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

Optimal. Leaf size=100 \[ -\frac {6 b^2 (c+d x)^{11/2} (b c-a d)}{11 d^4}+\frac {2 b (c+d x)^{9/2} (b c-a d)^2}{3 d^4}-\frac {2 (c+d x)^{7/2} (b c-a d)^3}{7 d^4}+\frac {2 b^3 (c+d x)^{13/2}}{13 d^4} \]

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Rubi [A]  time = 0.03, antiderivative size = 100, 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 {6 b^2 (c+d x)^{11/2} (b c-a d)}{11 d^4}+\frac {2 b (c+d x)^{9/2} (b c-a d)^2}{3 d^4}-\frac {2 (c+d x)^{7/2} (b c-a d)^3}{7 d^4}+\frac {2 b^3 (c+d x)^{13/2}}{13 d^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^3*(c + d*x)^(5/2),x]

[Out]

(-2*(b*c - a*d)^3*(c + d*x)^(7/2))/(7*d^4) + (2*b*(b*c - a*d)^2*(c + d*x)^(9/2))/(3*d^4) - (6*b^2*(b*c - a*d)*
(c + d*x)^(11/2))/(11*d^4) + (2*b^3*(c + d*x)^(13/2))/(13*d^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])

Rubi steps

\begin {align*} \int (a+b x)^3 (c+d x)^{5/2} \, dx &=\int \left (\frac {(-b c+a d)^3 (c+d x)^{5/2}}{d^3}+\frac {3 b (b c-a d)^2 (c+d x)^{7/2}}{d^3}-\frac {3 b^2 (b c-a d) (c+d x)^{9/2}}{d^3}+\frac {b^3 (c+d x)^{11/2}}{d^3}\right ) \, dx\\ &=-\frac {2 (b c-a d)^3 (c+d x)^{7/2}}{7 d^4}+\frac {2 b (b c-a d)^2 (c+d x)^{9/2}}{3 d^4}-\frac {6 b^2 (b c-a d) (c+d x)^{11/2}}{11 d^4}+\frac {2 b^3 (c+d x)^{13/2}}{13 d^4}\\ \end {align*}

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Mathematica [A]  time = 0.07, size = 79, normalized size = 0.79 \begin {gather*} \frac {2 (c+d x)^{7/2} \left (-819 b^2 (c+d x)^2 (b c-a d)+1001 b (c+d x) (b c-a d)^2-429 (b c-a d)^3+231 b^3 (c+d x)^3\right )}{3003 d^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(c + d*x)^(7/2)*(-429*(b*c - a*d)^3 + 1001*b*(b*c - a*d)^2*(c + d*x) - 819*b^2*(b*c - a*d)*(c + d*x)^2 + 23
1*b^3*(c + d*x)^3))/(3003*d^4)

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IntegrateAlgebraic [A]  time = 0.06, size = 132, normalized size = 1.32 \begin {gather*} \frac {2 (c+d x)^{7/2} \left (429 a^3 d^3+1001 a^2 b d^2 (c+d x)-1287 a^2 b c d^2+1287 a b^2 c^2 d+819 a b^2 d (c+d x)^2-2002 a b^2 c d (c+d x)-429 b^3 c^3+1001 b^3 c^2 (c+d x)+231 b^3 (c+d x)^3-819 b^3 c (c+d x)^2\right )}{3003 d^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(c + d*x)^(7/2)*(-429*b^3*c^3 + 1287*a*b^2*c^2*d - 1287*a^2*b*c*d^2 + 429*a^3*d^3 + 1001*b^3*c^2*(c + d*x)
- 2002*a*b^2*c*d*(c + d*x) + 1001*a^2*b*d^2*(c + d*x) - 819*b^3*c*(c + d*x)^2 + 819*a*b^2*d*(c + d*x)^2 + 231*
b^3*(c + d*x)^3))/(3003*d^4)

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fricas [B]  time = 1.54, size = 268, normalized size = 2.68 \begin {gather*} \frac {2 \, {\left (231 \, b^{3} d^{6} x^{6} - 16 \, b^{3} c^{6} + 104 \, a b^{2} c^{5} d - 286 \, a^{2} b c^{4} d^{2} + 429 \, a^{3} c^{3} d^{3} + 63 \, {\left (9 \, b^{3} c d^{5} + 13 \, a b^{2} d^{6}\right )} x^{5} + 7 \, {\left (53 \, b^{3} c^{2} d^{4} + 299 \, a b^{2} c d^{5} + 143 \, a^{2} b d^{6}\right )} x^{4} + {\left (5 \, b^{3} c^{3} d^{3} + 1469 \, a b^{2} c^{2} d^{4} + 2717 \, a^{2} b c d^{5} + 429 \, a^{3} d^{6}\right )} x^{3} - 3 \, {\left (2 \, b^{3} c^{4} d^{2} - 13 \, a b^{2} c^{3} d^{3} - 715 \, a^{2} b c^{2} d^{4} - 429 \, a^{3} c d^{5}\right )} x^{2} + {\left (8 \, b^{3} c^{5} d - 52 \, a b^{2} c^{4} d^{2} + 143 \, a^{2} b c^{3} d^{3} + 1287 \, a^{3} c^{2} d^{4}\right )} x\right )} \sqrt {d x + c}}{3003 \, d^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3003*(231*b^3*d^6*x^6 - 16*b^3*c^6 + 104*a*b^2*c^5*d - 286*a^2*b*c^4*d^2 + 429*a^3*c^3*d^3 + 63*(9*b^3*c*d^5
+ 13*a*b^2*d^6)*x^5 + 7*(53*b^3*c^2*d^4 + 299*a*b^2*c*d^5 + 143*a^2*b*d^6)*x^4 + (5*b^3*c^3*d^3 + 1469*a*b^2*
c^2*d^4 + 2717*a^2*b*c*d^5 + 429*a^3*d^6)*x^3 - 3*(2*b^3*c^4*d^2 - 13*a*b^2*c^3*d^3 - 715*a^2*b*c^2*d^4 - 429*
a^3*c*d^5)*x^2 + (8*b^3*c^5*d - 52*a*b^2*c^4*d^2 + 143*a^2*b*c^3*d^3 + 1287*a^3*c^2*d^4)*x)*sqrt(d*x + c)/d^4

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giac [B]  time = 1.58, size = 857, normalized size = 8.57

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15015*(15015*sqrt(d*x + c)*a^3*c^3 + 15015*((d*x + c)^(3/2) - 3*sqrt(d*x + c)*c)*a^3*c^2 + 15015*((d*x + c)^
(3/2) - 3*sqrt(d*x + c)*c)*a^2*b*c^3/d + 3003*(3*(d*x + c)^(5/2) - 10*(d*x + c)^(3/2)*c + 15*sqrt(d*x + c)*c^2
)*a^3*c + 3003*(3*(d*x + c)^(5/2) - 10*(d*x + c)^(3/2)*c + 15*sqrt(d*x + c)*c^2)*a*b^2*c^3/d^2 + 9009*(3*(d*x
+ c)^(5/2) - 10*(d*x + c)^(3/2)*c + 15*sqrt(d*x + c)*c^2)*a^2*b*c^2/d + 429*(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 + 429*(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)*b^3*c^3/d^3 + 3861*(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^2*c^2/d^2 + 3861*(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*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^3*c^2/d^3 + 429*(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*b^
2*c/d^2 + 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)*a^2*b/d + 65*(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 - 693*sqrt(d*x + c)*c^5)*b^3*c/d^3 + 65*(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 -
693*sqrt(d*x + c)*c^5)*a*b^2/d^2 + 5*(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
^3/d^3)/d

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maple [A]  time = 0.01, size = 116, normalized size = 1.16 \begin {gather*} \frac {2 \left (d x +c \right )^{\frac {7}{2}} \left (231 b^{3} x^{3} d^{3}+819 a \,b^{2} d^{3} x^{2}-126 b^{3} c \,d^{2} x^{2}+1001 a^{2} b \,d^{3} x -364 a \,b^{2} c \,d^{2} x +56 b^{3} c^{2} d x +429 a^{3} d^{3}-286 a^{2} b c \,d^{2}+104 a \,b^{2} c^{2} d -16 b^{3} c^{3}\right )}{3003 d^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3003*(d*x+c)^(7/2)*(231*b^3*d^3*x^3+819*a*b^2*d^3*x^2-126*b^3*c*d^2*x^2+1001*a^2*b*d^3*x-364*a*b^2*c*d^2*x+5
6*b^3*c^2*d*x+429*a^3*d^3-286*a^2*b*c*d^2+104*a*b^2*c^2*d-16*b^3*c^3)/d^4

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maxima [A]  time = 1.40, size = 118, normalized size = 1.18 \begin {gather*} \frac {2 \, {\left (231 \, {\left (d x + c\right )}^{\frac {13}{2}} b^{3} - 819 \, {\left (b^{3} c - a b^{2} d\right )} {\left (d x + c\right )}^{\frac {11}{2}} + 1001 \, {\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} {\left (d x + c\right )}^{\frac {9}{2}} - 429 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} {\left (d x + c\right )}^{\frac {7}{2}}\right )}}{3003 \, d^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3003*(231*(d*x + c)^(13/2)*b^3 - 819*(b^3*c - a*b^2*d)*(d*x + c)^(11/2) + 1001*(b^3*c^2 - 2*a*b^2*c*d + a^2*
b*d^2)*(d*x + c)^(9/2) - 429*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*(d*x + c)^(7/2))/d^4

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mupad [B]  time = 0.08, size = 87, normalized size = 0.87 \begin {gather*} \frac {2\,b^3\,{\left (c+d\,x\right )}^{13/2}}{13\,d^4}-\frac {\left (6\,b^3\,c-6\,a\,b^2\,d\right )\,{\left (c+d\,x\right )}^{11/2}}{11\,d^4}+\frac {2\,{\left (a\,d-b\,c\right )}^3\,{\left (c+d\,x\right )}^{7/2}}{7\,d^4}+\frac {2\,b\,{\left (a\,d-b\,c\right )}^2\,{\left (c+d\,x\right )}^{9/2}}{3\,d^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^3*(c + d*x)^(5/2),x)

[Out]

(2*b^3*(c + d*x)^(13/2))/(13*d^4) - ((6*b^3*c - 6*a*b^2*d)*(c + d*x)^(11/2))/(11*d^4) + (2*(a*d - b*c)^3*(c +
d*x)^(7/2))/(7*d^4) + (2*b*(a*d - b*c)^2*(c + d*x)^(9/2))/(3*d^4)

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sympy [A]  time = 4.61, size = 549, normalized size = 5.49 \begin {gather*} \begin {cases} \frac {2 a^{3} c^{3} \sqrt {c + d x}}{7 d} + \frac {6 a^{3} c^{2} x \sqrt {c + d x}}{7} + \frac {6 a^{3} c d x^{2} \sqrt {c + d x}}{7} + \frac {2 a^{3} d^{2} x^{3} \sqrt {c + d x}}{7} - \frac {4 a^{2} b c^{4} \sqrt {c + d x}}{21 d^{2}} + \frac {2 a^{2} b c^{3} x \sqrt {c + d x}}{21 d} + \frac {10 a^{2} b c^{2} x^{2} \sqrt {c + d x}}{7} + \frac {38 a^{2} b c d x^{3} \sqrt {c + d x}}{21} + \frac {2 a^{2} b d^{2} x^{4} \sqrt {c + d x}}{3} + \frac {16 a b^{2} c^{5} \sqrt {c + d x}}{231 d^{3}} - \frac {8 a b^{2} c^{4} x \sqrt {c + d x}}{231 d^{2}} + \frac {2 a b^{2} c^{3} x^{2} \sqrt {c + d x}}{77 d} + \frac {226 a b^{2} c^{2} x^{3} \sqrt {c + d x}}{231} + \frac {46 a b^{2} c d x^{4} \sqrt {c + d x}}{33} + \frac {6 a b^{2} d^{2} x^{5} \sqrt {c + d x}}{11} - \frac {32 b^{3} c^{6} \sqrt {c + d x}}{3003 d^{4}} + \frac {16 b^{3} c^{5} x \sqrt {c + d x}}{3003 d^{3}} - \frac {4 b^{3} c^{4} x^{2} \sqrt {c + d x}}{1001 d^{2}} + \frac {10 b^{3} c^{3} x^{3} \sqrt {c + d x}}{3003 d} + \frac {106 b^{3} c^{2} x^{4} \sqrt {c + d x}}{429} + \frac {54 b^{3} c d x^{5} \sqrt {c + d x}}{143} + \frac {2 b^{3} d^{2} x^{6} \sqrt {c + d x}}{13} & \text {for}\: d \neq 0 \\c^{\frac {5}{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)**3*(d*x+c)**(5/2),x)

[Out]

Piecewise((2*a**3*c**3*sqrt(c + d*x)/(7*d) + 6*a**3*c**2*x*sqrt(c + d*x)/7 + 6*a**3*c*d*x**2*sqrt(c + d*x)/7 +
2*a**3*d**2*x**3*sqrt(c + d*x)/7 - 4*a**2*b*c**4*sqrt(c + d*x)/(21*d**2) + 2*a**2*b*c**3*x*sqrt(c + d*x)/(21*
d) + 10*a**2*b*c**2*x**2*sqrt(c + d*x)/7 + 38*a**2*b*c*d*x**3*sqrt(c + d*x)/21 + 2*a**2*b*d**2*x**4*sqrt(c + d
*x)/3 + 16*a*b**2*c**5*sqrt(c + d*x)/(231*d**3) - 8*a*b**2*c**4*x*sqrt(c + d*x)/(231*d**2) + 2*a*b**2*c**3*x**
2*sqrt(c + d*x)/(77*d) + 226*a*b**2*c**2*x**3*sqrt(c + d*x)/231 + 46*a*b**2*c*d*x**4*sqrt(c + d*x)/33 + 6*a*b*
*2*d**2*x**5*sqrt(c + d*x)/11 - 32*b**3*c**6*sqrt(c + d*x)/(3003*d**4) + 16*b**3*c**5*x*sqrt(c + d*x)/(3003*d*
*3) - 4*b**3*c**4*x**2*sqrt(c + d*x)/(1001*d**2) + 10*b**3*c**3*x**3*sqrt(c + d*x)/(3003*d) + 106*b**3*c**2*x*
*4*sqrt(c + d*x)/429 + 54*b**3*c*d*x**5*sqrt(c + d*x)/143 + 2*b**3*d**2*x**6*sqrt(c + d*x)/13, Ne(d, 0)), (c**
(5/2)*(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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