3.356 \(\int \sqrt {d+e x} (b x+c x^2)^3 \, dx\)

Optimal. Leaf size=248 \[ \frac {6 c (d+e x)^{11/2} \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{11 e^7}-\frac {2 (d+e x)^{9/2} (2 c d-b e) \left (b^2 e^2-10 b c d e+10 c^2 d^2\right )}{9 e^7}+\frac {6 d (d+e x)^{7/2} (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{7 e^7}-\frac {6 c^2 (d+e x)^{13/2} (2 c d-b e)}{13 e^7}+\frac {2 d^3 (d+e x)^{3/2} (c d-b e)^3}{3 e^7}-\frac {6 d^2 (d+e x)^{5/2} (c d-b e)^2 (2 c d-b e)}{5 e^7}+\frac {2 c^3 (d+e x)^{15/2}}{15 e^7} \]

[Out]

2/3*d^3*(-b*e+c*d)^3*(e*x+d)^(3/2)/e^7-6/5*d^2*(-b*e+c*d)^2*(-b*e+2*c*d)*(e*x+d)^(5/2)/e^7+6/7*d*(-b*e+c*d)*(b
^2*e^2-5*b*c*d*e+5*c^2*d^2)*(e*x+d)^(7/2)/e^7-2/9*(-b*e+2*c*d)*(b^2*e^2-10*b*c*d*e+10*c^2*d^2)*(e*x+d)^(9/2)/e
^7+6/11*c*(b^2*e^2-5*b*c*d*e+5*c^2*d^2)*(e*x+d)^(11/2)/e^7-6/13*c^2*(-b*e+2*c*d)*(e*x+d)^(13/2)/e^7+2/15*c^3*(
e*x+d)^(15/2)/e^7

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Rubi [A]  time = 0.11, antiderivative size = 248, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {698} \[ \frac {6 c (d+e x)^{11/2} \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{11 e^7}-\frac {2 (d+e x)^{9/2} (2 c d-b e) \left (b^2 e^2-10 b c d e+10 c^2 d^2\right )}{9 e^7}+\frac {6 d (d+e x)^{7/2} (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{7 e^7}-\frac {6 c^2 (d+e x)^{13/2} (2 c d-b e)}{13 e^7}-\frac {6 d^2 (d+e x)^{5/2} (c d-b e)^2 (2 c d-b e)}{5 e^7}+\frac {2 d^3 (d+e x)^{3/2} (c d-b e)^3}{3 e^7}+\frac {2 c^3 (d+e x)^{15/2}}{15 e^7} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[d + e*x]*(b*x + c*x^2)^3,x]

[Out]

(2*d^3*(c*d - b*e)^3*(d + e*x)^(3/2))/(3*e^7) - (6*d^2*(c*d - b*e)^2*(2*c*d - b*e)*(d + e*x)^(5/2))/(5*e^7) +
(6*d*(c*d - b*e)*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2)*(d + e*x)^(7/2))/(7*e^7) - (2*(2*c*d - b*e)*(10*c^2*d^2 - 1
0*b*c*d*e + b^2*e^2)*(d + e*x)^(9/2))/(9*e^7) + (6*c*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2)*(d + e*x)^(11/2))/(11*e
^7) - (6*c^2*(2*c*d - b*e)*(d + e*x)^(13/2))/(13*e^7) + (2*c^3*(d + e*x)^(15/2))/(15*e^7)

Rule 698

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

Rubi steps

\begin {align*} \int \sqrt {d+e x} \left (b x+c x^2\right )^3 \, dx &=\int \left (\frac {d^3 (c d-b e)^3 \sqrt {d+e x}}{e^6}-\frac {3 d^2 (c d-b e)^2 (2 c d-b e) (d+e x)^{3/2}}{e^6}+\frac {3 d (c d-b e) \left (5 c^2 d^2-5 b c d e+b^2 e^2\right ) (d+e x)^{5/2}}{e^6}+\frac {(2 c d-b e) \left (-10 c^2 d^2+10 b c d e-b^2 e^2\right ) (d+e x)^{7/2}}{e^6}+\frac {3 c \left (5 c^2 d^2-5 b c d e+b^2 e^2\right ) (d+e x)^{9/2}}{e^6}-\frac {3 c^2 (2 c d-b e) (d+e x)^{11/2}}{e^6}+\frac {c^3 (d+e x)^{13/2}}{e^6}\right ) \, dx\\ &=\frac {2 d^3 (c d-b e)^3 (d+e x)^{3/2}}{3 e^7}-\frac {6 d^2 (c d-b e)^2 (2 c d-b e) (d+e x)^{5/2}}{5 e^7}+\frac {6 d (c d-b e) \left (5 c^2 d^2-5 b c d e+b^2 e^2\right ) (d+e x)^{7/2}}{7 e^7}-\frac {2 (2 c d-b e) \left (10 c^2 d^2-10 b c d e+b^2 e^2\right ) (d+e x)^{9/2}}{9 e^7}+\frac {6 c \left (5 c^2 d^2-5 b c d e+b^2 e^2\right ) (d+e x)^{11/2}}{11 e^7}-\frac {6 c^2 (2 c d-b e) (d+e x)^{13/2}}{13 e^7}+\frac {2 c^3 (d+e x)^{15/2}}{15 e^7}\\ \end {align*}

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Mathematica [A]  time = 0.13, size = 206, normalized size = 0.83 \[ \frac {2 (d+e x)^{3/2} \left (12285 c (d+e x)^4 \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )-5005 (d+e x)^3 (2 c d-b e) \left (b^2 e^2-10 b c d e+10 c^2 d^2\right )+19305 d (d+e x)^2 (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )-10395 c^2 (d+e x)^5 (2 c d-b e)+15015 d^3 (c d-b e)^3-27027 d^2 (d+e x) (c d-b e)^2 (2 c d-b e)+3003 c^3 (d+e x)^6\right )}{45045 e^7} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[d + e*x]*(b*x + c*x^2)^3,x]

[Out]

(2*(d + e*x)^(3/2)*(15015*d^3*(c*d - b*e)^3 - 27027*d^2*(c*d - b*e)^2*(2*c*d - b*e)*(d + e*x) + 19305*d*(c*d -
b*e)*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2)*(d + e*x)^2 - 5005*(2*c*d - b*e)*(10*c^2*d^2 - 10*b*c*d*e + b^2*e^2)*(
d + e*x)^3 + 12285*c*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2)*(d + e*x)^4 - 10395*c^2*(2*c*d - b*e)*(d + e*x)^5 + 300
3*c^3*(d + e*x)^6))/(45045*e^7)

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fricas [A]  time = 0.90, size = 321, normalized size = 1.29 \[ \frac {2 \, {\left (3003 \, c^{3} e^{7} x^{7} + 1024 \, c^{3} d^{7} - 3840 \, b c^{2} d^{6} e + 4992 \, b^{2} c d^{5} e^{2} - 2288 \, b^{3} d^{4} e^{3} + 231 \, {\left (c^{3} d e^{6} + 45 \, b c^{2} e^{7}\right )} x^{6} - 63 \, {\left (4 \, c^{3} d^{2} e^{5} - 15 \, b c^{2} d e^{6} - 195 \, b^{2} c e^{7}\right )} x^{5} + 35 \, {\left (8 \, c^{3} d^{3} e^{4} - 30 \, b c^{2} d^{2} e^{5} + 39 \, b^{2} c d e^{6} + 143 \, b^{3} e^{7}\right )} x^{4} - 5 \, {\left (64 \, c^{3} d^{4} e^{3} - 240 \, b c^{2} d^{3} e^{4} + 312 \, b^{2} c d^{2} e^{5} - 143 \, b^{3} d e^{6}\right )} x^{3} + 6 \, {\left (64 \, c^{3} d^{5} e^{2} - 240 \, b c^{2} d^{4} e^{3} + 312 \, b^{2} c d^{3} e^{4} - 143 \, b^{3} d^{2} e^{5}\right )} x^{2} - 8 \, {\left (64 \, c^{3} d^{6} e - 240 \, b c^{2} d^{5} e^{2} + 312 \, b^{2} c d^{4} e^{3} - 143 \, b^{3} d^{3} e^{4}\right )} x\right )} \sqrt {e x + d}}{45045 \, e^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(1/2)*(c*x^2+b*x)^3,x, algorithm="fricas")

[Out]

2/45045*(3003*c^3*e^7*x^7 + 1024*c^3*d^7 - 3840*b*c^2*d^6*e + 4992*b^2*c*d^5*e^2 - 2288*b^3*d^4*e^3 + 231*(c^3
*d*e^6 + 45*b*c^2*e^7)*x^6 - 63*(4*c^3*d^2*e^5 - 15*b*c^2*d*e^6 - 195*b^2*c*e^7)*x^5 + 35*(8*c^3*d^3*e^4 - 30*
b*c^2*d^2*e^5 + 39*b^2*c*d*e^6 + 143*b^3*e^7)*x^4 - 5*(64*c^3*d^4*e^3 - 240*b*c^2*d^3*e^4 + 312*b^2*c*d^2*e^5
- 143*b^3*d*e^6)*x^3 + 6*(64*c^3*d^5*e^2 - 240*b*c^2*d^4*e^3 + 312*b^2*c*d^3*e^4 - 143*b^3*d^2*e^5)*x^2 - 8*(6
4*c^3*d^6*e - 240*b*c^2*d^5*e^2 + 312*b^2*c*d^4*e^3 - 143*b^3*d^3*e^4)*x)*sqrt(e*x + d)/e^7

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giac [B]  time = 0.23, size = 661, normalized size = 2.67 \[ \frac {2}{45045} \, {\left (1287 \, {\left (5 \, {\left (x e + d\right )}^{\frac {7}{2}} - 21 \, {\left (x e + d\right )}^{\frac {5}{2}} d + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {x e + d} d^{3}\right )} b^{3} d e^{\left (-3\right )} + 429 \, {\left (35 \, {\left (x e + d\right )}^{\frac {9}{2}} - 180 \, {\left (x e + d\right )}^{\frac {7}{2}} d + 378 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {x e + d} d^{4}\right )} b^{2} c d e^{\left (-4\right )} + 195 \, {\left (63 \, {\left (x e + d\right )}^{\frac {11}{2}} - 385 \, {\left (x e + d\right )}^{\frac {9}{2}} d + 990 \, {\left (x e + d\right )}^{\frac {7}{2}} d^{2} - 1386 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{3} + 1155 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{4} - 693 \, \sqrt {x e + d} d^{5}\right )} b c^{2} d e^{\left (-5\right )} + 15 \, {\left (231 \, {\left (x e + d\right )}^{\frac {13}{2}} - 1638 \, {\left (x e + d\right )}^{\frac {11}{2}} d + 5005 \, {\left (x e + d\right )}^{\frac {9}{2}} d^{2} - 8580 \, {\left (x e + d\right )}^{\frac {7}{2}} d^{3} + 9009 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{4} - 6006 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{5} + 3003 \, \sqrt {x e + d} d^{6}\right )} c^{3} d e^{\left (-6\right )} + 143 \, {\left (35 \, {\left (x e + d\right )}^{\frac {9}{2}} - 180 \, {\left (x e + d\right )}^{\frac {7}{2}} d + 378 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {x e + d} d^{4}\right )} b^{3} e^{\left (-3\right )} + 195 \, {\left (63 \, {\left (x e + d\right )}^{\frac {11}{2}} - 385 \, {\left (x e + d\right )}^{\frac {9}{2}} d + 990 \, {\left (x e + d\right )}^{\frac {7}{2}} d^{2} - 1386 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{3} + 1155 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{4} - 693 \, \sqrt {x e + d} d^{5}\right )} b^{2} c e^{\left (-4\right )} + 45 \, {\left (231 \, {\left (x e + d\right )}^{\frac {13}{2}} - 1638 \, {\left (x e + d\right )}^{\frac {11}{2}} d + 5005 \, {\left (x e + d\right )}^{\frac {9}{2}} d^{2} - 8580 \, {\left (x e + d\right )}^{\frac {7}{2}} d^{3} + 9009 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{4} - 6006 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{5} + 3003 \, \sqrt {x e + d} d^{6}\right )} b c^{2} e^{\left (-5\right )} + 7 \, {\left (429 \, {\left (x e + d\right )}^{\frac {15}{2}} - 3465 \, {\left (x e + d\right )}^{\frac {13}{2}} d + 12285 \, {\left (x e + d\right )}^{\frac {11}{2}} d^{2} - 25025 \, {\left (x e + d\right )}^{\frac {9}{2}} d^{3} + 32175 \, {\left (x e + d\right )}^{\frac {7}{2}} d^{4} - 27027 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{5} + 15015 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{6} - 6435 \, \sqrt {x e + d} d^{7}\right )} c^{3} e^{\left (-6\right )}\right )} e^{\left (-1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(1/2)*(c*x^2+b*x)^3,x, algorithm="giac")

[Out]

2/45045*(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
*e^(-3) + 429*(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*c*d*e^(-4) + 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^2*d*e^(-5) + 15*(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^3*d*e^(-6) + 143*(35*(x*e + d)^(9/2) - 1
80*(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*e^(-3) +
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^2*c*e^(-4) + 45*(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*c^2*e^(-5) + 7*(429*(x*e + d)^(15/2) - 3465*(x*e + d)^(13/2)*d + 12285*(x*e + d)^(1
1/2)*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)*c^3*e^(-6))*e^(-1)

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maple [A]  time = 0.04, size = 286, normalized size = 1.15 \[ -\frac {2 \left (e x +d \right )^{\frac {3}{2}} \left (-3003 c^{3} x^{6} e^{6}-10395 b \,c^{2} e^{6} x^{5}+2772 c^{3} d \,e^{5} x^{5}-12285 b^{2} c \,e^{6} x^{4}+9450 b \,c^{2} d \,e^{5} x^{4}-2520 c^{3} d^{2} e^{4} x^{4}-5005 b^{3} e^{6} x^{3}+10920 b^{2} c d \,e^{5} x^{3}-8400 b \,c^{2} d^{2} e^{4} x^{3}+2240 c^{3} d^{3} e^{3} x^{3}+4290 b^{3} d \,e^{5} x^{2}-9360 b^{2} c \,d^{2} e^{4} x^{2}+7200 b \,c^{2} d^{3} e^{3} x^{2}-1920 c^{3} d^{4} e^{2} x^{2}-3432 b^{3} d^{2} e^{4} x +7488 b^{2} c \,d^{3} e^{3} x -5760 b \,c^{2} d^{4} e^{2} x +1536 c^{3} d^{5} e x +2288 b^{3} d^{3} e^{3}-4992 b^{2} c \,d^{4} e^{2}+3840 b \,c^{2} d^{5} e -1024 c^{3} d^{6}\right )}{45045 e^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(1/2)*(c*x^2+b*x)^3,x)

[Out]

-2/45045*(e*x+d)^(3/2)*(-3003*c^3*e^6*x^6-10395*b*c^2*e^6*x^5+2772*c^3*d*e^5*x^5-12285*b^2*c*e^6*x^4+9450*b*c^
2*d*e^5*x^4-2520*c^3*d^2*e^4*x^4-5005*b^3*e^6*x^3+10920*b^2*c*d*e^5*x^3-8400*b*c^2*d^2*e^4*x^3+2240*c^3*d^3*e^
3*x^3+4290*b^3*d*e^5*x^2-9360*b^2*c*d^2*e^4*x^2+7200*b*c^2*d^3*e^3*x^2-1920*c^3*d^4*e^2*x^2-3432*b^3*d^2*e^4*x
+7488*b^2*c*d^3*e^3*x-5760*b*c^2*d^4*e^2*x+1536*c^3*d^5*e*x+2288*b^3*d^3*e^3-4992*b^2*c*d^4*e^2+3840*b*c^2*d^5
*e-1024*c^3*d^6)/e^7

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maxima [A]  time = 1.40, size = 271, normalized size = 1.09 \[ \frac {2 \, {\left (3003 \, {\left (e x + d\right )}^{\frac {15}{2}} c^{3} - 10395 \, {\left (2 \, c^{3} d - b c^{2} e\right )} {\left (e x + d\right )}^{\frac {13}{2}} + 12285 \, {\left (5 \, c^{3} d^{2} - 5 \, b c^{2} d e + b^{2} c e^{2}\right )} {\left (e x + d\right )}^{\frac {11}{2}} - 5005 \, {\left (20 \, c^{3} d^{3} - 30 \, b c^{2} d^{2} e + 12 \, b^{2} c d e^{2} - b^{3} e^{3}\right )} {\left (e x + d\right )}^{\frac {9}{2}} + 19305 \, {\left (5 \, c^{3} d^{4} - 10 \, b c^{2} d^{3} e + 6 \, b^{2} c d^{2} e^{2} - b^{3} d e^{3}\right )} {\left (e x + d\right )}^{\frac {7}{2}} - 27027 \, {\left (2 \, c^{3} d^{5} - 5 \, b c^{2} d^{4} e + 4 \, b^{2} c d^{3} e^{2} - b^{3} d^{2} e^{3}\right )} {\left (e x + d\right )}^{\frac {5}{2}} + 15015 \, {\left (c^{3} d^{6} - 3 \, b c^{2} d^{5} e + 3 \, b^{2} c d^{4} e^{2} - b^{3} d^{3} e^{3}\right )} {\left (e x + d\right )}^{\frac {3}{2}}\right )}}{45045 \, e^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(1/2)*(c*x^2+b*x)^3,x, algorithm="maxima")

[Out]

2/45045*(3003*(e*x + d)^(15/2)*c^3 - 10395*(2*c^3*d - b*c^2*e)*(e*x + d)^(13/2) + 12285*(5*c^3*d^2 - 5*b*c^2*d
*e + b^2*c*e^2)*(e*x + d)^(11/2) - 5005*(20*c^3*d^3 - 30*b*c^2*d^2*e + 12*b^2*c*d*e^2 - b^3*e^3)*(e*x + d)^(9/
2) + 19305*(5*c^3*d^4 - 10*b*c^2*d^3*e + 6*b^2*c*d^2*e^2 - b^3*d*e^3)*(e*x + d)^(7/2) - 27027*(2*c^3*d^5 - 5*b
*c^2*d^4*e + 4*b^2*c*d^3*e^2 - b^3*d^2*e^3)*(e*x + d)^(5/2) + 15015*(c^3*d^6 - 3*b*c^2*d^5*e + 3*b^2*c*d^4*e^2
- b^3*d^3*e^3)*(e*x + d)^(3/2))/e^7

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mupad [B]  time = 0.21, size = 239, normalized size = 0.96 \[ \frac {{\left (d+e\,x\right )}^{9/2}\,\left (2\,b^3\,e^3-24\,b^2\,c\,d\,e^2+60\,b\,c^2\,d^2\,e-40\,c^3\,d^3\right )}{9\,e^7}+\frac {2\,c^3\,{\left (d+e\,x\right )}^{15/2}}{15\,e^7}-\frac {\left (12\,c^3\,d-6\,b\,c^2\,e\right )\,{\left (d+e\,x\right )}^{13/2}}{13\,e^7}+\frac {{\left (d+e\,x\right )}^{11/2}\,\left (6\,b^2\,c\,e^2-30\,b\,c^2\,d\,e+30\,c^3\,d^2\right )}{11\,e^7}+\frac {{\left (d+e\,x\right )}^{7/2}\,\left (-6\,b^3\,d\,e^3+36\,b^2\,c\,d^2\,e^2-60\,b\,c^2\,d^3\,e+30\,c^3\,d^4\right )}{7\,e^7}-\frac {2\,d^3\,{\left (b\,e-c\,d\right )}^3\,{\left (d+e\,x\right )}^{3/2}}{3\,e^7}+\frac {6\,d^2\,{\left (b\,e-c\,d\right )}^2\,\left (b\,e-2\,c\,d\right )\,{\left (d+e\,x\right )}^{5/2}}{5\,e^7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x + c*x^2)^3*(d + e*x)^(1/2),x)

[Out]

((d + e*x)^(9/2)*(2*b^3*e^3 - 40*c^3*d^3 + 60*b*c^2*d^2*e - 24*b^2*c*d*e^2))/(9*e^7) + (2*c^3*(d + e*x)^(15/2)
)/(15*e^7) - ((12*c^3*d - 6*b*c^2*e)*(d + e*x)^(13/2))/(13*e^7) + ((d + e*x)^(11/2)*(30*c^3*d^2 + 6*b^2*c*e^2
- 30*b*c^2*d*e))/(11*e^7) + ((d + e*x)^(7/2)*(30*c^3*d^4 - 6*b^3*d*e^3 + 36*b^2*c*d^2*e^2 - 60*b*c^2*d^3*e))/(
7*e^7) - (2*d^3*(b*e - c*d)^3*(d + e*x)^(3/2))/(3*e^7) + (6*d^2*(b*e - c*d)^2*(b*e - 2*c*d)*(d + e*x)^(5/2))/(
5*e^7)

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sympy [A]  time = 5.88, size = 326, normalized size = 1.31 \[ \frac {2 \left (\frac {c^{3} \left (d + e x\right )^{\frac {15}{2}}}{15 e^{6}} + \frac {\left (d + e x\right )^{\frac {13}{2}} \left (3 b c^{2} e - 6 c^{3} d\right )}{13 e^{6}} + \frac {\left (d + e x\right )^{\frac {11}{2}} \left (3 b^{2} c e^{2} - 15 b c^{2} d e + 15 c^{3} d^{2}\right )}{11 e^{6}} + \frac {\left (d + e x\right )^{\frac {9}{2}} \left (b^{3} e^{3} - 12 b^{2} c d e^{2} + 30 b c^{2} d^{2} e - 20 c^{3} d^{3}\right )}{9 e^{6}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \left (- 3 b^{3} d e^{3} + 18 b^{2} c d^{2} e^{2} - 30 b c^{2} d^{3} e + 15 c^{3} d^{4}\right )}{7 e^{6}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \left (3 b^{3} d^{2} e^{3} - 12 b^{2} c d^{3} e^{2} + 15 b c^{2} d^{4} e - 6 c^{3} d^{5}\right )}{5 e^{6}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \left (- b^{3} d^{3} e^{3} + 3 b^{2} c d^{4} e^{2} - 3 b c^{2} d^{5} e + c^{3} d^{6}\right )}{3 e^{6}}\right )}{e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**(1/2)*(c*x**2+b*x)**3,x)

[Out]

2*(c**3*(d + e*x)**(15/2)/(15*e**6) + (d + e*x)**(13/2)*(3*b*c**2*e - 6*c**3*d)/(13*e**6) + (d + e*x)**(11/2)*
(3*b**2*c*e**2 - 15*b*c**2*d*e + 15*c**3*d**2)/(11*e**6) + (d + e*x)**(9/2)*(b**3*e**3 - 12*b**2*c*d*e**2 + 30
*b*c**2*d**2*e - 20*c**3*d**3)/(9*e**6) + (d + e*x)**(7/2)*(-3*b**3*d*e**3 + 18*b**2*c*d**2*e**2 - 30*b*c**2*d
**3*e + 15*c**3*d**4)/(7*e**6) + (d + e*x)**(5/2)*(3*b**3*d**2*e**3 - 12*b**2*c*d**3*e**2 + 15*b*c**2*d**4*e -
6*c**3*d**5)/(5*e**6) + (d + e*x)**(3/2)*(-b**3*d**3*e**3 + 3*b**2*c*d**4*e**2 - 3*b*c**2*d**5*e + c**3*d**6)
/(3*e**6))/e

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