448*b^4*d^3*e*x+5*a^4*e^4+8*a^3*b*
d*e^3+16*a^2*b^2*d^2*e^2+64*a*b^3*d^3*e-128*b^4*d^4)*((b*x+a)^2)^(3/2)/e^5/(b*x+a)^3

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maxima [A]  time = 0.73, size = 348, normalized size = 1.35 \begin {gather*} -\frac {2 \, {\left (35 \, b^{3} e^{3} x^{3} + 16 \, b^{3} d^{3} + 8 \, a b^{2} d^{2} e + 6 \, a^{2} b d e^{2} + 5 \, a^{3} e^{3} + 35 \, {\left (2 \, b^{3} d e^{2} + a b^{2} e^{3}\right )} x^{2} + 7 \, {\left (8 \, b^{3} d^{2} e + 4 \, a b^{2} d e^{2} + 3 \, a^{2} b e^{3}\right )} x\right )} a}{35 \, {\left (e^{7} x^{3} + 3 \, d e^{6} x^{2} + 3 \, d^{2} e^{5} x + d^{3} e^{4}\right )} \sqrt {e x + d}} + \frac {2 \, {\left (35 \, b^{3} e^{4} x^{4} + 128 \, b^{3} d^{4} - 48 \, a b^{2} d^{3} e - 8 \, a^{2} b d^{2} e^{2} - 2 \, a^{3} d e^{3} + 35 \, {\left (8 \, b^{3} d e^{3} - 3 \, a b^{2} e^{4}\right )} x^{3} + 35 \, {\left (16 \, b^{3} d^{2} e^{2} - 6 \, a b^{2} d e^{3} - a^{2} b e^{4}\right )} x^{2} + 7 \, {\left (64 \, b^{3} d^{3} e - 24 \, a b^{2} d^{2} e^{2} - 4 \, a^{2} b d e^{3} - a^{3} e^{4}\right )} x\right )} b}{35 \, {\left (e^{8} x^{3} + 3 \, d e^{7} x^{2} + 3 \, d^{2} e^{6} x + d^{3} e^{5}\right )} \sqrt {e x + d}} \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)^(9/2),x, algorithm="maxima")

[Out]

-2/35*(35*b^3*e^3*x^3 + 16*b^3*d^3 + 8*a*b^2*d^2*e + 6*a^2*b*d*e^2 + 5*a^3*e^3 + 35*(2*b^3*d*e^2 + a*b^2*e^3)*
x^2 + 7*(8*b^3*d^2*e + 4*a*b^2*d*e^2 + 3*a^2*b*e^3)*x)*a/((e^7*x^3 + 3*d*e^6*x^2 + 3*d^2*e^5*x + d^3*e^4)*sqrt
(e*x + d)) + 2/35*(35*b^3*e^4*x^4 + 128*b^3*d^4 - 48*a*b^2*d^3*e - 8*a^2*b*d^2*e^2 - 2*a^3*d*e^3 + 35*(8*b^3*d
*e^3 - 3*a*b^2*e^4)*x^3 + 35*(16*b^3*d^2*e^2 - 6*a*b^2*d*e^3 - a^2*b*e^4)*x^2 + 7*(64*b^3*d^3*e - 24*a*b^2*d^2
*e^2 - 4*a^2*b*d*e^3 - a^3*e^4)*x)*b/((e^8*x^3 + 3*d*e^7*x^2 + 3*d^2*e^6*x + d^3*e^5)*sqrt(e*x + d))

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mupad [B]  time = 3.00, size = 309, normalized size = 1.20 \begin {gather*} -\frac {\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}\,\left (\frac {10\,a^4\,e^4+16\,a^3\,b\,d\,e^3+32\,a^2\,b^2\,d^2\,e^2+128\,a\,b^3\,d^3\,e-256\,b^4\,d^4}{35\,b\,e^8}-\frac {2\,b^3\,x^4}{e^4}+\frac {x\,\left (56\,a^3\,b\,e^4+112\,a^2\,b^2\,d\,e^3+448\,a\,b^3\,d^2\,e^2-896\,b^4\,d^3\,e\right )}{35\,b\,e^8}+\frac {8\,b^2\,x^3\,\left (a\,e-2\,b\,d\right )}{e^5}+\frac {4\,b\,x^2\,\left (a^2\,e^2+4\,a\,b\,d\,e-8\,b^2\,d^2\right )}{e^6}\right )}{x^4\,\sqrt {d+e\,x}+\frac {a\,d^3\,\sqrt {d+e\,x}}{b\,e^3}+\frac {x^3\,\left (35\,a\,e^8+105\,b\,d\,e^7\right )\,\sqrt {d+e\,x}}{35\,b\,e^8}+\frac {3\,d\,x^2\,\left (a\,e+b\,d\right )\,\sqrt {d+e\,x}}{b\,e^2}+\frac {d^2\,x\,\left (3\,a\,e+b\,d\right )\,\sqrt {d+e\,x}}{b\,e^3}} \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)^(9/2),x)

[Out]

-((a^2 + b^2*x^2 + 2*a*b*x)^(1/2)*((10*a^4*e^4 - 256*b^4*d^4 + 32*a^2*b^2*d^2*e^2 + 128*a*b^3*d^3*e + 16*a^3*b
*d*e^3)/(35*b*e^8) - (2*b^3*x^4)/e^4 + (x*(56*a^3*b*e^4 - 896*b^4*d^3*e + 448*a*b^3*d^2*e^2 + 112*a^2*b^2*d*e^
3))/(35*b*e^8) + (8*b^2*x^3*(a*e - 2*b*d))/e^5 + (4*b*x^2*(a^2*e^2 - 8*b^2*d^2 + 4*a*b*d*e))/e^6))/(x^4*(d + e
*x)^(1/2) + (a*d^3*(d + e*x)^(1/2))/(b*e^3) + (x^3*(35*a*e^8 + 105*b*d*e^7)*(d + e*x)^(1/2))/(35*b*e^8) + (3*d
*x^2*(a*e + b*d)*(d + e*x)^(1/2))/(b*e^2) + (d^2*x*(3*a*e + b*d)*(d + e*x)^(1/2))/(b*e^3))

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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)**(9/2),x)

[Out]

Timed out

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