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

Optimal. Leaf size=152 \[ -\frac {10 b^4 (d+e x)^{7/2} (b d-a e)}{7 e^6}+\frac {4 b^3 (d+e x)^{5/2} (b d-a e)^2}{e^6}-\frac {20 b^2 (d+e x)^{3/2} (b d-a e)^3}{3 e^6}+\frac {10 b \sqrt {d+e x} (b d-a e)^4}{e^6}+\frac {2 (b d-a e)^5}{e^6 \sqrt {d+e x}}+\frac {2 b^5 (d+e x)^{9/2}}{9 e^6} \]

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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 {10 b^4 (d+e x)^{7/2} (b d-a e)}{7 e^6}+\frac {4 b^3 (d+e x)^{5/2} (b d-a e)^2}{e^6}-\frac {20 b^2 (d+e x)^{3/2} (b d-a e)^3}{3 e^6}+\frac {10 b \sqrt {d+e x} (b d-a e)^4}{e^6}+\frac {2 (b d-a e)^5}{e^6 \sqrt {d+e x}}+\frac {2 b^5 (d+e x)^{9/2}}{9 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)^(3/2),x]

[Out]

(2*(b*d - a*e)^5)/(e^6*Sqrt[d + e*x]) + (10*b*(b*d - a*e)^4*Sqrt[d + e*x])/e^6 - (20*b^2*(b*d - a*e)^3*(d + e*
x)^(3/2))/(3*e^6) + (4*b^3*(b*d - a*e)^2*(d + e*x)^(5/2))/e^6 - (10*b^4*(b*d - a*e)*(d + e*x)^(7/2))/(7*e^6) +
(2*b^5*(d + e*x)^(9/2))/(9*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)^{3/2}} \, dx &=\int \frac {(a+b x)^5}{(d+e x)^{3/2}} \, dx\\ &=\int \left (\frac {(-b d+a e)^5}{e^5 (d+e x)^{3/2}}+\frac {5 b (b d-a e)^4}{e^5 \sqrt {d+e x}}-\frac {10 b^2 (b d-a e)^3 \sqrt {d+e x}}{e^5}+\frac {10 b^3 (b d-a e)^2 (d+e x)^{3/2}}{e^5}-\frac {5 b^4 (b d-a e) (d+e x)^{5/2}}{e^5}+\frac {b^5 (d+e x)^{7/2}}{e^5}\right ) \, dx\\ &=\frac {2 (b d-a e)^5}{e^6 \sqrt {d+e x}}+\frac {10 b (b d-a e)^4 \sqrt {d+e x}}{e^6}-\frac {20 b^2 (b d-a e)^3 (d+e x)^{3/2}}{3 e^6}+\frac {4 b^3 (b d-a e)^2 (d+e x)^{5/2}}{e^6}-\frac {10 b^4 (b d-a e) (d+e x)^{7/2}}{7 e^6}+\frac {2 b^5 (d+e x)^{9/2}}{9 e^6}\\ \end {align*}

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Mathematica [A]  time = 0.09, size = 123, normalized size = 0.81 \begin {gather*} \frac {2 \left (-45 b^4 (d+e x)^4 (b d-a e)+126 b^3 (d+e x)^3 (b d-a e)^2-210 b^2 (d+e x)^2 (b d-a e)^3+315 b (d+e x) (b d-a e)^4+63 (b d-a e)^5+7 b^5 (d+e x)^5\right )}{63 e^6 \sqrt {d+e x}} \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)^(3/2),x]

[Out]

(2*(63*(b*d - a*e)^5 + 315*b*(b*d - a*e)^4*(d + e*x) - 210*b^2*(b*d - a*e)^3*(d + e*x)^2 + 126*b^3*(b*d - a*e)
^2*(d + e*x)^3 - 45*b^4*(b*d - a*e)*(d + e*x)^4 + 7*b^5*(d + e*x)^5))/(63*e^6*Sqrt[d + e*x])

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IntegrateAlgebraic [B]  time = 0.08, size = 315, normalized size = 2.07 \begin {gather*} \frac {2 \left (-63 a^5 e^5+315 a^4 b e^4 (d+e x)+315 a^4 b d e^4-630 a^3 b^2 d^2 e^3+210 a^3 b^2 e^3 (d+e x)^2-1260 a^3 b^2 d e^3 (d+e x)+630 a^2 b^3 d^3 e^2+1890 a^2 b^3 d^2 e^2 (d+e x)+126 a^2 b^3 e^2 (d+e x)^3-630 a^2 b^3 d e^2 (d+e x)^2-315 a b^4 d^4 e-1260 a b^4 d^3 e (d+e x)+630 a b^4 d^2 e (d+e x)^2+45 a b^4 e (d+e x)^4-252 a b^4 d e (d+e x)^3+63 b^5 d^5+315 b^5 d^4 (d+e x)-210 b^5 d^3 (d+e x)^2+126 b^5 d^2 (d+e x)^3+7 b^5 (d+e x)^5-45 b^5 d (d+e x)^4\right )}{63 e^6 \sqrt {d+e x}} \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)^(3/2),x]

[Out]

(2*(63*b^5*d^5 - 315*a*b^4*d^4*e + 630*a^2*b^3*d^3*e^2 - 630*a^3*b^2*d^2*e^3 + 315*a^4*b*d*e^4 - 63*a^5*e^5 +
315*b^5*d^4*(d + e*x) - 1260*a*b^4*d^3*e*(d + e*x) + 1890*a^2*b^3*d^2*e^2*(d + e*x) - 1260*a^3*b^2*d*e^3*(d +
e*x) + 315*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 + 126*b^5*d^2*(d + e*x)^3 - 252*a*b^4*d*e*(d + e*x)^3 + 126*a^2*b^3*e^2
*(d + e*x)^3 - 45*b^5*d*(d + e*x)^4 + 45*a*b^4*e*(d + e*x)^4 + 7*b^5*(d + e*x)^5))/(63*e^6*Sqrt[d + e*x])

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fricas [B]  time = 0.43, size = 271, normalized size = 1.78 \begin {gather*} \frac {2 \, {\left (7 \, b^{5} e^{5} x^{5} + 256 \, b^{5} d^{5} - 1152 \, a b^{4} d^{4} e + 2016 \, a^{2} b^{3} d^{3} e^{2} - 1680 \, a^{3} b^{2} d^{2} e^{3} + 630 \, a^{4} b d e^{4} - 63 \, a^{5} e^{5} - 5 \, {\left (2 \, b^{5} d e^{4} - 9 \, a b^{4} e^{5}\right )} x^{4} + 2 \, {\left (8 \, b^{5} d^{2} e^{3} - 36 \, a b^{4} d e^{4} + 63 \, a^{2} b^{3} e^{5}\right )} x^{3} - 2 \, {\left (16 \, b^{5} d^{3} e^{2} - 72 \, a b^{4} d^{2} e^{3} + 126 \, a^{2} b^{3} d e^{4} - 105 \, a^{3} b^{2} e^{5}\right )} x^{2} + {\left (128 \, b^{5} d^{4} e - 576 \, a b^{4} d^{3} e^{2} + 1008 \, a^{2} b^{3} d^{2} e^{3} - 840 \, a^{3} b^{2} d e^{4} + 315 \, a^{4} b e^{5}\right )} x\right )} \sqrt {e x + d}}{63 \, {\left (e^{7} x + d 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)^(3/2),x, algorithm="fricas")

[Out]

2/63*(7*b^5*e^5*x^5 + 256*b^5*d^5 - 1152*a*b^4*d^4*e + 2016*a^2*b^3*d^3*e^2 - 1680*a^3*b^2*d^2*e^3 + 630*a^4*b
*d*e^4 - 63*a^5*e^5 - 5*(2*b^5*d*e^4 - 9*a*b^4*e^5)*x^4 + 2*(8*b^5*d^2*e^3 - 36*a*b^4*d*e^4 + 63*a^2*b^3*e^5)*
x^3 - 2*(16*b^5*d^3*e^2 - 72*a*b^4*d^2*e^3 + 126*a^2*b^3*d*e^4 - 105*a^3*b^2*e^5)*x^2 + (128*b^5*d^4*e - 576*a
*b^4*d^3*e^2 + 1008*a^2*b^3*d^2*e^3 - 840*a^3*b^2*d*e^4 + 315*a^4*b*e^5)*x)*sqrt(e*x + d)/(e^7*x + d*e^6)

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giac [B]  time = 0.20, size = 346, normalized size = 2.28 \begin {gather*} \frac {2}{63} \, {\left (7 \, {\left (x e + d\right )}^{\frac {9}{2}} b^{5} e^{48} - 45 \, {\left (x e + d\right )}^{\frac {7}{2}} b^{5} d e^{48} + 126 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{5} d^{2} e^{48} - 210 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{5} d^{3} e^{48} + 315 \, \sqrt {x e + d} b^{5} d^{4} e^{48} + 45 \, {\left (x e + d\right )}^{\frac {7}{2}} a b^{4} e^{49} - 252 \, {\left (x e + d\right )}^{\frac {5}{2}} a b^{4} d e^{49} + 630 \, {\left (x e + d\right )}^{\frac {3}{2}} a b^{4} d^{2} e^{49} - 1260 \, \sqrt {x e + d} a b^{4} d^{3} e^{49} + 126 \, {\left (x e + d\right )}^{\frac {5}{2}} a^{2} b^{3} e^{50} - 630 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{2} b^{3} d e^{50} + 1890 \, \sqrt {x e + d} a^{2} b^{3} d^{2} e^{50} + 210 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{3} b^{2} e^{51} - 1260 \, \sqrt {x e + d} a^{3} b^{2} d e^{51} + 315 \, \sqrt {x e + d} a^{4} b e^{52}\right )} e^{\left (-54\right )} + \frac {2 \, {\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}\right )} e^{\left (-6\right )}}{\sqrt {x e + 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)^2/(e*x+d)^(3/2),x, algorithm="giac")

[Out]

2/63*(7*(x*e + d)^(9/2)*b^5*e^48 - 45*(x*e + d)^(7/2)*b^5*d*e^48 + 126*(x*e + d)^(5/2)*b^5*d^2*e^48 - 210*(x*e
+ d)^(3/2)*b^5*d^3*e^48 + 315*sqrt(x*e + d)*b^5*d^4*e^48 + 45*(x*e + d)^(7/2)*a*b^4*e^49 - 252*(x*e + d)^(5/2
)*a*b^4*d*e^49 + 630*(x*e + d)^(3/2)*a*b^4*d^2*e^49 - 1260*sqrt(x*e + d)*a*b^4*d^3*e^49 + 126*(x*e + d)^(5/2)*
a^2*b^3*e^50 - 630*(x*e + d)^(3/2)*a^2*b^3*d*e^50 + 1890*sqrt(x*e + d)*a^2*b^3*d^2*e^50 + 210*(x*e + d)^(3/2)*
a^3*b^2*e^51 - 1260*sqrt(x*e + d)*a^3*b^2*d*e^51 + 315*sqrt(x*e + d)*a^4*b*e^52)*e^(-54) + 2*(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)*e^(-6)/sqrt(x*e + d)

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maple [B]  time = 0.05, size = 273, normalized size = 1.80 \begin {gather*} -\frac {2 \left (-7 b^{5} e^{5} x^{5}-45 a \,b^{4} e^{5} x^{4}+10 b^{5} d \,e^{4} x^{4}-126 a^{2} b^{3} e^{5} x^{3}+72 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}+252 a^{2} b^{3} d \,e^{4} x^{2}-144 a \,b^{4} d^{2} e^{3} x^{2}+32 b^{5} d^{3} e^{2} x^{2}-315 a^{4} b \,e^{5} x +840 a^{3} b^{2} d \,e^{4} x -1008 a^{2} b^{3} d^{2} e^{3} x +576 a \,b^{4} d^{3} e^{2} x -128 b^{5} d^{4} e x +63 a^{5} e^{5}-630 a^{4} b d \,e^{4}+1680 a^{3} b^{2} d^{2} e^{3}-2016 a^{2} b^{3} d^{3} e^{2}+1152 a \,b^{4} d^{4} e -256 b^{5} d^{5}\right )}{63 \sqrt {e x +d}\, 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)^(3/2),x)

[Out]

-2/63*(-7*b^5*e^5*x^5-45*a*b^4*e^5*x^4+10*b^5*d*e^4*x^4-126*a^2*b^3*e^5*x^3+72*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+252*a^2*b^3*d*e^4*x^2-144*a*b^4*d^2*e^3*x^2+32*b^5*d^3*e^2*x^2-315*a^4*b*e^5*x+840*a^3
*b^2*d*e^4*x-1008*a^2*b^3*d^2*e^3*x+576*a*b^4*d^3*e^2*x-128*b^5*d^4*e*x+63*a^5*e^5-630*a^4*b*d*e^4+1680*a^3*b^
2*d^2*e^3-2016*a^2*b^3*d^3*e^2+1152*a*b^4*d^4*e-256*b^5*d^5)/(e*x+d)^(1/2)/e^6

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maxima [A]  time = 0.52, size = 267, normalized size = 1.76 \begin {gather*} \frac {2 \, {\left (\frac {7 \, {\left (e x + d\right )}^{\frac {9}{2}} b^{5} - 45 \, {\left (b^{5} d - a b^{4} e\right )} {\left (e x + d\right )}^{\frac {7}{2}} + 126 \, {\left (b^{5} d^{2} - 2 \, a b^{4} d e + a^{2} b^{3} e^{2}\right )} {\left (e x + d\right )}^{\frac {5}{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 )} {\left (e x + d\right )}^{\frac {3}{2}} + 315 \, {\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 )} \sqrt {e x + d}}{e^{5}} + \frac {63 \, {\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}\right )}}{\sqrt {e x + d} e^{5}}\right )}}{63 \, 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)^(3/2),x, algorithm="maxima")

[Out]

2/63*((7*(e*x + d)^(9/2)*b^5 - 45*(b^5*d - a*b^4*e)*(e*x + d)^(7/2) + 126*(b^5*d^2 - 2*a*b^4*d*e + a^2*b^3*e^2
)*(e*x + d)^(5/2) - 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)*(e*x + d)^(3/2) + 315*(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)*sqrt(e*x + d))/e^5 + 63*(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)/(sqrt(e*x + d)*e^5))/e

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mupad [B]  time = 2.04, size = 192, normalized size = 1.26 \begin {gather*} \frac {2\,b^5\,{\left (d+e\,x\right )}^{9/2}}{9\,e^6}-\frac {\left (10\,b^5\,d-10\,a\,b^4\,e\right )\,{\left (d+e\,x\right )}^{7/2}}{7\,e^6}-\frac {2\,a^5\,e^5-10\,a^4\,b\,d\,e^4+20\,a^3\,b^2\,d^2\,e^3-20\,a^2\,b^3\,d^3\,e^2+10\,a\,b^4\,d^4\,e-2\,b^5\,d^5}{e^6\,\sqrt {d+e\,x}}+\frac {20\,b^2\,{\left (a\,e-b\,d\right )}^3\,{\left (d+e\,x\right )}^{3/2}}{3\,e^6}+\frac {4\,b^3\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{5/2}}{e^6}+\frac {10\,b\,{\left (a\,e-b\,d\right )}^4\,\sqrt {d+e\,x}}{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)^(3/2),x)

[Out]

(2*b^5*(d + e*x)^(9/2))/(9*e^6) - ((10*b^5*d - 10*a*b^4*e)*(d + e*x)^(7/2))/(7*e^6) - (2*a^5*e^5 - 2*b^5*d^5 -
20*a^2*b^3*d^3*e^2 + 20*a^3*b^2*d^2*e^3 + 10*a*b^4*d^4*e - 10*a^4*b*d*e^4)/(e^6*(d + e*x)^(1/2)) + (20*b^2*(a
*e - b*d)^3*(d + e*x)^(3/2))/(3*e^6) + (4*b^3*(a*e - b*d)^2*(d + e*x)^(5/2))/e^6 + (10*b*(a*e - b*d)^4*(d + e*
x)^(1/2))/e^6

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sympy [A]  time = 49.19, size = 243, normalized size = 1.60 \begin {gather*} \frac {2 b^{5} \left (d + e x\right )^{\frac {9}{2}}}{9 e^{6}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \left (10 a b^{4} e - 10 b^{5} d\right )}{7 e^{6}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \left (20 a^{2} b^{3} e^{2} - 40 a b^{4} d e + 20 b^{5} d^{2}\right )}{5 e^{6}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \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 )}{3 e^{6}} + \frac {\sqrt {d + e x} \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 )}{e^{6}} - \frac {2 \left (a e - b d\right )^{5}}{e^{6} \sqrt {d + e x}} \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)**(3/2),x)

[Out]

2*b**5*(d + e*x)**(9/2)/(9*e**6) + (d + e*x)**(7/2)*(10*a*b**4*e - 10*b**5*d)/(7*e**6) + (d + e*x)**(5/2)*(20*
a**2*b**3*e**2 - 40*a*b**4*d*e + 20*b**5*d**2)/(5*e**6) + (d + e*x)**(3/2)*(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)/(3*e**6) + sqrt(d + e*x)*(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)/e**6 - 2*(a*e - b*d)**5/(e**6*sqrt(d + e*x))

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