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

Optimal. Leaf size=132 \[ \frac {2 (d+e x)^{9/2} \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{9 e^4}-\frac {2 (d+e x)^{7/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right )}{7 e^4}-\frac {6 c (d+e x)^{11/2} (2 c d-b e)}{11 e^4}+\frac {4 c^2 (d+e x)^{13/2}}{13 e^4} \]

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Rubi [A]  time = 0.08, antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {771} \begin {gather*} \frac {2 (d+e x)^{9/2} \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{9 e^4}-\frac {2 (d+e x)^{7/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right )}{7 e^4}-\frac {6 c (d+e x)^{11/2} (2 c d-b e)}{11 e^4}+\frac {4 c^2 (d+e x)^{13/2}}{13 e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 771

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> In
t[ExpandIntegrand[(d + e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && N
eQ[b^2 - 4*a*c, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

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

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Mathematica [A]  time = 0.14, size = 111, normalized size = 0.84 \begin {gather*} \frac {2 (d+e x)^{7/2} \left (13 c e \left (22 a e (7 e x-2 d)+3 b \left (8 d^2-28 d e x+63 e^2 x^2\right )\right )+143 b e^2 (9 a e-2 b d+7 b e x)-6 c^2 \left (16 d^3-56 d^2 e x+126 d e^2 x^2-231 e^3 x^3\right )\right )}{9009 e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(d + e*x)^(7/2)*(143*b*e^2*(-2*b*d + 9*a*e + 7*b*e*x) - 6*c^2*(16*d^3 - 56*d^2*e*x + 126*d*e^2*x^2 - 231*e^
3*x^3) + 13*c*e*(22*a*e*(-2*d + 7*e*x) + 3*b*(8*d^2 - 28*d*e*x + 63*e^2*x^2))))/(9009*e^4)

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IntegrateAlgebraic [A]  time = 0.10, size = 143, normalized size = 1.08 \begin {gather*} \frac {2 (d+e x)^{7/2} \left (1287 a b e^3+2002 a c e^2 (d+e x)-2574 a c d e^2+1001 b^2 e^2 (d+e x)-1287 b^2 d e^2+3861 b c d^2 e-6006 b c d e (d+e x)+2457 b c e (d+e x)^2-2574 c^2 d^3+6006 c^2 d^2 (d+e x)-4914 c^2 d (d+e x)^2+1386 c^2 (d+e x)^3\right )}{9009 e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(d + e*x)^(7/2)*(-2574*c^2*d^3 + 3861*b*c*d^2*e - 1287*b^2*d*e^2 - 2574*a*c*d*e^2 + 1287*a*b*e^3 + 6006*c^2
*d^2*(d + e*x) - 6006*b*c*d*e*(d + e*x) + 1001*b^2*e^2*(d + e*x) + 2002*a*c*e^2*(d + e*x) - 4914*c^2*d*(d + e*
x)^2 + 2457*b*c*e*(d + e*x)^2 + 1386*c^2*(d + e*x)^3))/(9009*e^4)

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fricas [B]  time = 0.40, size = 272, normalized size = 2.06 \begin {gather*} \frac {2 \, {\left (1386 \, c^{2} e^{6} x^{6} - 96 \, c^{2} d^{6} + 312 \, b c d^{5} e + 1287 \, a b d^{3} e^{3} - 286 \, {\left (b^{2} + 2 \, a c\right )} d^{4} e^{2} + 189 \, {\left (18 \, c^{2} d e^{5} + 13 \, b c e^{6}\right )} x^{5} + 7 \, {\left (318 \, c^{2} d^{2} e^{4} + 897 \, b c d e^{5} + 143 \, {\left (b^{2} + 2 \, a c\right )} e^{6}\right )} x^{4} + {\left (30 \, c^{2} d^{3} e^{3} + 4407 \, b c d^{2} e^{4} + 1287 \, a b e^{6} + 2717 \, {\left (b^{2} + 2 \, a c\right )} d e^{5}\right )} x^{3} - 3 \, {\left (12 \, c^{2} d^{4} e^{2} - 39 \, b c d^{3} e^{3} - 1287 \, a b d e^{5} - 715 \, {\left (b^{2} + 2 \, a c\right )} d^{2} e^{4}\right )} x^{2} + {\left (48 \, c^{2} d^{5} e - 156 \, b c d^{4} e^{2} + 3861 \, a b d^{2} e^{4} + 143 \, {\left (b^{2} + 2 \, a c\right )} d^{3} e^{3}\right )} x\right )} \sqrt {e x + d}}{9009 \, e^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/9009*(1386*c^2*e^6*x^6 - 96*c^2*d^6 + 312*b*c*d^5*e + 1287*a*b*d^3*e^3 - 286*(b^2 + 2*a*c)*d^4*e^2 + 189*(18
*c^2*d*e^5 + 13*b*c*e^6)*x^5 + 7*(318*c^2*d^2*e^4 + 897*b*c*d*e^5 + 143*(b^2 + 2*a*c)*e^6)*x^4 + (30*c^2*d^3*e
^3 + 4407*b*c*d^2*e^4 + 1287*a*b*e^6 + 2717*(b^2 + 2*a*c)*d*e^5)*x^3 - 3*(12*c^2*d^4*e^2 - 39*b*c*d^3*e^3 - 12
87*a*b*d*e^5 - 715*(b^2 + 2*a*c)*d^2*e^4)*x^2 + (48*c^2*d^5*e - 156*b*c*d^4*e^2 + 3861*a*b*d^2*e^4 + 143*(b^2
+ 2*a*c)*d^3*e^3)*x)*sqrt(e*x + d)/e^4

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giac [B]  time = 0.23, size = 1087, normalized size = 8.23

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/45045*(15015*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*b^2*d^3*e^(-1) + 30030*((x*e + d)^(3/2) - 3*sqrt(x*e + d)
*d)*a*c*d^3*e^(-1) + 9009*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*b*c*d^3*e^(-2) + 2
574*(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)*c^2*d^3*e^(-3)
+ 9009*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*b^2*d^2*e^(-1) + 18018*(3*(x*e + d)^(
5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*a*c*d^2*e^(-1) + 11583*(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*c*d^2*e^(-2) + 858*(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)*c^2*d^2*e^(-3) + 45045
*sqrt(x*e + d)*a*b*d^3 + 45045*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*a*b*d^2 + 3861*(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^2*d*e^(-1) + 7722*(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*c*d*e^(-1) + 1287*(35*(x*e + d)^(9/2) - 18
0*(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*c*d*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)*c^2*d*e^(-3) + 9009*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d +
15*sqrt(x*e + d)*d^2)*a*b*d + 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)*b^2*e^(-1) + 286*(35*(x*e + d)^(9/2) - 180*(x*e + d)^(7/2)*d + 37
8*(x*e + d)^(5/2)*d^2 - 420*(x*e + d)^(3/2)*d^3 + 315*sqrt(x*e + d)*d^4)*a*c*e^(-1) + 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)*b*c*e^(-2) + 30*(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)*c^
2*e^(-3) + 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)*a*b
)*e^(-1)

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maple [A]  time = 0.05, size = 123, normalized size = 0.93 \begin {gather*} \frac {2 \left (e x +d \right )^{\frac {7}{2}} \left (1386 c^{2} x^{3} e^{3}+2457 b c \,e^{3} x^{2}-756 c^{2} d \,e^{2} x^{2}+2002 a c \,e^{3} x +1001 b^{2} e^{3} x -1092 b c d \,e^{2} x +336 c^{2} d^{2} e x +1287 a b \,e^{3}-572 a c d \,e^{2}-286 b^{2} d \,e^{2}+312 b c \,d^{2} e -96 c^{2} d^{3}\right )}{9009 e^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/9009*(e*x+d)^(7/2)*(1386*c^2*e^3*x^3+2457*b*c*e^3*x^2-756*c^2*d*e^2*x^2+2002*a*c*e^3*x+1001*b^2*e^3*x-1092*b
*c*d*e^2*x+336*c^2*d^2*e*x+1287*a*b*e^3-572*a*c*d*e^2-286*b^2*d*e^2+312*b*c*d^2*e-96*c^2*d^3)/e^4

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maxima [A]  time = 0.55, size = 121, normalized size = 0.92 \begin {gather*} \frac {2 \, {\left (1386 \, {\left (e x + d\right )}^{\frac {13}{2}} c^{2} - 2457 \, {\left (2 \, c^{2} d - b c e\right )} {\left (e x + d\right )}^{\frac {11}{2}} + 1001 \, {\left (6 \, c^{2} d^{2} - 6 \, b c d e + {\left (b^{2} + 2 \, a c\right )} e^{2}\right )} {\left (e x + d\right )}^{\frac {9}{2}} - 1287 \, {\left (2 \, c^{2} d^{3} - 3 \, b c d^{2} e - a b e^{3} + {\left (b^{2} + 2 \, a c\right )} d e^{2}\right )} {\left (e x + d\right )}^{\frac {7}{2}}\right )}}{9009 \, e^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/9009*(1386*(e*x + d)^(13/2)*c^2 - 2457*(2*c^2*d - b*c*e)*(e*x + d)^(11/2) + 1001*(6*c^2*d^2 - 6*b*c*d*e + (b
^2 + 2*a*c)*e^2)*(e*x + d)^(9/2) - 1287*(2*c^2*d^3 - 3*b*c*d^2*e - a*b*e^3 + (b^2 + 2*a*c)*d*e^2)*(e*x + d)^(7
/2))/e^4

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mupad [B]  time = 0.09, size = 118, normalized size = 0.89 \begin {gather*} \frac {4\,c^2\,{\left (d+e\,x\right )}^{13/2}}{13\,e^4}+\frac {{\left (d+e\,x\right )}^{9/2}\,\left (2\,b^2\,e^2-12\,b\,c\,d\,e+12\,c^2\,d^2+4\,a\,c\,e^2\right )}{9\,e^4}-\frac {\left (12\,c^2\,d-6\,b\,c\,e\right )\,{\left (d+e\,x\right )}^{11/2}}{11\,e^4}+\frac {2\,\left (b\,e-2\,c\,d\right )\,{\left (d+e\,x\right )}^{7/2}\,\left (c\,d^2-b\,d\,e+a\,e^2\right )}{7\,e^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(4*c^2*(d + e*x)^(13/2))/(13*e^4) + ((d + e*x)^(9/2)*(2*b^2*e^2 + 12*c^2*d^2 + 4*a*c*e^2 - 12*b*c*d*e))/(9*e^4
) - ((12*c^2*d - 6*b*c*e)*(d + e*x)^(11/2))/(11*e^4) + (2*(b*e - 2*c*d)*(d + e*x)^(7/2)*(a*e^2 + c*d^2 - b*d*e
))/(7*e^4)

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sympy [A]  time = 4.92, size = 643, normalized size = 4.87 \begin {gather*} \begin {cases} \frac {2 a b d^{3} \sqrt {d + e x}}{7 e} + \frac {6 a b d^{2} x \sqrt {d + e x}}{7} + \frac {6 a b d e x^{2} \sqrt {d + e x}}{7} + \frac {2 a b e^{2} x^{3} \sqrt {d + e x}}{7} - \frac {8 a c d^{4} \sqrt {d + e x}}{63 e^{2}} + \frac {4 a c d^{3} x \sqrt {d + e x}}{63 e} + \frac {20 a c d^{2} x^{2} \sqrt {d + e x}}{21} + \frac {76 a c d e x^{3} \sqrt {d + e x}}{63} + \frac {4 a c e^{2} x^{4} \sqrt {d + e x}}{9} - \frac {4 b^{2} d^{4} \sqrt {d + e x}}{63 e^{2}} + \frac {2 b^{2} d^{3} x \sqrt {d + e x}}{63 e} + \frac {10 b^{2} d^{2} x^{2} \sqrt {d + e x}}{21} + \frac {38 b^{2} d e x^{3} \sqrt {d + e x}}{63} + \frac {2 b^{2} e^{2} x^{4} \sqrt {d + e x}}{9} + \frac {16 b c d^{5} \sqrt {d + e x}}{231 e^{3}} - \frac {8 b c d^{4} x \sqrt {d + e x}}{231 e^{2}} + \frac {2 b c d^{3} x^{2} \sqrt {d + e x}}{77 e} + \frac {226 b c d^{2} x^{3} \sqrt {d + e x}}{231} + \frac {46 b c d e x^{4} \sqrt {d + e x}}{33} + \frac {6 b c e^{2} x^{5} \sqrt {d + e x}}{11} - \frac {64 c^{2} d^{6} \sqrt {d + e x}}{3003 e^{4}} + \frac {32 c^{2} d^{5} x \sqrt {d + e x}}{3003 e^{3}} - \frac {8 c^{2} d^{4} x^{2} \sqrt {d + e x}}{1001 e^{2}} + \frac {20 c^{2} d^{3} x^{3} \sqrt {d + e x}}{3003 e} + \frac {212 c^{2} d^{2} x^{4} \sqrt {d + e x}}{429} + \frac {108 c^{2} d e x^{5} \sqrt {d + e x}}{143} + \frac {4 c^{2} e^{2} x^{6} \sqrt {d + e x}}{13} & \text {for}\: e \neq 0 \\d^{\frac {5}{2}} \left (a b x + a c x^{2} + \frac {b^{2} x^{2}}{2} + b c x^{3} + \frac {c^{2} x^{4}}{2}\right ) & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((2*a*b*d**3*sqrt(d + e*x)/(7*e) + 6*a*b*d**2*x*sqrt(d + e*x)/7 + 6*a*b*d*e*x**2*sqrt(d + e*x)/7 + 2*
a*b*e**2*x**3*sqrt(d + e*x)/7 - 8*a*c*d**4*sqrt(d + e*x)/(63*e**2) + 4*a*c*d**3*x*sqrt(d + e*x)/(63*e) + 20*a*
c*d**2*x**2*sqrt(d + e*x)/21 + 76*a*c*d*e*x**3*sqrt(d + e*x)/63 + 4*a*c*e**2*x**4*sqrt(d + e*x)/9 - 4*b**2*d**
4*sqrt(d + e*x)/(63*e**2) + 2*b**2*d**3*x*sqrt(d + e*x)/(63*e) + 10*b**2*d**2*x**2*sqrt(d + e*x)/21 + 38*b**2*
d*e*x**3*sqrt(d + e*x)/63 + 2*b**2*e**2*x**4*sqrt(d + e*x)/9 + 16*b*c*d**5*sqrt(d + e*x)/(231*e**3) - 8*b*c*d*
*4*x*sqrt(d + e*x)/(231*e**2) + 2*b*c*d**3*x**2*sqrt(d + e*x)/(77*e) + 226*b*c*d**2*x**3*sqrt(d + e*x)/231 + 4
6*b*c*d*e*x**4*sqrt(d + e*x)/33 + 6*b*c*e**2*x**5*sqrt(d + e*x)/11 - 64*c**2*d**6*sqrt(d + e*x)/(3003*e**4) +
32*c**2*d**5*x*sqrt(d + e*x)/(3003*e**3) - 8*c**2*d**4*x**2*sqrt(d + e*x)/(1001*e**2) + 20*c**2*d**3*x**3*sqrt
(d + e*x)/(3003*e) + 212*c**2*d**2*x**4*sqrt(d + e*x)/429 + 108*c**2*d*e*x**5*sqrt(d + e*x)/143 + 4*c**2*e**2*
x**6*sqrt(d + e*x)/13, Ne(e, 0)), (d**(5/2)*(a*b*x + a*c*x**2 + b**2*x**2/2 + b*c*x**3 + c**2*x**4/2), True))

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