3.13.73 \(\int (d+e x)^3 (a^2+2 a b x+b^2 x^2)^{5/2} \, dx\)

Optimal. Leaf size=172 \[ \frac {3 e^2 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^7 (b d-a e)}{8 b^4}+\frac {3 e \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^6 (b d-a e)^2}{7 b^4}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^5 (b d-a e)^3}{6 b^4}+\frac {e^3 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^8}{9 b^4} \]

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Rubi [A]  time = 0.18, antiderivative size = 172, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {646, 43} \begin {gather*} \frac {3 e^2 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^7 (b d-a e)}{8 b^4}+\frac {3 e \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^6 (b d-a e)^2}{7 b^4}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^5 (b d-a e)^3}{6 b^4}+\frac {e^3 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^8}{9 b^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^3*(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

((b*d - a*e)^3*(a + b*x)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(6*b^4) + (3*e*(b*d - a*e)^2*(a + b*x)^6*Sqrt[a^2 +
2*a*b*x + b^2*x^2])/(7*b^4) + (3*e^2*(b*d - a*e)*(a + b*x)^7*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(8*b^4) + (e^3*(a
+ b*x)^8*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*b^4)

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 646

Int[((d_.) + (e_.)*(x_))^(m_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[(a + b*x + c*x^2)^Fra
cPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*FracPart[p])), Int[(d + e*x)^m*(b/2 + c*x)^(2*p), x], x] /; FreeQ[{a, b,
c, d, e, m, p}, x] && EqQ[b^2 - 4*a*c, 0] &&  !IntegerQ[p] && NeQ[2*c*d - b*e, 0]

Rubi steps

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

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Mathematica [A]  time = 0.08, size = 253, normalized size = 1.47 \begin {gather*} \frac {x \sqrt {(a+b x)^2} \left (126 a^5 \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right )+126 a^4 b x \left (10 d^3+20 d^2 e x+15 d e^2 x^2+4 e^3 x^3\right )+84 a^3 b^2 x^2 \left (20 d^3+45 d^2 e x+36 d e^2 x^2+10 e^3 x^3\right )+36 a^2 b^3 x^3 \left (35 d^3+84 d^2 e x+70 d e^2 x^2+20 e^3 x^3\right )+9 a b^4 x^4 \left (56 d^3+140 d^2 e x+120 d e^2 x^2+35 e^3 x^3\right )+b^5 x^5 \left (84 d^3+216 d^2 e x+189 d e^2 x^2+56 e^3 x^3\right )\right )}{504 (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^3*(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

(x*Sqrt[(a + b*x)^2]*(126*a^5*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x^3) + 126*a^4*b*x*(10*d^3 + 20*d^2*e*x +
15*d*e^2*x^2 + 4*e^3*x^3) + 84*a^3*b^2*x^2*(20*d^3 + 45*d^2*e*x + 36*d*e^2*x^2 + 10*e^3*x^3) + 36*a^2*b^3*x^3
*(35*d^3 + 84*d^2*e*x + 70*d*e^2*x^2 + 20*e^3*x^3) + 9*a*b^4*x^4*(56*d^3 + 140*d^2*e*x + 120*d*e^2*x^2 + 35*e^
3*x^3) + b^5*x^5*(84*d^3 + 216*d^2*e*x + 189*d*e^2*x^2 + 56*e^3*x^3)))/(504*(a + b*x))

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IntegrateAlgebraic [F]  time = 2.32, size = 0, normalized size = 0.00 \begin {gather*} \int (d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(d + e*x)^3*(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

Defer[IntegrateAlgebraic][(d + e*x)^3*(a^2 + 2*a*b*x + b^2*x^2)^(5/2), x]

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fricas [B]  time = 0.40, size = 277, normalized size = 1.61 \begin {gather*} \frac {1}{9} \, b^{5} e^{3} x^{9} + a^{5} d^{3} x + \frac {1}{8} \, {\left (3 \, b^{5} d e^{2} + 5 \, a b^{4} e^{3}\right )} x^{8} + \frac {1}{7} \, {\left (3 \, b^{5} d^{2} e + 15 \, a b^{4} d e^{2} + 10 \, a^{2} b^{3} e^{3}\right )} x^{7} + \frac {1}{6} \, {\left (b^{5} d^{3} + 15 \, a b^{4} d^{2} e + 30 \, a^{2} b^{3} d e^{2} + 10 \, a^{3} b^{2} e^{3}\right )} x^{6} + {\left (a b^{4} d^{3} + 6 \, a^{2} b^{3} d^{2} e + 6 \, a^{3} b^{2} d e^{2} + a^{4} b e^{3}\right )} x^{5} + \frac {1}{4} \, {\left (10 \, a^{2} b^{3} d^{3} + 30 \, a^{3} b^{2} d^{2} e + 15 \, a^{4} b d e^{2} + a^{5} e^{3}\right )} x^{4} + \frac {1}{3} \, {\left (10 \, a^{3} b^{2} d^{3} + 15 \, a^{4} b d^{2} e + 3 \, a^{5} d e^{2}\right )} x^{3} + \frac {1}{2} \, {\left (5 \, a^{4} b d^{3} + 3 \, a^{5} d^{2} e\right )} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3*(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="fricas")

[Out]

1/9*b^5*e^3*x^9 + a^5*d^3*x + 1/8*(3*b^5*d*e^2 + 5*a*b^4*e^3)*x^8 + 1/7*(3*b^5*d^2*e + 15*a*b^4*d*e^2 + 10*a^2
*b^3*e^3)*x^7 + 1/6*(b^5*d^3 + 15*a*b^4*d^2*e + 30*a^2*b^3*d*e^2 + 10*a^3*b^2*e^3)*x^6 + (a*b^4*d^3 + 6*a^2*b^
3*d^2*e + 6*a^3*b^2*d*e^2 + a^4*b*e^3)*x^5 + 1/4*(10*a^2*b^3*d^3 + 30*a^3*b^2*d^2*e + 15*a^4*b*d*e^2 + a^5*e^3
)*x^4 + 1/3*(10*a^3*b^2*d^3 + 15*a^4*b*d^2*e + 3*a^5*d*e^2)*x^3 + 1/2*(5*a^4*b*d^3 + 3*a^5*d^2*e)*x^2

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giac [B]  time = 0.18, size = 441, normalized size = 2.56 \begin {gather*} \frac {1}{9} \, b^{5} x^{9} e^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {3}{8} \, b^{5} d x^{8} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {3}{7} \, b^{5} d^{2} x^{7} e \mathrm {sgn}\left (b x + a\right ) + \frac {1}{6} \, b^{5} d^{3} x^{6} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{8} \, a b^{4} x^{8} e^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {15}{7} \, a b^{4} d x^{7} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{2} \, a b^{4} d^{2} x^{6} e \mathrm {sgn}\left (b x + a\right ) + a b^{4} d^{3} x^{5} \mathrm {sgn}\left (b x + a\right ) + \frac {10}{7} \, a^{2} b^{3} x^{7} e^{3} \mathrm {sgn}\left (b x + a\right ) + 5 \, a^{2} b^{3} d x^{6} e^{2} \mathrm {sgn}\left (b x + a\right ) + 6 \, a^{2} b^{3} d^{2} x^{5} e \mathrm {sgn}\left (b x + a\right ) + \frac {5}{2} \, a^{2} b^{3} d^{3} x^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{3} \, a^{3} b^{2} x^{6} e^{3} \mathrm {sgn}\left (b x + a\right ) + 6 \, a^{3} b^{2} d x^{5} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {15}{2} \, a^{3} b^{2} d^{2} x^{4} e \mathrm {sgn}\left (b x + a\right ) + \frac {10}{3} \, a^{3} b^{2} d^{3} x^{3} \mathrm {sgn}\left (b x + a\right ) + a^{4} b x^{5} e^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {15}{4} \, a^{4} b d x^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) + 5 \, a^{4} b d^{2} x^{3} e \mathrm {sgn}\left (b x + a\right ) + \frac {5}{2} \, a^{4} b d^{3} x^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{4} \, a^{5} x^{4} e^{3} \mathrm {sgn}\left (b x + a\right ) + a^{5} d x^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {3}{2} \, a^{5} d^{2} x^{2} e \mathrm {sgn}\left (b x + a\right ) + a^{5} d^{3} x \mathrm {sgn}\left (b x + a\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3*(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="giac")

[Out]

1/9*b^5*x^9*e^3*sgn(b*x + a) + 3/8*b^5*d*x^8*e^2*sgn(b*x + a) + 3/7*b^5*d^2*x^7*e*sgn(b*x + a) + 1/6*b^5*d^3*x
^6*sgn(b*x + a) + 5/8*a*b^4*x^8*e^3*sgn(b*x + a) + 15/7*a*b^4*d*x^7*e^2*sgn(b*x + a) + 5/2*a*b^4*d^2*x^6*e*sgn
(b*x + a) + a*b^4*d^3*x^5*sgn(b*x + a) + 10/7*a^2*b^3*x^7*e^3*sgn(b*x + a) + 5*a^2*b^3*d*x^6*e^2*sgn(b*x + a)
+ 6*a^2*b^3*d^2*x^5*e*sgn(b*x + a) + 5/2*a^2*b^3*d^3*x^4*sgn(b*x + a) + 5/3*a^3*b^2*x^6*e^3*sgn(b*x + a) + 6*a
^3*b^2*d*x^5*e^2*sgn(b*x + a) + 15/2*a^3*b^2*d^2*x^4*e*sgn(b*x + a) + 10/3*a^3*b^2*d^3*x^3*sgn(b*x + a) + a^4*
b*x^5*e^3*sgn(b*x + a) + 15/4*a^4*b*d*x^4*e^2*sgn(b*x + a) + 5*a^4*b*d^2*x^3*e*sgn(b*x + a) + 5/2*a^4*b*d^3*x^
2*sgn(b*x + a) + 1/4*a^5*x^4*e^3*sgn(b*x + a) + a^5*d*x^3*e^2*sgn(b*x + a) + 3/2*a^5*d^2*x^2*e*sgn(b*x + a) +
a^5*d^3*x*sgn(b*x + a)

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maple [B]  time = 0.05, size = 322, normalized size = 1.87 \begin {gather*} \frac {\left (56 e^{3} b^{5} x^{8}+315 x^{7} e^{3} a \,b^{4}+189 x^{7} d \,e^{2} b^{5}+720 x^{6} e^{3} a^{2} b^{3}+1080 x^{6} d \,e^{2} a \,b^{4}+216 x^{6} d^{2} e \,b^{5}+840 x^{5} e^{3} a^{3} b^{2}+2520 x^{5} d \,e^{2} a^{2} b^{3}+1260 x^{5} d^{2} e a \,b^{4}+84 x^{5} d^{3} b^{5}+504 a^{4} b \,e^{3} x^{4}+3024 a^{3} b^{2} d \,e^{2} x^{4}+3024 a^{2} b^{3} d^{2} e \,x^{4}+504 a \,b^{4} d^{3} x^{4}+126 x^{3} e^{3} a^{5}+1890 x^{3} d \,e^{2} a^{4} b +3780 x^{3} d^{2} e \,a^{3} b^{2}+1260 x^{3} d^{3} a^{2} b^{3}+504 x^{2} d \,e^{2} a^{5}+2520 x^{2} d^{2} e \,a^{4} b +1680 x^{2} d^{3} a^{3} b^{2}+756 x \,d^{2} e \,a^{5}+1260 x \,d^{3} a^{4} b +504 d^{3} a^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}} x}{504 \left (b x +a \right )^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^3*(b^2*x^2+2*a*b*x+a^2)^(5/2),x)

[Out]

1/504*x*(56*b^5*e^3*x^8+315*a*b^4*e^3*x^7+189*b^5*d*e^2*x^7+720*a^2*b^3*e^3*x^6+1080*a*b^4*d*e^2*x^6+216*b^5*d
^2*e*x^6+840*a^3*b^2*e^3*x^5+2520*a^2*b^3*d*e^2*x^5+1260*a*b^4*d^2*e*x^5+84*b^5*d^3*x^5+504*a^4*b*e^3*x^4+3024
*a^3*b^2*d*e^2*x^4+3024*a^2*b^3*d^2*e*x^4+504*a*b^4*d^3*x^4+126*a^5*e^3*x^3+1890*a^4*b*d*e^2*x^3+3780*a^3*b^2*
d^2*e*x^3+1260*a^2*b^3*d^3*x^3+504*a^5*d*e^2*x^2+2520*a^4*b*d^2*e*x^2+1680*a^3*b^2*d^3*x^2+756*a^5*d^2*e*x+126
0*a^4*b*d^3*x+504*a^5*d^3)*((b*x+a)^2)^(5/2)/(b*x+a)^5

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maxima [B]  time = 1.16, size = 400, normalized size = 2.33 \begin {gather*} \frac {1}{6} \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} d^{3} x - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a d^{2} e x}{2 \, b} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{2} d e^{2} x}{2 \, b^{2}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{3} e^{3} x}{6 \, b^{3}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} e^{3} x^{2}}{9 \, b^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a d^{3}}{6 \, b} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{2} d^{2} e}{2 \, b^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{3} d e^{2}}{2 \, b^{3}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{4} e^{3}}{6 \, b^{4}} + \frac {3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} d e^{2} x}{8 \, b^{2}} - \frac {11 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} a e^{3} x}{72 \, b^{3}} + \frac {3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} d^{2} e}{7 \, b^{2}} - \frac {27 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} a d e^{2}}{56 \, b^{3}} + \frac {83 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} a^{2} e^{3}}{504 \, b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3*(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="maxima")

[Out]

1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*d^3*x - 1/2*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a*d^2*e*x/b + 1/2*(b^2*x^2 + 2
*a*b*x + a^2)^(5/2)*a^2*d*e^2*x/b^2 - 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^3*e^3*x/b^3 + 1/9*(b^2*x^2 + 2*a*b
*x + a^2)^(7/2)*e^3*x^2/b^2 + 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a*d^3/b - 1/2*(b^2*x^2 + 2*a*b*x + a^2)^(5/2
)*a^2*d^2*e/b^2 + 1/2*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^3*d*e^2/b^3 - 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^4*
e^3/b^4 + 3/8*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*d*e^2*x/b^2 - 11/72*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a*e^3*x/b^3
+ 3/7*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*d^2*e/b^2 - 27/56*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a*d*e^2/b^3 + 83/504*(
b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^2*e^3/b^4

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (d+e\,x\right )}^3\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^3*(a^2 + b^2*x^2 + 2*a*b*x)^(5/2),x)

[Out]

int((d + e*x)^3*(a^2 + b^2*x^2 + 2*a*b*x)^(5/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (d + e x\right )^{3} \left (\left (a + b x\right )^{2}\right )^{\frac {5}{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**3*(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

Integral((d + e*x)**3*((a + b*x)**2)**(5/2), x)

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