3.19.17 \(\int (a+b x) (d+e x)^{7/2} (a^2+2 a b x+b^2 x^2) \, dx\)

Optimal. Leaf size=100 \[ -\frac {6 b^2 (d+e x)^{13/2} (b d-a e)}{13 e^4}+\frac {6 b (d+e x)^{11/2} (b d-a e)^2}{11 e^4}-\frac {2 (d+e x)^{9/2} (b d-a e)^3}{9 e^4}+\frac {2 b^3 (d+e x)^{15/2}}{15 e^4} \]

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Rubi [A]  time = 0.04, antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {27, 43} \begin {gather*} -\frac {6 b^2 (d+e x)^{13/2} (b d-a e)}{13 e^4}+\frac {6 b (d+e x)^{11/2} (b d-a e)^2}{11 e^4}-\frac {2 (d+e x)^{9/2} (b d-a e)^3}{9 e^4}+\frac {2 b^3 (d+e x)^{15/2}}{15 e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)*(d + e*x)^(7/2)*(a^2 + 2*a*b*x + b^2*x^2),x]

[Out]

(-2*(b*d - a*e)^3*(d + e*x)^(9/2))/(9*e^4) + (6*b*(b*d - a*e)^2*(d + e*x)^(11/2))/(11*e^4) - (6*b^2*(b*d - a*e
)*(d + e*x)^(13/2))/(13*e^4) + (2*b^3*(d + e*x)^(15/2))/(15*e^4)

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 (a+b x) (d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right ) \, dx &=\int (a+b x)^3 (d+e x)^{7/2} \, dx\\ &=\int \left (\frac {(-b d+a e)^3 (d+e x)^{7/2}}{e^3}+\frac {3 b (b d-a e)^2 (d+e x)^{9/2}}{e^3}-\frac {3 b^2 (b d-a e) (d+e x)^{11/2}}{e^3}+\frac {b^3 (d+e x)^{13/2}}{e^3}\right ) \, dx\\ &=-\frac {2 (b d-a e)^3 (d+e x)^{9/2}}{9 e^4}+\frac {6 b (b d-a e)^2 (d+e x)^{11/2}}{11 e^4}-\frac {6 b^2 (b d-a e) (d+e x)^{13/2}}{13 e^4}+\frac {2 b^3 (d+e x)^{15/2}}{15 e^4}\\ \end {align*}

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Mathematica [A]  time = 0.08, size = 79, normalized size = 0.79 \begin {gather*} \frac {2 (d+e x)^{9/2} \left (-1485 b^2 (d+e x)^2 (b d-a e)+1755 b (d+e x) (b d-a e)^2-715 (b d-a e)^3+429 b^3 (d+e x)^3\right )}{6435 e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)*(d + e*x)^(7/2)*(a^2 + 2*a*b*x + b^2*x^2),x]

[Out]

(2*(d + e*x)^(9/2)*(-715*(b*d - a*e)^3 + 1755*b*(b*d - a*e)^2*(d + e*x) - 1485*b^2*(b*d - a*e)*(d + e*x)^2 + 4
29*b^3*(d + e*x)^3))/(6435*e^4)

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IntegrateAlgebraic [A]  time = 0.07, size = 132, normalized size = 1.32 \begin {gather*} \frac {2 (d+e x)^{9/2} \left (715 a^3 e^3+1755 a^2 b e^2 (d+e x)-2145 a^2 b d e^2+2145 a b^2 d^2 e+1485 a b^2 e (d+e x)^2-3510 a b^2 d e (d+e x)-715 b^3 d^3+1755 b^3 d^2 (d+e x)+429 b^3 (d+e x)^3-1485 b^3 d (d+e x)^2\right )}{6435 e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(a + b*x)*(d + e*x)^(7/2)*(a^2 + 2*a*b*x + b^2*x^2),x]

[Out]

(2*(d + e*x)^(9/2)*(-715*b^3*d^3 + 2145*a*b^2*d^2*e - 2145*a^2*b*d*e^2 + 715*a^3*e^3 + 1755*b^3*d^2*(d + e*x)
- 3510*a*b^2*d*e*(d + e*x) + 1755*a^2*b*e^2*(d + e*x) - 1485*b^3*d*(d + e*x)^2 + 1485*a*b^2*e*(d + e*x)^2 + 42
9*b^3*(d + e*x)^3))/(6435*e^4)

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fricas [B]  time = 0.42, size = 320, normalized size = 3.20 \begin {gather*} \frac {2 \, {\left (429 \, b^{3} e^{7} x^{7} - 16 \, b^{3} d^{7} + 120 \, a b^{2} d^{6} e - 390 \, a^{2} b d^{5} e^{2} + 715 \, a^{3} d^{4} e^{3} + 33 \, {\left (46 \, b^{3} d e^{6} + 45 \, a b^{2} e^{7}\right )} x^{6} + 9 \, {\left (206 \, b^{3} d^{2} e^{5} + 600 \, a b^{2} d e^{6} + 195 \, a^{2} b e^{7}\right )} x^{5} + 5 \, {\left (160 \, b^{3} d^{3} e^{4} + 1374 \, a b^{2} d^{2} e^{5} + 1326 \, a^{2} b d e^{6} + 143 \, a^{3} e^{7}\right )} x^{4} + 5 \, {\left (b^{3} d^{4} e^{3} + 636 \, a b^{2} d^{3} e^{4} + 1794 \, a^{2} b d^{2} e^{5} + 572 \, a^{3} d e^{6}\right )} x^{3} - 3 \, {\left (2 \, b^{3} d^{5} e^{2} - 15 \, a b^{2} d^{4} e^{3} - 1560 \, a^{2} b d^{3} e^{4} - 1430 \, a^{3} d^{2} e^{5}\right )} x^{2} + {\left (8 \, b^{3} d^{6} e - 60 \, a b^{2} d^{5} e^{2} + 195 \, a^{2} b d^{4} e^{3} + 2860 \, a^{3} d^{3} e^{4}\right )} x\right )} \sqrt {e x + d}}{6435 \, e^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)^(7/2)*(b^2*x^2+2*a*b*x+a^2),x, algorithm="fricas")

[Out]

2/6435*(429*b^3*e^7*x^7 - 16*b^3*d^7 + 120*a*b^2*d^6*e - 390*a^2*b*d^5*e^2 + 715*a^3*d^4*e^3 + 33*(46*b^3*d*e^
6 + 45*a*b^2*e^7)*x^6 + 9*(206*b^3*d^2*e^5 + 600*a*b^2*d*e^6 + 195*a^2*b*e^7)*x^5 + 5*(160*b^3*d^3*e^4 + 1374*
a*b^2*d^2*e^5 + 1326*a^2*b*d*e^6 + 143*a^3*e^7)*x^4 + 5*(b^3*d^4*e^3 + 636*a*b^2*d^3*e^4 + 1794*a^2*b*d^2*e^5
+ 572*a^3*d*e^6)*x^3 - 3*(2*b^3*d^5*e^2 - 15*a*b^2*d^4*e^3 - 1560*a^2*b*d^3*e^4 - 1430*a^3*d^2*e^5)*x^2 + (8*b
^3*d^6*e - 60*a*b^2*d^5*e^2 + 195*a^2*b*d^4*e^3 + 2860*a^3*d^3*e^4)*x)*sqrt(e*x + d)/e^4

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giac [B]  time = 0.26, size = 1270, normalized size = 12.70

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)^(7/2)*(b^2*x^2+2*a*b*x+a^2),x, algorithm="giac")

[Out]

2/45045*(45045*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*a^2*b*d^4*e^(-1) + 9009*(3*(x*e + d)^(5/2) - 10*(x*e + d)
^(3/2)*d + 15*sqrt(x*e + d)*d^2)*a*b^2*d^4*e^(-2) + 1287*(5*(x*e + d)^(7/2) - 21*(x*e + d)^(5/2)*d + 35*(x*e +
d)^(3/2)*d^2 - 35*sqrt(x*e + d)*d^3)*b^3*d^4*e^(-3) + 36036*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sq
rt(x*e + d)*d^2)*a^2*b*d^3*e^(-1) + 15444*(5*(x*e + d)^(7/2) - 21*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2 -
35*sqrt(x*e + d)*d^3)*a*b^2*d^3*e^(-2) + 572*(35*(x*e + d)^(9/2) - 180*(x*e + d)^(7/2)*d + 378*(x*e + d)^(5/2
)*d^2 - 420*(x*e + d)^(3/2)*d^3 + 315*sqrt(x*e + d)*d^4)*b^3*d^3*e^(-3) + 45045*sqrt(x*e + d)*a^3*d^4 + 60060*
((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*a^3*d^3 + 23166*(5*(x*e + d)^(7/2) - 21*(x*e + d)^(5/2)*d + 35*(x*e + d)
^(3/2)*d^2 - 35*sqrt(x*e + d)*d^3)*a^2*b*d^2*e^(-1) + 2574*(35*(x*e + d)^(9/2) - 180*(x*e + d)^(7/2)*d + 378*(
x*e + d)^(5/2)*d^2 - 420*(x*e + d)^(3/2)*d^3 + 315*sqrt(x*e + d)*d^4)*a*b^2*d^2*e^(-2) + 390*(63*(x*e + d)^(11
/2) - 385*(x*e + d)^(9/2)*d + 990*(x*e + d)^(7/2)*d^2 - 1386*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4 -
693*sqrt(x*e + d)*d^5)*b^3*d^2*e^(-3) + 18018*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2
)*a^3*d^2 + 1716*(35*(x*e + d)^(9/2) - 180*(x*e + d)^(7/2)*d + 378*(x*e + d)^(5/2)*d^2 - 420*(x*e + d)^(3/2)*d
^3 + 315*sqrt(x*e + d)*d^4)*a^2*b*d*e^(-1) + 780*(63*(x*e + d)^(11/2) - 385*(x*e + d)^(9/2)*d + 990*(x*e + d)^
(7/2)*d^2 - 1386*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4 - 693*sqrt(x*e + d)*d^5)*a*b^2*d*e^(-2) + 60*(
231*(x*e + d)^(13/2) - 1638*(x*e + d)^(11/2)*d + 5005*(x*e + d)^(9/2)*d^2 - 8580*(x*e + d)^(7/2)*d^3 + 9009*(x
*e + d)^(5/2)*d^4 - 6006*(x*e + d)^(3/2)*d^5 + 3003*sqrt(x*e + d)*d^6)*b^3*d*e^(-3) + 5148*(5*(x*e + d)^(7/2)
- 21*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2 - 35*sqrt(x*e + d)*d^3)*a^3*d + 195*(63*(x*e + d)^(11/2) - 385
*(x*e + d)^(9/2)*d + 990*(x*e + d)^(7/2)*d^2 - 1386*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4 - 693*sqrt(
x*e + d)*d^5)*a^2*b*e^(-1) + 45*(231*(x*e + d)^(13/2) - 1638*(x*e + d)^(11/2)*d + 5005*(x*e + d)^(9/2)*d^2 - 8
580*(x*e + d)^(7/2)*d^3 + 9009*(x*e + d)^(5/2)*d^4 - 6006*(x*e + d)^(3/2)*d^5 + 3003*sqrt(x*e + d)*d^6)*a*b^2*
e^(-2) + 7*(429*(x*e + d)^(15/2) - 3465*(x*e + d)^(13/2)*d + 12285*(x*e + d)^(11/2)*d^2 - 25025*(x*e + d)^(9/2
)*d^3 + 32175*(x*e + d)^(7/2)*d^4 - 27027*(x*e + d)^(5/2)*d^5 + 15015*(x*e + d)^(3/2)*d^6 - 6435*sqrt(x*e + d)
*d^7)*b^3*e^(-3) + 143*(35*(x*e + d)^(9/2) - 180*(x*e + d)^(7/2)*d + 378*(x*e + d)^(5/2)*d^2 - 420*(x*e + d)^(
3/2)*d^3 + 315*sqrt(x*e + d)*d^4)*a^3)*e^(-1)

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maple [A]  time = 0.05, size = 116, normalized size = 1.16 \begin {gather*} \frac {2 \left (e x +d \right )^{\frac {9}{2}} \left (429 b^{3} e^{3} x^{3}+1485 a \,b^{2} e^{3} x^{2}-198 b^{3} d \,e^{2} x^{2}+1755 a^{2} b \,e^{3} x -540 a \,b^{2} d \,e^{2} x +72 b^{3} d^{2} e x +715 a^{3} e^{3}-390 a^{2} b d \,e^{2}+120 a \,b^{2} d^{2} e -16 b^{3} d^{3}\right )}{6435 e^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)*(e*x+d)^(7/2)*(b^2*x^2+2*a*b*x+a^2),x)

[Out]

2/6435*(e*x+d)^(9/2)*(429*b^3*e^3*x^3+1485*a*b^2*e^3*x^2-198*b^3*d*e^2*x^2+1755*a^2*b*e^3*x-540*a*b^2*d*e^2*x+
72*b^3*d^2*e*x+715*a^3*e^3-390*a^2*b*d*e^2+120*a*b^2*d^2*e-16*b^3*d^3)/e^4

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maxima [A]  time = 0.60, size = 118, normalized size = 1.18 \begin {gather*} \frac {2 \, {\left (429 \, {\left (e x + d\right )}^{\frac {15}{2}} b^{3} - 1485 \, {\left (b^{3} d - a b^{2} e\right )} {\left (e x + d\right )}^{\frac {13}{2}} + 1755 \, {\left (b^{3} d^{2} - 2 \, a b^{2} d e + a^{2} b e^{2}\right )} {\left (e x + d\right )}^{\frac {11}{2}} - 715 \, {\left (b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}\right )} {\left (e x + d\right )}^{\frac {9}{2}}\right )}}{6435 \, e^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)^(7/2)*(b^2*x^2+2*a*b*x+a^2),x, algorithm="maxima")

[Out]

2/6435*(429*(e*x + d)^(15/2)*b^3 - 1485*(b^3*d - a*b^2*e)*(e*x + d)^(13/2) + 1755*(b^3*d^2 - 2*a*b^2*d*e + a^2
*b*e^2)*(e*x + d)^(11/2) - 715*(b^3*d^3 - 3*a*b^2*d^2*e + 3*a^2*b*d*e^2 - a^3*e^3)*(e*x + d)^(9/2))/e^4

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mupad [B]  time = 0.08, size = 87, normalized size = 0.87 \begin {gather*} \frac {2\,b^3\,{\left (d+e\,x\right )}^{15/2}}{15\,e^4}-\frac {\left (6\,b^3\,d-6\,a\,b^2\,e\right )\,{\left (d+e\,x\right )}^{13/2}}{13\,e^4}+\frac {2\,{\left (a\,e-b\,d\right )}^3\,{\left (d+e\,x\right )}^{9/2}}{9\,e^4}+\frac {6\,b\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{11/2}}{11\,e^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)*(d + e*x)^(7/2)*(a^2 + b^2*x^2 + 2*a*b*x),x)

[Out]

(2*b^3*(d + e*x)^(15/2))/(15*e^4) - ((6*b^3*d - 6*a*b^2*e)*(d + e*x)^(13/2))/(13*e^4) + (2*(a*e - b*d)^3*(d +
e*x)^(9/2))/(9*e^4) + (6*b*(a*e - b*d)^2*(d + e*x)^(11/2))/(11*e^4)

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sympy [A]  time = 9.58, size = 654, normalized size = 6.54 \begin {gather*} \begin {cases} \frac {2 a^{3} d^{4} \sqrt {d + e x}}{9 e} + \frac {8 a^{3} d^{3} x \sqrt {d + e x}}{9} + \frac {4 a^{3} d^{2} e x^{2} \sqrt {d + e x}}{3} + \frac {8 a^{3} d e^{2} x^{3} \sqrt {d + e x}}{9} + \frac {2 a^{3} e^{3} x^{4} \sqrt {d + e x}}{9} - \frac {4 a^{2} b d^{5} \sqrt {d + e x}}{33 e^{2}} + \frac {2 a^{2} b d^{4} x \sqrt {d + e x}}{33 e} + \frac {16 a^{2} b d^{3} x^{2} \sqrt {d + e x}}{11} + \frac {92 a^{2} b d^{2} e x^{3} \sqrt {d + e x}}{33} + \frac {68 a^{2} b d e^{2} x^{4} \sqrt {d + e x}}{33} + \frac {6 a^{2} b e^{3} x^{5} \sqrt {d + e x}}{11} + \frac {16 a b^{2} d^{6} \sqrt {d + e x}}{429 e^{3}} - \frac {8 a b^{2} d^{5} x \sqrt {d + e x}}{429 e^{2}} + \frac {2 a b^{2} d^{4} x^{2} \sqrt {d + e x}}{143 e} + \frac {424 a b^{2} d^{3} x^{3} \sqrt {d + e x}}{429} + \frac {916 a b^{2} d^{2} e x^{4} \sqrt {d + e x}}{429} + \frac {240 a b^{2} d e^{2} x^{5} \sqrt {d + e x}}{143} + \frac {6 a b^{2} e^{3} x^{6} \sqrt {d + e x}}{13} - \frac {32 b^{3} d^{7} \sqrt {d + e x}}{6435 e^{4}} + \frac {16 b^{3} d^{6} x \sqrt {d + e x}}{6435 e^{3}} - \frac {4 b^{3} d^{5} x^{2} \sqrt {d + e x}}{2145 e^{2}} + \frac {2 b^{3} d^{4} x^{3} \sqrt {d + e x}}{1287 e} + \frac {320 b^{3} d^{3} x^{4} \sqrt {d + e x}}{1287} + \frac {412 b^{3} d^{2} e x^{5} \sqrt {d + e x}}{715} + \frac {92 b^{3} d e^{2} x^{6} \sqrt {d + e x}}{195} + \frac {2 b^{3} e^{3} x^{7} \sqrt {d + e x}}{15} & \text {for}\: e \neq 0 \\d^{\frac {7}{2}} \left (a^{3} x + \frac {3 a^{2} b x^{2}}{2} + a b^{2} x^{3} + \frac {b^{3} x^{4}}{4}\right ) & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)**(7/2)*(b**2*x**2+2*a*b*x+a**2),x)

[Out]

Piecewise((2*a**3*d**4*sqrt(d + e*x)/(9*e) + 8*a**3*d**3*x*sqrt(d + e*x)/9 + 4*a**3*d**2*e*x**2*sqrt(d + e*x)/
3 + 8*a**3*d*e**2*x**3*sqrt(d + e*x)/9 + 2*a**3*e**3*x**4*sqrt(d + e*x)/9 - 4*a**2*b*d**5*sqrt(d + e*x)/(33*e*
*2) + 2*a**2*b*d**4*x*sqrt(d + e*x)/(33*e) + 16*a**2*b*d**3*x**2*sqrt(d + e*x)/11 + 92*a**2*b*d**2*e*x**3*sqrt
(d + e*x)/33 + 68*a**2*b*d*e**2*x**4*sqrt(d + e*x)/33 + 6*a**2*b*e**3*x**5*sqrt(d + e*x)/11 + 16*a*b**2*d**6*s
qrt(d + e*x)/(429*e**3) - 8*a*b**2*d**5*x*sqrt(d + e*x)/(429*e**2) + 2*a*b**2*d**4*x**2*sqrt(d + e*x)/(143*e)
+ 424*a*b**2*d**3*x**3*sqrt(d + e*x)/429 + 916*a*b**2*d**2*e*x**4*sqrt(d + e*x)/429 + 240*a*b**2*d*e**2*x**5*s
qrt(d + e*x)/143 + 6*a*b**2*e**3*x**6*sqrt(d + e*x)/13 - 32*b**3*d**7*sqrt(d + e*x)/(6435*e**4) + 16*b**3*d**6
*x*sqrt(d + e*x)/(6435*e**3) - 4*b**3*d**5*x**2*sqrt(d + e*x)/(2145*e**2) + 2*b**3*d**4*x**3*sqrt(d + e*x)/(12
87*e) + 320*b**3*d**3*x**4*sqrt(d + e*x)/1287 + 412*b**3*d**2*e*x**5*sqrt(d + e*x)/715 + 92*b**3*d*e**2*x**6*s
qrt(d + e*x)/195 + 2*b**3*e**3*x**7*sqrt(d + e*x)/15, Ne(e, 0)), (d**(7/2)*(a**3*x + 3*a**2*b*x**2/2 + a*b**2*
x**3 + b**3*x**4/4), True))

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