3.4.47 \(\int (d+e x)^{3/2} (b x+c x^2)^2 \, dx\) [347]

Optimal. Leaf size=147 \[ \frac {2 d^2 (c d-b e)^2 (d+e x)^{5/2}}{5 e^5}-\frac {4 d (c d-b e) (2 c d-b e) (d+e x)^{7/2}}{7 e^5}+\frac {2 \left (6 c^2 d^2-6 b c d e+b^2 e^2\right ) (d+e x)^{9/2}}{9 e^5}-\frac {4 c (2 c d-b e) (d+e x)^{11/2}}{11 e^5}+\frac {2 c^2 (d+e x)^{13/2}}{13 e^5} \]

[Out]

2/5*d^2*(-b*e+c*d)^2*(e*x+d)^(5/2)/e^5-4/7*d*(-b*e+c*d)*(-b*e+2*c*d)*(e*x+d)^(7/2)/e^5+2/9*(b^2*e^2-6*b*c*d*e+
6*c^2*d^2)*(e*x+d)^(9/2)/e^5-4/11*c*(-b*e+2*c*d)*(e*x+d)^(11/2)/e^5+2/13*c^2*(e*x+d)^(13/2)/e^5

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Rubi [A]
time = 0.04, antiderivative size = 147, 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 = {712} \begin {gather*} \frac {2 (d+e x)^{9/2} \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )}{9 e^5}+\frac {2 d^2 (d+e x)^{5/2} (c d-b e)^2}{5 e^5}-\frac {4 c (d+e x)^{11/2} (2 c d-b e)}{11 e^5}-\frac {4 d (d+e x)^{7/2} (c d-b e) (2 c d-b e)}{7 e^5}+\frac {2 c^2 (d+e x)^{13/2}}{13 e^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^(3/2)*(b*x + c*x^2)^2,x]

[Out]

(2*d^2*(c*d - b*e)^2*(d + e*x)^(5/2))/(5*e^5) - (4*d*(c*d - b*e)*(2*c*d - b*e)*(d + e*x)^(7/2))/(7*e^5) + (2*(
6*c^2*d^2 - 6*b*c*d*e + b^2*e^2)*(d + e*x)^(9/2))/(9*e^5) - (4*c*(2*c*d - b*e)*(d + e*x)^(11/2))/(11*e^5) + (2
*c^2*(d + e*x)^(13/2))/(13*e^5)

Rule 712

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

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Mathematica [A]
time = 0.07, size = 125, normalized size = 0.85 \begin {gather*} \frac {2 (d+e x)^{5/2} \left (143 b^2 e^2 \left (8 d^2-20 d e x+35 e^2 x^2\right )+78 b c e \left (-16 d^3+40 d^2 e x-70 d e^2 x^2+105 e^3 x^3\right )+3 c^2 \left (128 d^4-320 d^3 e x+560 d^2 e^2 x^2-840 d e^3 x^3+1155 e^4 x^4\right )\right )}{45045 e^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(d + e*x)^(5/2)*(143*b^2*e^2*(8*d^2 - 20*d*e*x + 35*e^2*x^2) + 78*b*c*e*(-16*d^3 + 40*d^2*e*x - 70*d*e^2*x^
2 + 105*e^3*x^3) + 3*c^2*(128*d^4 - 320*d^3*e*x + 560*d^2*e^2*x^2 - 840*d*e^3*x^3 + 1155*e^4*x^4)))/(45045*e^5
)

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Maple [A]
time = 0.43, size = 144, normalized size = 0.98

method	result	size
gosper	\(\frac {2 \left (e x +d \right )^{\frac {5}{2}} \left (3465 c^{2} x^{4} e^{4}+8190 b c \,e^{4} x^{3}-2520 c^{2} d \,e^{3} x^{3}+5005 b^{2} e^{4} x^{2}-5460 b c d \,e^{3} x^{2}+1680 c^{2} d^{2} e^{2} x^{2}-2860 b^{2} d \,e^{3} x +3120 b c \,d^{2} e^{2} x -960 c^{2} d^{3} e x +1144 d^{2} e^{2} b^{2}-1248 b c \,d^{3} e +384 c^{2} d^{4}\right )}{45045 e^{5}}\)	\(141\)
derivativedivides	\(\frac {\frac {2 c^{2} \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {2 \left (-2 c^{2} d +2 \left (b e -c d \right ) c \right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (d^{2} c^{2}-4 d \left (b e -c d \right ) c +\left (b e -c d \right )^{2}\right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (2 d^{2} \left (b e -c d \right ) c -2 d \left (b e -c d \right )^{2}\right ) \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 d^{2} \left (b e -c d \right )^{2} \left (e x +d \right )^{\frac {5}{2}}}{5}}{e^{5}}\)	\(144\)
default	\(\frac {\frac {2 c^{2} \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {2 \left (-2 c^{2} d +2 \left (b e -c d \right ) c \right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (d^{2} c^{2}-4 d \left (b e -c d \right ) c +\left (b e -c d \right )^{2}\right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (2 d^{2} \left (b e -c d \right ) c -2 d \left (b e -c d \right )^{2}\right ) \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 d^{2} \left (b e -c d \right )^{2} \left (e x +d \right )^{\frac {5}{2}}}{5}}{e^{5}}\)	\(144\)
trager	\(\frac {2 \left (3465 c^{2} e^{6} x^{6}+8190 b c \,e^{6} x^{5}+4410 c^{2} d \,e^{5} x^{5}+5005 b^{2} e^{6} x^{4}+10920 b c d \,e^{5} x^{4}+105 c^{2} d^{2} e^{4} x^{4}+7150 b^{2} d \,e^{5} x^{3}+390 b c \,d^{2} e^{4} x^{3}-120 c^{2} d^{3} e^{3} x^{3}+429 b^{2} d^{2} e^{4} x^{2}-468 b c \,d^{3} e^{3} x^{2}+144 c^{2} d^{4} e^{2} x^{2}-572 b^{2} d^{3} e^{3} x +624 b c \,d^{4} e^{2} x -192 c^{2} d^{5} e x +1144 b^{2} d^{4} e^{2}-1248 b c \,d^{5} e +384 c^{2} d^{6}\right ) \sqrt {e x +d}}{45045 e^{5}}\)	\(223\)
risch	\(\frac {2 \left (3465 c^{2} e^{6} x^{6}+8190 b c \,e^{6} x^{5}+4410 c^{2} d \,e^{5} x^{5}+5005 b^{2} e^{6} x^{4}+10920 b c d \,e^{5} x^{4}+105 c^{2} d^{2} e^{4} x^{4}+7150 b^{2} d \,e^{5} x^{3}+390 b c \,d^{2} e^{4} x^{3}-120 c^{2} d^{3} e^{3} x^{3}+429 b^{2} d^{2} e^{4} x^{2}-468 b c \,d^{3} e^{3} x^{2}+144 c^{2} d^{4} e^{2} x^{2}-572 b^{2} d^{3} e^{3} x +624 b c \,d^{4} e^{2} x -192 c^{2} d^{5} e x +1144 b^{2} d^{4} e^{2}-1248 b c \,d^{5} e +384 c^{2} d^{6}\right ) \sqrt {e x +d}}{45045 e^{5}}\)	\(223\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(3/2)*(c*x^2+b*x)^2,x,method=_RETURNVERBOSE)

[Out]

2/e^5*(1/13*c^2*(e*x+d)^(13/2)+1/11*(-2*c^2*d+2*(b*e-c*d)*c)*(e*x+d)^(11/2)+1/9*(d^2*c^2-4*d*(b*e-c*d)*c+(b*e-
c*d)^2)*(e*x+d)^(9/2)+1/7*(2*d^2*(b*e-c*d)*c-2*d*(b*e-c*d)^2)*(e*x+d)^(7/2)+1/5*d^2*(b*e-c*d)^2*(e*x+d)^(5/2))

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Maxima [A]
time = 0.28, size = 144, normalized size = 0.98 \begin {gather*} \frac {2}{45045} \, {\left (3465 \, {\left (x e + d\right )}^{\frac {13}{2}} c^{2} - 8190 \, {\left (2 \, c^{2} d - b c e\right )} {\left (x e + d\right )}^{\frac {11}{2}} + 5005 \, {\left (6 \, c^{2} d^{2} - 6 \, b c d e + b^{2} e^{2}\right )} {\left (x e + d\right )}^{\frac {9}{2}} - 12870 \, {\left (2 \, c^{2} d^{3} - 3 \, b c d^{2} e + b^{2} d e^{2}\right )} {\left (x e + d\right )}^{\frac {7}{2}} + 9009 \, {\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2}\right )} {\left (x e + d\right )}^{\frac {5}{2}}\right )} e^{\left (-5\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/45045*(3465*(x*e + d)^(13/2)*c^2 - 8190*(2*c^2*d - b*c*e)*(x*e + d)^(11/2) + 5005*(6*c^2*d^2 - 6*b*c*d*e + b
^2*e^2)*(x*e + d)^(9/2) - 12870*(2*c^2*d^3 - 3*b*c*d^2*e + b^2*d*e^2)*(x*e + d)^(7/2) + 9009*(c^2*d^4 - 2*b*c*
d^3*e + b^2*d^2*e^2)*(x*e + d)^(5/2))*e^(-5)

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Fricas [A]
time = 1.52, size = 205, normalized size = 1.39 \begin {gather*} \frac {2}{45045} \, {\left (384 \, c^{2} d^{6} + 35 \, {\left (99 \, c^{2} x^{6} + 234 \, b c x^{5} + 143 \, b^{2} x^{4}\right )} e^{6} + 10 \, {\left (441 \, c^{2} d x^{5} + 1092 \, b c d x^{4} + 715 \, b^{2} d x^{3}\right )} e^{5} + 3 \, {\left (35 \, c^{2} d^{2} x^{4} + 130 \, b c d^{2} x^{3} + 143 \, b^{2} d^{2} x^{2}\right )} e^{4} - 4 \, {\left (30 \, c^{2} d^{3} x^{3} + 117 \, b c d^{3} x^{2} + 143 \, b^{2} d^{3} x\right )} e^{3} + 8 \, {\left (18 \, c^{2} d^{4} x^{2} + 78 \, b c d^{4} x + 143 \, b^{2} d^{4}\right )} e^{2} - 96 \, {\left (2 \, c^{2} d^{5} x + 13 \, b c d^{5}\right )} e\right )} \sqrt {x e + d} e^{\left (-5\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/45045*(384*c^2*d^6 + 35*(99*c^2*x^6 + 234*b*c*x^5 + 143*b^2*x^4)*e^6 + 10*(441*c^2*d*x^5 + 1092*b*c*d*x^4 +
715*b^2*d*x^3)*e^5 + 3*(35*c^2*d^2*x^4 + 130*b*c*d^2*x^3 + 143*b^2*d^2*x^2)*e^4 - 4*(30*c^2*d^3*x^3 + 117*b*c*
d^3*x^2 + 143*b^2*d^3*x)*e^3 + 8*(18*c^2*d^4*x^2 + 78*b*c*d^4*x + 143*b^2*d^4)*e^2 - 96*(2*c^2*d^5*x + 13*b*c*
d^5)*e)*sqrt(x*e + d)*e^(-5)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 413 vs. \(2 (144) = 288\).
time = 7.69, size = 413, normalized size = 2.81 \begin {gather*} \frac {2 b^{2} d \left (\frac {d^{2} \left (d + e x\right )^{\frac {3}{2}}}{3} - \frac {2 d \left (d + e x\right )^{\frac {5}{2}}}{5} + \frac {\left (d + e x\right )^{\frac {7}{2}}}{7}\right )}{e^{3}} + \frac {2 b^{2} \left (- \frac {d^{3} \left (d + e x\right )^{\frac {3}{2}}}{3} + \frac {3 d^{2} \left (d + e x\right )^{\frac {5}{2}}}{5} - \frac {3 d \left (d + e x\right )^{\frac {7}{2}}}{7} + \frac {\left (d + e x\right )^{\frac {9}{2}}}{9}\right )}{e^{3}} + \frac {4 b c d \left (- \frac {d^{3} \left (d + e x\right )^{\frac {3}{2}}}{3} + \frac {3 d^{2} \left (d + e x\right )^{\frac {5}{2}}}{5} - \frac {3 d \left (d + e x\right )^{\frac {7}{2}}}{7} + \frac {\left (d + e x\right )^{\frac {9}{2}}}{9}\right )}{e^{4}} + \frac {4 b c \left (\frac {d^{4} \left (d + e x\right )^{\frac {3}{2}}}{3} - \frac {4 d^{3} \left (d + e x\right )^{\frac {5}{2}}}{5} + \frac {6 d^{2} \left (d + e x\right )^{\frac {7}{2}}}{7} - \frac {4 d \left (d + e x\right )^{\frac {9}{2}}}{9} + \frac {\left (d + e x\right )^{\frac {11}{2}}}{11}\right )}{e^{4}} + \frac {2 c^{2} d \left (\frac {d^{4} \left (d + e x\right )^{\frac {3}{2}}}{3} - \frac {4 d^{3} \left (d + e x\right )^{\frac {5}{2}}}{5} + \frac {6 d^{2} \left (d + e x\right )^{\frac {7}{2}}}{7} - \frac {4 d \left (d + e x\right )^{\frac {9}{2}}}{9} + \frac {\left (d + e x\right )^{\frac {11}{2}}}{11}\right )}{e^{5}} + \frac {2 c^{2} \left (- \frac {d^{5} \left (d + e x\right )^{\frac {3}{2}}}{3} + d^{4} \left (d + e x\right )^{\frac {5}{2}} - \frac {10 d^{3} \left (d + e x\right )^{\frac {7}{2}}}{7} + \frac {10 d^{2} \left (d + e x\right )^{\frac {9}{2}}}{9} - \frac {5 d \left (d + e x\right )^{\frac {11}{2}}}{11} + \frac {\left (d + e x\right )^{\frac {13}{2}}}{13}\right )}{e^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*b**2*d*(d**2*(d + e*x)**(3/2)/3 - 2*d*(d + e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e**3 + 2*b**2*(-d**3*(d + e*x
)**(3/2)/3 + 3*d**2*(d + e*x)**(5/2)/5 - 3*d*(d + e*x)**(7/2)/7 + (d + e*x)**(9/2)/9)/e**3 + 4*b*c*d*(-d**3*(d
+ e*x)**(3/2)/3 + 3*d**2*(d + e*x)**(5/2)/5 - 3*d*(d + e*x)**(7/2)/7 + (d + e*x)**(9/2)/9)/e**4 + 4*b*c*(d**4
*(d + e*x)**(3/2)/3 - 4*d**3*(d + e*x)**(5/2)/5 + 6*d**2*(d + e*x)**(7/2)/7 - 4*d*(d + e*x)**(9/2)/9 + (d + e*
x)**(11/2)/11)/e**4 + 2*c**2*d*(d**4*(d + e*x)**(3/2)/3 - 4*d**3*(d + e*x)**(5/2)/5 + 6*d**2*(d + e*x)**(7/2)/
7 - 4*d*(d + e*x)**(9/2)/9 + (d + e*x)**(11/2)/11)/e**5 + 2*c**2*(-d**5*(d + e*x)**(3/2)/3 + d**4*(d + e*x)**(
5/2) - 10*d**3*(d + e*x)**(7/2)/7 + 10*d**2*(d + e*x)**(9/2)/9 - 5*d*(d + e*x)**(11/2)/11 + (d + e*x)**(13/2)/
13)/e**5

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 626 vs. \(2 (131) = 262\).
time = 1.50, size = 626, normalized size = 4.26 \begin {gather*} \frac {2}{45045} \, {\left (3003 \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} - 10 \, {\left (x e + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {x e + d} d^{2}\right )} b^{2} d^{2} e^{\left (-2\right )} + 2574 \, {\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 c d^{2} e^{\left (-3\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 )} c^{2} d^{2} e^{\left (-4\right )} + 2574 \, {\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^{2} d e^{\left (-2\right )} + 572 \, {\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 c d e^{\left (-3\right )} + 130 \, {\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 )} c^{2} d e^{\left (-4\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^{2} e^{\left (-2\right )} + 130 \, {\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 e^{\left (-3\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^{2} e^{\left (-4\right )}\right )} e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/45045*(3003*(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^(-2) + 2574*(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^(-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)*c^2*d^2*e^(-4) + 2574*(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^(-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*c*d*e^(-3) + 130*(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^(-4)
+ 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*sq
rt(x*e + d)*d^4)*b^2*e^(-2) + 130*(63*(x*e + d)^(11/2) - 385*(x*e + d)^(9/2)*d + 990*(x*e + d)^(7/2)*d^2 - 138
6*(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^(-3) + 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^2*e^(-4))*e^(-1)

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Mupad [B]
time = 0.04, size = 138, normalized size = 0.94 \begin {gather*} \frac {2\,c^2\,{\left (d+e\,x\right )}^{13/2}}{13\,e^5}-\frac {{\left (d+e\,x\right )}^{7/2}\,\left (4\,b^2\,d\,e^2-12\,b\,c\,d^2\,e+8\,c^2\,d^3\right )}{7\,e^5}+\frac {{\left (d+e\,x\right )}^{9/2}\,\left (2\,b^2\,e^2-12\,b\,c\,d\,e+12\,c^2\,d^2\right )}{9\,e^5}-\frac {\left (8\,c^2\,d-4\,b\,c\,e\right )\,{\left (d+e\,x\right )}^{11/2}}{11\,e^5}+\frac {2\,d^2\,{\left (b\,e-c\,d\right )}^2\,{\left (d+e\,x\right )}^{5/2}}{5\,e^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*c^2*(d + e*x)^(13/2))/(13*e^5) - ((d + e*x)^(7/2)*(8*c^2*d^3 + 4*b^2*d*e^2 - 12*b*c*d^2*e))/(7*e^5) + ((d +
e*x)^(9/2)*(2*b^2*e^2 + 12*c^2*d^2 - 12*b*c*d*e))/(9*e^5) - ((8*c^2*d - 4*b*c*e)*(d + e*x)^(11/2))/(11*e^5) +
(2*d^2*(b*e - c*d)^2*(d + e*x)^(5/2))/(5*e^5)

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