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

Optimal. Leaf size=152 \[ -\frac {2 b^4 (d+e x)^{5/2} (b d-a e)}{e^6}+\frac {20 b^3 (d+e x)^{3/2} (b d-a e)^2}{3 e^6}-\frac {20 b^2 \sqrt {d+e x} (b d-a e)^3}{e^6}-\frac {10 b (b d-a e)^4}{e^6 \sqrt {d+e x}}+\frac {2 (b d-a e)^5}{3 e^6 (d+e x)^{3/2}}+\frac {2 b^5 (d+e x)^{7/2}}{7 e^6} \]

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Rubi [A]  time = 0.06, antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {27, 43} \begin {gather*} -\frac {2 b^4 (d+e x)^{5/2} (b d-a e)}{e^6}+\frac {20 b^3 (d+e x)^{3/2} (b d-a e)^2}{3 e^6}-\frac {20 b^2 \sqrt {d+e x} (b d-a e)^3}{e^6}-\frac {10 b (b d-a e)^4}{e^6 \sqrt {d+e x}}+\frac {2 (b d-a e)^5}{3 e^6 (d+e x)^{3/2}}+\frac {2 b^5 (d+e x)^{7/2}}{7 e^6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(b*d - a*e)^5)/(3*e^6*(d + e*x)^(3/2)) - (10*b*(b*d - a*e)^4)/(e^6*Sqrt[d + e*x]) - (20*b^2*(b*d - a*e)^3*S
qrt[d + e*x])/e^6 + (20*b^3*(b*d - a*e)^2*(d + e*x)^(3/2))/(3*e^6) - (2*b^4*(b*d - a*e)*(d + e*x)^(5/2))/e^6 +
(2*b^5*(d + e*x)^(7/2))/(7*e^6)

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

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])

Rubi steps

\begin {align*} \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^{5/2}} \, dx &=\int \frac {(a+b x)^5}{(d+e x)^{5/2}} \, dx\\ &=\int \left (\frac {(-b d+a e)^5}{e^5 (d+e x)^{5/2}}+\frac {5 b (b d-a e)^4}{e^5 (d+e x)^{3/2}}-\frac {10 b^2 (b d-a e)^3}{e^5 \sqrt {d+e x}}+\frac {10 b^3 (b d-a e)^2 \sqrt {d+e x}}{e^5}-\frac {5 b^4 (b d-a e) (d+e x)^{3/2}}{e^5}+\frac {b^5 (d+e x)^{5/2}}{e^5}\right ) \, dx\\ &=\frac {2 (b d-a e)^5}{3 e^6 (d+e x)^{3/2}}-\frac {10 b (b d-a e)^4}{e^6 \sqrt {d+e x}}-\frac {20 b^2 (b d-a e)^3 \sqrt {d+e x}}{e^6}+\frac {20 b^3 (b d-a e)^2 (d+e x)^{3/2}}{3 e^6}-\frac {2 b^4 (b d-a e) (d+e x)^{5/2}}{e^6}+\frac {2 b^5 (d+e x)^{7/2}}{7 e^6}\\ \end {align*}

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Mathematica [A]  time = 0.09, size = 123, normalized size = 0.81 \begin {gather*} \frac {2 \left (-21 b^4 (d+e x)^4 (b d-a e)+70 b^3 (d+e x)^3 (b d-a e)^2-210 b^2 (d+e x)^2 (b d-a e)^3-105 b (d+e x) (b d-a e)^4+7 (b d-a e)^5+3 b^5 (d+e x)^5\right )}{21 e^6 (d+e x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(7*(b*d - a*e)^5 - 105*b*(b*d - a*e)^4*(d + e*x) - 210*b^2*(b*d - a*e)^3*(d + e*x)^2 + 70*b^3*(b*d - a*e)^2
*(d + e*x)^3 - 21*b^4*(b*d - a*e)*(d + e*x)^4 + 3*b^5*(d + e*x)^5))/(21*e^6*(d + e*x)^(3/2))

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IntegrateAlgebraic [B]  time = 0.09, size = 315, normalized size = 2.07 \begin {gather*} \frac {2 \left (-7 a^5 e^5-105 a^4 b e^4 (d+e x)+35 a^4 b d e^4-70 a^3 b^2 d^2 e^3+210 a^3 b^2 e^3 (d+e x)^2+420 a^3 b^2 d e^3 (d+e x)+70 a^2 b^3 d^3 e^2-630 a^2 b^3 d^2 e^2 (d+e x)+70 a^2 b^3 e^2 (d+e x)^3-630 a^2 b^3 d e^2 (d+e x)^2-35 a b^4 d^4 e+420 a b^4 d^3 e (d+e x)+630 a b^4 d^2 e (d+e x)^2+21 a b^4 e (d+e x)^4-140 a b^4 d e (d+e x)^3+7 b^5 d^5-105 b^5 d^4 (d+e x)-210 b^5 d^3 (d+e x)^2+70 b^5 d^2 (d+e x)^3+3 b^5 (d+e x)^5-21 b^5 d (d+e x)^4\right )}{21 e^6 (d+e x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(7*b^5*d^5 - 35*a*b^4*d^4*e + 70*a^2*b^3*d^3*e^2 - 70*a^3*b^2*d^2*e^3 + 35*a^4*b*d*e^4 - 7*a^5*e^5 - 105*b^
5*d^4*(d + e*x) + 420*a*b^4*d^3*e*(d + e*x) - 630*a^2*b^3*d^2*e^2*(d + e*x) + 420*a^3*b^2*d*e^3*(d + e*x) - 10
5*a^4*b*e^4*(d + e*x) - 210*b^5*d^3*(d + e*x)^2 + 630*a*b^4*d^2*e*(d + e*x)^2 - 630*a^2*b^3*d*e^2*(d + e*x)^2
+ 210*a^3*b^2*e^3*(d + e*x)^2 + 70*b^5*d^2*(d + e*x)^3 - 140*a*b^4*d*e*(d + e*x)^3 + 70*a^2*b^3*e^2*(d + e*x)^
3 - 21*b^5*d*(d + e*x)^4 + 21*a*b^4*e*(d + e*x)^4 + 3*b^5*(d + e*x)^5))/(21*e^6*(d + e*x)^(3/2))

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fricas [B]  time = 0.43, size = 283, normalized size = 1.86 \begin {gather*} \frac {2 \, {\left (3 \, b^{5} e^{5} x^{5} - 256 \, b^{5} d^{5} + 896 \, a b^{4} d^{4} e - 1120 \, a^{2} b^{3} d^{3} e^{2} + 560 \, a^{3} b^{2} d^{2} e^{3} - 70 \, a^{4} b d e^{4} - 7 \, a^{5} e^{5} - 3 \, {\left (2 \, b^{5} d e^{4} - 7 \, a b^{4} e^{5}\right )} x^{4} + 2 \, {\left (8 \, b^{5} d^{2} e^{3} - 28 \, a b^{4} d e^{4} + 35 \, a^{2} b^{3} e^{5}\right )} x^{3} - 6 \, {\left (16 \, b^{5} d^{3} e^{2} - 56 \, a b^{4} d^{2} e^{3} + 70 \, a^{2} b^{3} d e^{4} - 35 \, a^{3} b^{2} e^{5}\right )} x^{2} - 3 \, {\left (128 \, b^{5} d^{4} e - 448 \, a b^{4} d^{3} e^{2} + 560 \, a^{2} b^{3} d^{2} e^{3} - 280 \, a^{3} b^{2} d e^{4} + 35 \, a^{4} b e^{5}\right )} x\right )} \sqrt {e x + d}}{21 \, {\left (e^{8} x^{2} + 2 \, d e^{7} x + d^{2} e^{6}\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)^2/(e*x+d)^(5/2),x, algorithm="fricas")

[Out]

2/21*(3*b^5*e^5*x^5 - 256*b^5*d^5 + 896*a*b^4*d^4*e - 1120*a^2*b^3*d^3*e^2 + 560*a^3*b^2*d^2*e^3 - 70*a^4*b*d*
e^4 - 7*a^5*e^5 - 3*(2*b^5*d*e^4 - 7*a*b^4*e^5)*x^4 + 2*(8*b^5*d^2*e^3 - 28*a*b^4*d*e^4 + 35*a^2*b^3*e^5)*x^3
- 6*(16*b^5*d^3*e^2 - 56*a*b^4*d^2*e^3 + 70*a^2*b^3*d*e^4 - 35*a^3*b^2*e^5)*x^2 - 3*(128*b^5*d^4*e - 448*a*b^4
*d^3*e^2 + 560*a^2*b^3*d^2*e^3 - 280*a^3*b^2*d*e^4 + 35*a^4*b*e^5)*x)*sqrt(e*x + d)/(e^8*x^2 + 2*d*e^7*x + d^2
*e^6)

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giac [B]  time = 0.21, size = 334, normalized size = 2.20 \begin {gather*} \frac {2}{21} \, {\left (3 \, {\left (x e + d\right )}^{\frac {7}{2}} b^{5} e^{36} - 21 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{5} d e^{36} + 70 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{5} d^{2} e^{36} - 210 \, \sqrt {x e + d} b^{5} d^{3} e^{36} + 21 \, {\left (x e + d\right )}^{\frac {5}{2}} a b^{4} e^{37} - 140 \, {\left (x e + d\right )}^{\frac {3}{2}} a b^{4} d e^{37} + 630 \, \sqrt {x e + d} a b^{4} d^{2} e^{37} + 70 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{2} b^{3} e^{38} - 630 \, \sqrt {x e + d} a^{2} b^{3} d e^{38} + 210 \, \sqrt {x e + d} a^{3} b^{2} e^{39}\right )} e^{\left (-42\right )} - \frac {2 \, {\left (15 \, {\left (x e + d\right )} b^{5} d^{4} - b^{5} d^{5} - 60 \, {\left (x e + d\right )} a b^{4} d^{3} e + 5 \, a b^{4} d^{4} e + 90 \, {\left (x e + d\right )} a^{2} b^{3} d^{2} e^{2} - 10 \, a^{2} b^{3} d^{3} e^{2} - 60 \, {\left (x e + d\right )} a^{3} b^{2} d e^{3} + 10 \, a^{3} b^{2} d^{2} e^{3} + 15 \, {\left (x e + d\right )} a^{4} b e^{4} - 5 \, a^{4} b d e^{4} + a^{5} e^{5}\right )} e^{\left (-6\right )}}{3 \, {\left (x e + d\right )}^{\frac {3}{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)^2/(e*x+d)^(5/2),x, algorithm="giac")

[Out]

2/21*(3*(x*e + d)^(7/2)*b^5*e^36 - 21*(x*e + d)^(5/2)*b^5*d*e^36 + 70*(x*e + d)^(3/2)*b^5*d^2*e^36 - 210*sqrt(
x*e + d)*b^5*d^3*e^36 + 21*(x*e + d)^(5/2)*a*b^4*e^37 - 140*(x*e + d)^(3/2)*a*b^4*d*e^37 + 630*sqrt(x*e + d)*a
*b^4*d^2*e^37 + 70*(x*e + d)^(3/2)*a^2*b^3*e^38 - 630*sqrt(x*e + d)*a^2*b^3*d*e^38 + 210*sqrt(x*e + d)*a^3*b^2
*e^39)*e^(-42) - 2/3*(15*(x*e + d)*b^5*d^4 - b^5*d^5 - 60*(x*e + d)*a*b^4*d^3*e + 5*a*b^4*d^4*e + 90*(x*e + d)
*a^2*b^3*d^2*e^2 - 10*a^2*b^3*d^3*e^2 - 60*(x*e + d)*a^3*b^2*d*e^3 + 10*a^3*b^2*d^2*e^3 + 15*(x*e + d)*a^4*b*e
^4 - 5*a^4*b*d*e^4 + a^5*e^5)*e^(-6)/(x*e + d)^(3/2)

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maple [B]  time = 0.06, size = 273, normalized size = 1.80 \begin {gather*} -\frac {2 \left (-3 b^{5} e^{5} x^{5}-21 a \,b^{4} e^{5} x^{4}+6 b^{5} d \,e^{4} x^{4}-70 a^{2} b^{3} e^{5} x^{3}+56 a \,b^{4} d \,e^{4} x^{3}-16 b^{5} d^{2} e^{3} x^{3}-210 a^{3} b^{2} e^{5} x^{2}+420 a^{2} b^{3} d \,e^{4} x^{2}-336 a \,b^{4} d^{2} e^{3} x^{2}+96 b^{5} d^{3} e^{2} x^{2}+105 a^{4} b \,e^{5} x -840 a^{3} b^{2} d \,e^{4} x +1680 a^{2} b^{3} d^{2} e^{3} x -1344 a \,b^{4} d^{3} e^{2} x +384 b^{5} d^{4} e x +7 a^{5} e^{5}+70 a^{4} b d \,e^{4}-560 a^{3} b^{2} d^{2} e^{3}+1120 a^{2} b^{3} d^{3} e^{2}-896 a \,b^{4} d^{4} e +256 b^{5} d^{5}\right )}{21 \left (e x +d \right )^{\frac {3}{2}} e^{6}} \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)^2/(e*x+d)^(5/2),x)

[Out]

-2/21*(-3*b^5*e^5*x^5-21*a*b^4*e^5*x^4+6*b^5*d*e^4*x^4-70*a^2*b^3*e^5*x^3+56*a*b^4*d*e^4*x^3-16*b^5*d^2*e^3*x^
3-210*a^3*b^2*e^5*x^2+420*a^2*b^3*d*e^4*x^2-336*a*b^4*d^2*e^3*x^2+96*b^5*d^3*e^2*x^2+105*a^4*b*e^5*x-840*a^3*b
^2*d*e^4*x+1680*a^2*b^3*d^2*e^3*x-1344*a*b^4*d^3*e^2*x+384*b^5*d^4*e*x+7*a^5*e^5+70*a^4*b*d*e^4-560*a^3*b^2*d^
2*e^3+1120*a^2*b^3*d^3*e^2-896*a*b^4*d^4*e+256*b^5*d^5)/(e*x+d)^(3/2)/e^6

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maxima [A]  time = 0.52, size = 265, normalized size = 1.74 \begin {gather*} \frac {2 \, {\left (\frac {3 \, {\left (e x + d\right )}^{\frac {7}{2}} b^{5} - 21 \, {\left (b^{5} d - a b^{4} e\right )} {\left (e x + d\right )}^{\frac {5}{2}} + 70 \, {\left (b^{5} d^{2} - 2 \, a b^{4} d e + a^{2} b^{3} e^{2}\right )} {\left (e x + d\right )}^{\frac {3}{2}} - 210 \, {\left (b^{5} d^{3} - 3 \, a b^{4} d^{2} e + 3 \, a^{2} b^{3} d e^{2} - a^{3} b^{2} e^{3}\right )} \sqrt {e x + d}}{e^{5}} + \frac {7 \, {\left (b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5} - 15 \, {\left (b^{5} d^{4} - 4 \, a b^{4} d^{3} e + 6 \, a^{2} b^{3} d^{2} e^{2} - 4 \, a^{3} b^{2} d e^{3} + a^{4} b e^{4}\right )} {\left (e x + d\right )}\right )}}{{\left (e x + d\right )}^{\frac {3}{2}} e^{5}}\right )}}{21 \, e} \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)^2/(e*x+d)^(5/2),x, algorithm="maxima")

[Out]

2/21*((3*(e*x + d)^(7/2)*b^5 - 21*(b^5*d - a*b^4*e)*(e*x + d)^(5/2) + 70*(b^5*d^2 - 2*a*b^4*d*e + a^2*b^3*e^2)
*(e*x + d)^(3/2) - 210*(b^5*d^3 - 3*a*b^4*d^2*e + 3*a^2*b^3*d*e^2 - a^3*b^2*e^3)*sqrt(e*x + d))/e^5 + 7*(b^5*d
^5 - 5*a*b^4*d^4*e + 10*a^2*b^3*d^3*e^2 - 10*a^3*b^2*d^2*e^3 + 5*a^4*b*d*e^4 - a^5*e^5 - 15*(b^5*d^4 - 4*a*b^4
*d^3*e + 6*a^2*b^3*d^2*e^2 - 4*a^3*b^2*d*e^3 + a^4*b*e^4)*(e*x + d))/((e*x + d)^(3/2)*e^5))/e

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mupad [B]  time = 2.02, size = 229, normalized size = 1.51 \begin {gather*} \frac {2\,b^5\,{\left (d+e\,x\right )}^{7/2}}{7\,e^6}-\frac {\left (10\,b^5\,d-10\,a\,b^4\,e\right )\,{\left (d+e\,x\right )}^{5/2}}{5\,e^6}-\frac {\left (d+e\,x\right )\,\left (10\,a^4\,b\,e^4-40\,a^3\,b^2\,d\,e^3+60\,a^2\,b^3\,d^2\,e^2-40\,a\,b^4\,d^3\,e+10\,b^5\,d^4\right )+\frac {2\,a^5\,e^5}{3}-\frac {2\,b^5\,d^5}{3}-\frac {20\,a^2\,b^3\,d^3\,e^2}{3}+\frac {20\,a^3\,b^2\,d^2\,e^3}{3}+\frac {10\,a\,b^4\,d^4\,e}{3}-\frac {10\,a^4\,b\,d\,e^4}{3}}{e^6\,{\left (d+e\,x\right )}^{3/2}}+\frac {20\,b^2\,{\left (a\,e-b\,d\right )}^3\,\sqrt {d+e\,x}}{e^6}+\frac {20\,b^3\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{3/2}}{3\,e^6} \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)^2)/(d + e*x)^(5/2),x)

[Out]

(2*b^5*(d + e*x)^(7/2))/(7*e^6) - ((10*b^5*d - 10*a*b^4*e)*(d + e*x)^(5/2))/(5*e^6) - ((d + e*x)*(10*b^5*d^4 +
10*a^4*b*e^4 - 40*a^3*b^2*d*e^3 + 60*a^2*b^3*d^2*e^2 - 40*a*b^4*d^3*e) + (2*a^5*e^5)/3 - (2*b^5*d^5)/3 - (20*
a^2*b^3*d^3*e^2)/3 + (20*a^3*b^2*d^2*e^3)/3 + (10*a*b^4*d^4*e)/3 - (10*a^4*b*d*e^4)/3)/(e^6*(d + e*x)^(3/2)) +
(20*b^2*(a*e - b*d)^3*(d + e*x)^(1/2))/e^6 + (20*b^3*(a*e - b*d)^2*(d + e*x)^(3/2))/(3*e^6)

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sympy [A]  time = 59.86, size = 196, normalized size = 1.29 \begin {gather*} \frac {2 b^{5} \left (d + e x\right )^{\frac {7}{2}}}{7 e^{6}} - \frac {10 b \left (a e - b d\right )^{4}}{e^{6} \sqrt {d + e x}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \left (10 a b^{4} e - 10 b^{5} d\right )}{5 e^{6}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \left (20 a^{2} b^{3} e^{2} - 40 a b^{4} d e + 20 b^{5} d^{2}\right )}{3 e^{6}} + \frac {\sqrt {d + e x} \left (20 a^{3} b^{2} e^{3} - 60 a^{2} b^{3} d e^{2} + 60 a b^{4} d^{2} e - 20 b^{5} d^{3}\right )}{e^{6}} - \frac {2 \left (a e - b d\right )^{5}}{3 e^{6} \left (d + e x\right )^{\frac {3}{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)**2/(e*x+d)**(5/2),x)

[Out]

2*b**5*(d + e*x)**(7/2)/(7*e**6) - 10*b*(a*e - b*d)**4/(e**6*sqrt(d + e*x)) + (d + e*x)**(5/2)*(10*a*b**4*e -
10*b**5*d)/(5*e**6) + (d + e*x)**(3/2)*(20*a**2*b**3*e**2 - 40*a*b**4*d*e + 20*b**5*d**2)/(3*e**6) + sqrt(d +
e*x)*(20*a**3*b**2*e**3 - 60*a**2*b**3*d*e**2 + 60*a*b**4*d**2*e - 20*b**5*d**3)/e**6 - 2*(a*e - b*d)**5/(3*e*
*6*(d + e*x)**(3/2))

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