3.13.94 \(\int (a+b x)^4 (c+d x)^{5/2} \, dx\)
Optimal. Leaf size=129 \[ -\frac {8 b^3 (c+d x)^{13/2} (b c-a d)}{13 d^5}+\frac {12 b^2 (c+d x)^{11/2} (b c-a d)^2}{11 d^5}-\frac {8 b (c+d x)^{9/2} (b c-a d)^3}{9 d^5}+\frac {2 (c+d x)^{7/2} (b c-a d)^4}{7 d^5}+\frac {2 b^4 (c+d x)^{15/2}}{15 d^5} \]
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Rubi [A] time = 0.04, antiderivative size = 129, normalized size of antiderivative = 1.00,
number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used =
{43} \begin {gather*} -\frac {8 b^3 (c+d x)^{13/2} (b c-a d)}{13 d^5}+\frac {12 b^2 (c+d x)^{11/2} (b c-a d)^2}{11 d^5}-\frac {8 b (c+d x)^{9/2} (b c-a d)^3}{9 d^5}+\frac {2 (c+d x)^{7/2} (b c-a d)^4}{7 d^5}+\frac {2 b^4 (c+d x)^{15/2}}{15 d^5} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(a + b*x)^4*(c + d*x)^(5/2),x]
[Out]
(2*(b*c - a*d)^4*(c + d*x)^(7/2))/(7*d^5) - (8*b*(b*c - a*d)^3*(c + d*x)^(9/2))/(9*d^5) + (12*b^2*(b*c - a*d)^
2*(c + d*x)^(11/2))/(11*d^5) - (8*b^3*(b*c - a*d)*(c + d*x)^(13/2))/(13*d^5) + (2*b^4*(c + d*x)^(15/2))/(15*d^
5)
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)^4 (c+d x)^{5/2} \, dx &=\int \left (\frac {(-b c+a d)^4 (c+d x)^{5/2}}{d^4}-\frac {4 b (b c-a d)^3 (c+d x)^{7/2}}{d^4}+\frac {6 b^2 (b c-a d)^2 (c+d x)^{9/2}}{d^4}-\frac {4 b^3 (b c-a d) (c+d x)^{11/2}}{d^4}+\frac {b^4 (c+d x)^{13/2}}{d^4}\right ) \, dx\\ &=\frac {2 (b c-a d)^4 (c+d x)^{7/2}}{7 d^5}-\frac {8 b (b c-a d)^3 (c+d x)^{9/2}}{9 d^5}+\frac {12 b^2 (b c-a d)^2 (c+d x)^{11/2}}{11 d^5}-\frac {8 b^3 (b c-a d) (c+d x)^{13/2}}{13 d^5}+\frac {2 b^4 (c+d x)^{15/2}}{15 d^5}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 101, normalized size = 0.78 \begin {gather*} \frac {2 (c+d x)^{7/2} \left (-13860 b^3 (c+d x)^3 (b c-a d)+24570 b^2 (c+d x)^2 (b c-a d)^2-20020 b (c+d x) (b c-a d)^3+6435 (b c-a d)^4+3003 b^4 (c+d x)^4\right )}{45045 d^5} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(a + b*x)^4*(c + d*x)^(5/2),x]
[Out]
(2*(c + d*x)^(7/2)*(6435*(b*c - a*d)^4 - 20020*b*(b*c - a*d)^3*(c + d*x) + 24570*b^2*(b*c - a*d)^2*(c + d*x)^2
- 13860*b^3*(b*c - a*d)*(c + d*x)^3 + 3003*b^4*(c + d*x)^4))/(45045*d^5)
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IntegrateAlgebraic [A] time = 0.08, size = 213, normalized size = 1.65 \begin {gather*} \frac {2 (c+d x)^{7/2} \left (6435 a^4 d^4+20020 a^3 b d^3 (c+d x)-25740 a^3 b c d^3+38610 a^2 b^2 c^2 d^2+24570 a^2 b^2 d^2 (c+d x)^2-60060 a^2 b^2 c d^2 (c+d x)-25740 a b^3 c^3 d+60060 a b^3 c^2 d (c+d x)+13860 a b^3 d (c+d x)^3-49140 a b^3 c d (c+d x)^2+6435 b^4 c^4-20020 b^4 c^3 (c+d x)+24570 b^4 c^2 (c+d x)^2+3003 b^4 (c+d x)^4-13860 b^4 c (c+d x)^3\right )}{45045 d^5} \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[(a + b*x)^4*(c + d*x)^(5/2),x]
[Out]
(2*(c + d*x)^(7/2)*(6435*b^4*c^4 - 25740*a*b^3*c^3*d + 38610*a^2*b^2*c^2*d^2 - 25740*a^3*b*c*d^3 + 6435*a^4*d^
4 - 20020*b^4*c^3*(c + d*x) + 60060*a*b^3*c^2*d*(c + d*x) - 60060*a^2*b^2*c*d^2*(c + d*x) + 20020*a^3*b*d^3*(c
+ d*x) + 24570*b^4*c^2*(c + d*x)^2 - 49140*a*b^3*c*d*(c + d*x)^2 + 24570*a^2*b^2*d^2*(c + d*x)^2 - 13860*b^4*
c*(c + d*x)^3 + 13860*a*b^3*d*(c + d*x)^3 + 3003*b^4*(c + d*x)^4))/(45045*d^5)
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fricas [B] time = 1.38, size = 377, normalized size = 2.92 \begin {gather*} \frac {2 \, {\left (3003 \, b^{4} d^{7} x^{7} + 128 \, b^{4} c^{7} - 960 \, a b^{3} c^{6} d + 3120 \, a^{2} b^{2} c^{5} d^{2} - 5720 \, a^{3} b c^{4} d^{3} + 6435 \, a^{4} c^{3} d^{4} + 231 \, {\left (31 \, b^{4} c d^{6} + 60 \, a b^{3} d^{7}\right )} x^{6} + 63 \, {\left (71 \, b^{4} c^{2} d^{5} + 540 \, a b^{3} c d^{6} + 390 \, a^{2} b^{2} d^{7}\right )} x^{5} + 35 \, {\left (b^{4} c^{3} d^{4} + 636 \, a b^{3} c^{2} d^{5} + 1794 \, a^{2} b^{2} c d^{6} + 572 \, a^{3} b d^{7}\right )} x^{4} - 5 \, {\left (8 \, b^{4} c^{4} d^{3} - 60 \, a b^{3} c^{3} d^{4} - 8814 \, a^{2} b^{2} c^{2} d^{5} - 10868 \, a^{3} b c d^{6} - 1287 \, a^{4} d^{7}\right )} x^{3} + 3 \, {\left (16 \, b^{4} c^{5} d^{2} - 120 \, a b^{3} c^{4} d^{3} + 390 \, a^{2} b^{2} c^{3} d^{4} + 14300 \, a^{3} b c^{2} d^{5} + 6435 \, a^{4} c d^{6}\right )} x^{2} - {\left (64 \, b^{4} c^{6} d - 480 \, a b^{3} c^{5} d^{2} + 1560 \, a^{2} b^{2} c^{4} d^{3} - 2860 \, a^{3} b c^{3} d^{4} - 19305 \, a^{4} c^{2} d^{5}\right )} x\right )} \sqrt {d x + c}}{45045 \, d^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)^4*(d*x+c)^(5/2),x, algorithm="fricas")
[Out]
2/45045*(3003*b^4*d^7*x^7 + 128*b^4*c^7 - 960*a*b^3*c^6*d + 3120*a^2*b^2*c^5*d^2 - 5720*a^3*b*c^4*d^3 + 6435*a
^4*c^3*d^4 + 231*(31*b^4*c*d^6 + 60*a*b^3*d^7)*x^6 + 63*(71*b^4*c^2*d^5 + 540*a*b^3*c*d^6 + 390*a^2*b^2*d^7)*x
^5 + 35*(b^4*c^3*d^4 + 636*a*b^3*c^2*d^5 + 1794*a^2*b^2*c*d^6 + 572*a^3*b*d^7)*x^4 - 5*(8*b^4*c^4*d^3 - 60*a*b
^3*c^3*d^4 - 8814*a^2*b^2*c^2*d^5 - 10868*a^3*b*c*d^6 - 1287*a^4*d^7)*x^3 + 3*(16*b^4*c^5*d^2 - 120*a*b^3*c^4*
d^3 + 390*a^2*b^2*c^3*d^4 + 14300*a^3*b*c^2*d^5 + 6435*a^4*c*d^6)*x^2 - (64*b^4*c^6*d - 480*a*b^3*c^5*d^2 + 15
60*a^2*b^2*c^4*d^3 - 2860*a^3*b*c^3*d^4 - 19305*a^4*c^2*d^5)*x)*sqrt(d*x + c)/d^5
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giac [B] time = 1.45, size = 1204, normalized size = 9.33
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)^4*(d*x+c)^(5/2),x, algorithm="giac")
[Out]
2/45045*(45045*sqrt(d*x + c)*a^4*c^3 + 45045*((d*x + c)^(3/2) - 3*sqrt(d*x + c)*c)*a^4*c^2 + 60060*((d*x + c)^
(3/2) - 3*sqrt(d*x + c)*c)*a^3*b*c^3/d + 9009*(3*(d*x + c)^(5/2) - 10*(d*x + c)^(3/2)*c + 15*sqrt(d*x + c)*c^2
)*a^4*c + 18018*(3*(d*x + c)^(5/2) - 10*(d*x + c)^(3/2)*c + 15*sqrt(d*x + c)*c^2)*a^2*b^2*c^3/d^2 + 36036*(3*(
d*x + c)^(5/2) - 10*(d*x + c)^(3/2)*c + 15*sqrt(d*x + c)*c^2)*a^3*b*c^2/d + 1287*(5*(d*x + c)^(7/2) - 21*(d*x
+ c)^(5/2)*c + 35*(d*x + c)^(3/2)*c^2 - 35*sqrt(d*x + c)*c^3)*a^4 + 5148*(5*(d*x + c)^(7/2) - 21*(d*x + c)^(5/
2)*c + 35*(d*x + c)^(3/2)*c^2 - 35*sqrt(d*x + c)*c^3)*a*b^3*c^3/d^3 + 23166*(5*(d*x + c)^(7/2) - 21*(d*x + c)^
(5/2)*c + 35*(d*x + c)^(3/2)*c^2 - 35*sqrt(d*x + c)*c^3)*a^2*b^2*c^2/d^2 + 15444*(5*(d*x + c)^(7/2) - 21*(d*x
+ c)^(5/2)*c + 35*(d*x + c)^(3/2)*c^2 - 35*sqrt(d*x + c)*c^3)*a^3*b*c/d + 143*(35*(d*x + c)^(9/2) - 180*(d*x +
c)^(7/2)*c + 378*(d*x + c)^(5/2)*c^2 - 420*(d*x + c)^(3/2)*c^3 + 315*sqrt(d*x + c)*c^4)*b^4*c^3/d^4 + 1716*(3
5*(d*x + c)^(9/2) - 180*(d*x + c)^(7/2)*c + 378*(d*x + c)^(5/2)*c^2 - 420*(d*x + c)^(3/2)*c^3 + 315*sqrt(d*x +
c)*c^4)*a*b^3*c^2/d^3 + 2574*(35*(d*x + c)^(9/2) - 180*(d*x + c)^(7/2)*c + 378*(d*x + c)^(5/2)*c^2 - 420*(d*x
+ c)^(3/2)*c^3 + 315*sqrt(d*x + c)*c^4)*a^2*b^2*c/d^2 + 572*(35*(d*x + c)^(9/2) - 180*(d*x + c)^(7/2)*c + 378
*(d*x + c)^(5/2)*c^2 - 420*(d*x + c)^(3/2)*c^3 + 315*sqrt(d*x + c)*c^4)*a^3*b/d + 195*(63*(d*x + c)^(11/2) - 3
85*(d*x + c)^(9/2)*c + 990*(d*x + c)^(7/2)*c^2 - 1386*(d*x + c)^(5/2)*c^3 + 1155*(d*x + c)^(3/2)*c^4 - 693*sqr
t(d*x + c)*c^5)*b^4*c^2/d^4 + 780*(63*(d*x + c)^(11/2) - 385*(d*x + c)^(9/2)*c + 990*(d*x + c)^(7/2)*c^2 - 138
6*(d*x + c)^(5/2)*c^3 + 1155*(d*x + c)^(3/2)*c^4 - 693*sqrt(d*x + c)*c^5)*a*b^3*c/d^3 + 390*(63*(d*x + c)^(11/
2) - 385*(d*x + c)^(9/2)*c + 990*(d*x + c)^(7/2)*c^2 - 1386*(d*x + c)^(5/2)*c^3 + 1155*(d*x + c)^(3/2)*c^4 - 6
93*sqrt(d*x + c)*c^5)*a^2*b^2/d^2 + 45*(231*(d*x + c)^(13/2) - 1638*(d*x + c)^(11/2)*c + 5005*(d*x + c)^(9/2)*
c^2 - 8580*(d*x + c)^(7/2)*c^3 + 9009*(d*x + c)^(5/2)*c^4 - 6006*(d*x + c)^(3/2)*c^5 + 3003*sqrt(d*x + c)*c^6)
*b^4*c/d^4 + 60*(231*(d*x + c)^(13/2) - 1638*(d*x + c)^(11/2)*c + 5005*(d*x + c)^(9/2)*c^2 - 8580*(d*x + c)^(7
/2)*c^3 + 9009*(d*x + c)^(5/2)*c^4 - 6006*(d*x + c)^(3/2)*c^5 + 3003*sqrt(d*x + c)*c^6)*a*b^3/d^3 + 7*(429*(d*
x + c)^(15/2) - 3465*(d*x + c)^(13/2)*c + 12285*(d*x + c)^(11/2)*c^2 - 25025*(d*x + c)^(9/2)*c^3 + 32175*(d*x
+ c)^(7/2)*c^4 - 27027*(d*x + c)^(5/2)*c^5 + 15015*(d*x + c)^(3/2)*c^6 - 6435*sqrt(d*x + c)*c^7)*b^4/d^4)/d
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maple [A] time = 0.01, size = 186, normalized size = 1.44 \begin {gather*} \frac {2 \left (d x +c \right )^{\frac {7}{2}} \left (3003 b^{4} x^{4} d^{4}+13860 a \,b^{3} d^{4} x^{3}-1848 b^{4} c \,d^{3} x^{3}+24570 a^{2} b^{2} d^{4} x^{2}-7560 a \,b^{3} c \,d^{3} x^{2}+1008 b^{4} c^{2} d^{2} x^{2}+20020 a^{3} b \,d^{4} x -10920 a^{2} b^{2} c \,d^{3} x +3360 a \,b^{3} c^{2} d^{2} x -448 b^{4} c^{3} d x +6435 a^{4} d^{4}-5720 a^{3} b c \,d^{3}+3120 a^{2} b^{2} c^{2} d^{2}-960 a \,b^{3} c^{3} d +128 b^{4} c^{4}\right )}{45045 d^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((b*x+a)^4*(d*x+c)^(5/2),x)
[Out]
2/45045*(d*x+c)^(7/2)*(3003*b^4*d^4*x^4+13860*a*b^3*d^4*x^3-1848*b^4*c*d^3*x^3+24570*a^2*b^2*d^4*x^2-7560*a*b^
3*c*d^3*x^2+1008*b^4*c^2*d^2*x^2+20020*a^3*b*d^4*x-10920*a^2*b^2*c*d^3*x+3360*a*b^3*c^2*d^2*x-448*b^4*c^3*d*x+
6435*a^4*d^4-5720*a^3*b*c*d^3+3120*a^2*b^2*c^2*d^2-960*a*b^3*c^3*d+128*b^4*c^4)/d^5
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maxima [A] time = 1.33, size = 181, normalized size = 1.40 \begin {gather*} \frac {2 \, {\left (3003 \, {\left (d x + c\right )}^{\frac {15}{2}} b^{4} - 13860 \, {\left (b^{4} c - a b^{3} d\right )} {\left (d x + c\right )}^{\frac {13}{2}} + 24570 \, {\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} {\left (d x + c\right )}^{\frac {11}{2}} - 20020 \, {\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} {\left (d x + c\right )}^{\frac {9}{2}} + 6435 \, {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} {\left (d x + c\right )}^{\frac {7}{2}}\right )}}{45045 \, d^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)^4*(d*x+c)^(5/2),x, algorithm="maxima")
[Out]
2/45045*(3003*(d*x + c)^(15/2)*b^4 - 13860*(b^4*c - a*b^3*d)*(d*x + c)^(13/2) + 24570*(b^4*c^2 - 2*a*b^3*c*d +
a^2*b^2*d^2)*(d*x + c)^(11/2) - 20020*(b^4*c^3 - 3*a*b^3*c^2*d + 3*a^2*b^2*c*d^2 - a^3*b*d^3)*(d*x + c)^(9/2)
+ 6435*(b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*(d*x + c)^(7/2))/d^5
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mupad [B] time = 0.23, size = 112, normalized size = 0.87 \begin {gather*} \frac {2\,b^4\,{\left (c+d\,x\right )}^{15/2}}{15\,d^5}-\frac {\left (8\,b^4\,c-8\,a\,b^3\,d\right )\,{\left (c+d\,x\right )}^{13/2}}{13\,d^5}+\frac {2\,{\left (a\,d-b\,c\right )}^4\,{\left (c+d\,x\right )}^{7/2}}{7\,d^5}+\frac {12\,b^2\,{\left (a\,d-b\,c\right )}^2\,{\left (c+d\,x\right )}^{11/2}}{11\,d^5}+\frac {8\,b\,{\left (a\,d-b\,c\right )}^3\,{\left (c+d\,x\right )}^{9/2}}{9\,d^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a + b*x)^4*(c + d*x)^(5/2),x)
[Out]
(2*b^4*(c + d*x)^(15/2))/(15*d^5) - ((8*b^4*c - 8*a*b^3*d)*(c + d*x)^(13/2))/(13*d^5) + (2*(a*d - b*c)^4*(c +
d*x)^(7/2))/(7*d^5) + (12*b^2*(a*d - b*c)^2*(c + d*x)^(11/2))/(11*d^5) + (8*b*(a*d - b*c)^3*(c + d*x)^(9/2))/(
9*d^5)
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sympy [A] time = 33.64, size = 960, normalized size = 7.44
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)**4*(d*x+c)**(5/2),x)
[Out]
a**4*c**2*Piecewise((sqrt(c)*x, Eq(d, 0)), (2*(c + d*x)**(3/2)/(3*d), True)) + 4*a**4*c*(-c*(c + d*x)**(3/2)/3
+ (c + d*x)**(5/2)/5)/d + 2*a**4*(c**2*(c + d*x)**(3/2)/3 - 2*c*(c + d*x)**(5/2)/5 + (c + d*x)**(7/2)/7)/d +
8*a**3*b*c**2*(-c*(c + d*x)**(3/2)/3 + (c + d*x)**(5/2)/5)/d**2 + 16*a**3*b*c*(c**2*(c + d*x)**(3/2)/3 - 2*c*(
c + d*x)**(5/2)/5 + (c + d*x)**(7/2)/7)/d**2 + 8*a**3*b*(-c**3*(c + d*x)**(3/2)/3 + 3*c**2*(c + d*x)**(5/2)/5
- 3*c*(c + d*x)**(7/2)/7 + (c + d*x)**(9/2)/9)/d**2 + 12*a**2*b**2*c**2*(c**2*(c + d*x)**(3/2)/3 - 2*c*(c + d*
x)**(5/2)/5 + (c + d*x)**(7/2)/7)/d**3 + 24*a**2*b**2*c*(-c**3*(c + d*x)**(3/2)/3 + 3*c**2*(c + d*x)**(5/2)/5
- 3*c*(c + d*x)**(7/2)/7 + (c + d*x)**(9/2)/9)/d**3 + 12*a**2*b**2*(c**4*(c + d*x)**(3/2)/3 - 4*c**3*(c + d*x)
**(5/2)/5 + 6*c**2*(c + d*x)**(7/2)/7 - 4*c*(c + d*x)**(9/2)/9 + (c + d*x)**(11/2)/11)/d**3 + 8*a*b**3*c**2*(-
c**3*(c + d*x)**(3/2)/3 + 3*c**2*(c + d*x)**(5/2)/5 - 3*c*(c + d*x)**(7/2)/7 + (c + d*x)**(9/2)/9)/d**4 + 16*a
*b**3*c*(c**4*(c + d*x)**(3/2)/3 - 4*c**3*(c + d*x)**(5/2)/5 + 6*c**2*(c + d*x)**(7/2)/7 - 4*c*(c + d*x)**(9/2
)/9 + (c + d*x)**(11/2)/11)/d**4 + 8*a*b**3*(-c**5*(c + d*x)**(3/2)/3 + c**4*(c + d*x)**(5/2) - 10*c**3*(c + d
*x)**(7/2)/7 + 10*c**2*(c + d*x)**(9/2)/9 - 5*c*(c + d*x)**(11/2)/11 + (c + d*x)**(13/2)/13)/d**4 + 2*b**4*c**
2*(c**4*(c + d*x)**(3/2)/3 - 4*c**3*(c + d*x)**(5/2)/5 + 6*c**2*(c + d*x)**(7/2)/7 - 4*c*(c + d*x)**(9/2)/9 +
(c + d*x)**(11/2)/11)/d**5 + 4*b**4*c*(-c**5*(c + d*x)**(3/2)/3 + c**4*(c + d*x)**(5/2) - 10*c**3*(c + d*x)**(
7/2)/7 + 10*c**2*(c + d*x)**(9/2)/9 - 5*c*(c + d*x)**(11/2)/11 + (c + d*x)**(13/2)/13)/d**5 + 2*b**4*(c**6*(c
+ d*x)**(3/2)/3 - 6*c**5*(c + d*x)**(5/2)/5 + 15*c**4*(c + d*x)**(7/2)/7 - 20*c**3*(c + d*x)**(9/2)/9 + 15*c**
2*(c + d*x)**(11/2)/11 - 6*c*(c + d*x)**(13/2)/13 + (c + d*x)**(15/2)/15)/d**5
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