3.4.59 \(\int \frac {(b x+c x^2)^3}{(d+e x)^{5/2}} \, dx\) [359]

Optimal. Leaf size=244 \[ -\frac {2 d^3 (c d-b e)^3}{3 e^7 (d+e x)^{3/2}}+\frac {6 d^2 (c d-b e)^2 (2 c d-b e)}{e^7 \sqrt {d+e x}}+\frac {6 d (c d-b e) \left (5 c^2 d^2-5 b c d e+b^2 e^2\right ) \sqrt {d+e x}}{e^7}-\frac {2 (2 c d-b e) \left (10 c^2 d^2-10 b c d e+b^2 e^2\right ) (d+e x)^{3/2}}{3 e^7}+\frac {6 c \left (5 c^2 d^2-5 b c d e+b^2 e^2\right ) (d+e x)^{5/2}}{5 e^7}-\frac {6 c^2 (2 c d-b e) (d+e x)^{7/2}}{7 e^7}+\frac {2 c^3 (d+e x)^{9/2}}{9 e^7} \]

[Out]

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

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Rubi [A]
time = 0.07, antiderivative size = 244, 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 {6 c (d+e x)^{5/2} \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{5 e^7}-\frac {2 (d+e x)^{3/2} (2 c d-b e) \left (b^2 e^2-10 b c d e+10 c^2 d^2\right )}{3 e^7}+\frac {6 d \sqrt {d+e x} (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{e^7}-\frac {6 c^2 (d+e x)^{7/2} (2 c d-b e)}{7 e^7}-\frac {2 d^3 (c d-b e)^3}{3 e^7 (d+e x)^{3/2}}+\frac {6 d^2 (c d-b e)^2 (2 c d-b e)}{e^7 \sqrt {d+e x}}+\frac {2 c^3 (d+e x)^{9/2}}{9 e^7} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*d^3*(c*d - b*e)^3)/(3*e^7*(d + e*x)^(3/2)) + (6*d^2*(c*d - b*e)^2*(2*c*d - b*e))/(e^7*Sqrt[d + e*x]) + (6*
d*(c*d - b*e)*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2)*Sqrt[d + e*x])/e^7 - (2*(2*c*d - b*e)*(10*c^2*d^2 - 10*b*c*d*e
+ b^2*e^2)*(d + e*x)^(3/2))/(3*e^7) + (6*c*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2)*(d + e*x)^(5/2))/(5*e^7) - (6*c^
2*(2*c*d - b*e)*(d + e*x)^(7/2))/(7*e^7) + (2*c^3*(d + e*x)^(9/2))/(9*e^7)

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

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Mathematica [A]
time = 0.11, size = 231, normalized size = 0.95 \begin {gather*} \frac {2 \left (105 b^3 e^3 \left (-16 d^3-24 d^2 e x-6 d e^2 x^2+e^3 x^3\right )+63 b^2 c e^2 \left (128 d^4+192 d^3 e x+48 d^2 e^2 x^2-8 d e^3 x^3+3 e^4 x^4\right )-45 b c^2 e \left (256 d^5+384 d^4 e x+96 d^3 e^2 x^2-16 d^2 e^3 x^3+6 d e^4 x^4-3 e^5 x^5\right )+5 c^3 \left (1024 d^6+1536 d^5 e x+384 d^4 e^2 x^2-64 d^3 e^3 x^3+24 d^2 e^4 x^4-12 d e^5 x^5+7 e^6 x^6\right )\right )}{315 e^7 (d+e x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(105*b^3*e^3*(-16*d^3 - 24*d^2*e*x - 6*d*e^2*x^2 + e^3*x^3) + 63*b^2*c*e^2*(128*d^4 + 192*d^3*e*x + 48*d^2*
e^2*x^2 - 8*d*e^3*x^3 + 3*e^4*x^4) - 45*b*c^2*e*(256*d^5 + 384*d^4*e*x + 96*d^3*e^2*x^2 - 16*d^2*e^3*x^3 + 6*d
*e^4*x^4 - 3*e^5*x^5) + 5*c^3*(1024*d^6 + 1536*d^5*e*x + 384*d^4*e^2*x^2 - 64*d^3*e^3*x^3 + 24*d^2*e^4*x^4 - 1
2*d*e^5*x^5 + 7*e^6*x^6)))/(315*e^7*(d + e*x)^(3/2))

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Maple [A]
time = 0.44, size = 319, normalized size = 1.31

method	result	size
risch	\(-\frac {2 \left (-35 c^{3} x^{4} e^{4}-135 b \,c^{2} e^{4} x^{3}+130 c^{3} d \,e^{3} x^{3}-189 b^{2} c \,e^{4} x^{2}+540 b \,c^{2} d \,e^{3} x^{2}-345 c^{3} d^{2} e^{2} x^{2}-105 b^{3} e^{4} x +882 b^{2} c d \,e^{3} x -1665 b \,c^{2} d^{2} e^{2} x +880 c^{3} d^{3} e x +840 b^{3} d \,e^{3}-4599 b^{2} c \,d^{2} e^{2}+7110 e \,d^{3} b \,c^{2}-3335 d^{4} c^{3}\right ) \sqrt {e x +d}}{315 e^{7}}-\frac {2 \left (9 b \,e^{2} x -18 c d e x +8 b d e -17 c \,d^{2}\right ) d^{2} \left (b^{2} e^{2}-2 b c d e +d^{2} c^{2}\right )}{3 e^{7} \left (e x +d \right )^{\frac {3}{2}}}\)	\(232\)
gosper	\(-\frac {2 \left (-35 c^{3} x^{6} e^{6}-135 b \,c^{2} e^{6} x^{5}+60 c^{3} d \,e^{5} x^{5}-189 b^{2} c \,e^{6} x^{4}+270 b \,c^{2} d \,e^{5} x^{4}-120 c^{3} d^{2} e^{4} x^{4}-105 b^{3} e^{6} x^{3}+504 b^{2} c d \,e^{5} x^{3}-720 b \,c^{2} d^{2} e^{4} x^{3}+320 c^{3} d^{3} e^{3} x^{3}+630 b^{3} d \,e^{5} x^{2}-3024 b^{2} c \,d^{2} e^{4} x^{2}+4320 b \,c^{2} d^{3} e^{3} x^{2}-1920 c^{3} d^{4} e^{2} x^{2}+2520 b^{3} d^{2} e^{4} x -12096 b^{2} c \,d^{3} e^{3} x +17280 b \,c^{2} d^{4} e^{2} x -7680 c^{3} d^{5} e x +1680 b^{3} d^{3} e^{3}-8064 b^{2} c \,d^{4} e^{2}+11520 b \,c^{2} d^{5} e -5120 c^{3} d^{6}\right )}{315 \left (e x +d \right )^{\frac {3}{2}} e^{7}}\)	\(286\)
trager	\(-\frac {2 \left (-35 c^{3} x^{6} e^{6}-135 b \,c^{2} e^{6} x^{5}+60 c^{3} d \,e^{5} x^{5}-189 b^{2} c \,e^{6} x^{4}+270 b \,c^{2} d \,e^{5} x^{4}-120 c^{3} d^{2} e^{4} x^{4}-105 b^{3} e^{6} x^{3}+504 b^{2} c d \,e^{5} x^{3}-720 b \,c^{2} d^{2} e^{4} x^{3}+320 c^{3} d^{3} e^{3} x^{3}+630 b^{3} d \,e^{5} x^{2}-3024 b^{2} c \,d^{2} e^{4} x^{2}+4320 b \,c^{2} d^{3} e^{3} x^{2}-1920 c^{3} d^{4} e^{2} x^{2}+2520 b^{3} d^{2} e^{4} x -12096 b^{2} c \,d^{3} e^{3} x +17280 b \,c^{2} d^{4} e^{2} x -7680 c^{3} d^{5} e x +1680 b^{3} d^{3} e^{3}-8064 b^{2} c \,d^{4} e^{2}+11520 b \,c^{2} d^{5} e -5120 c^{3} d^{6}\right )}{315 \left (e x +d \right )^{\frac {3}{2}} e^{7}}\)	\(286\)
derivativedivides	\(\frac {\frac {2 c^{3} \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {6 b \,c^{2} e \left (e x +d \right )^{\frac {7}{2}}}{7}-\frac {12 c^{3} d \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {6 b^{2} c \,e^{2} \left (e x +d \right )^{\frac {5}{2}}}{5}-6 b \,c^{2} d e \left (e x +d \right )^{\frac {5}{2}}+6 c^{3} d^{2} \left (e x +d \right )^{\frac {5}{2}}+\frac {2 b^{3} e^{3} \left (e x +d \right )^{\frac {3}{2}}}{3}-8 b^{2} c d \,e^{2} \left (e x +d \right )^{\frac {3}{2}}+20 b \,c^{2} d^{2} e \left (e x +d \right )^{\frac {3}{2}}-\frac {40 c^{3} d^{3} \left (e x +d \right )^{\frac {3}{2}}}{3}-6 b^{3} d \,e^{3} \sqrt {e x +d}+36 b^{2} c \,d^{2} e^{2} \sqrt {e x +d}-60 b \,c^{2} d^{3} e \sqrt {e x +d}+30 c^{3} d^{4} \sqrt {e x +d}-\frac {6 d^{2} \left (b^{3} e^{3}-4 b^{2} d \,e^{2} c +5 b \,c^{2} d^{2} e -2 c^{3} d^{3}\right )}{\sqrt {e x +d}}+\frac {2 d^{3} \left (b^{3} e^{3}-3 b^{2} d \,e^{2} c +3 b \,c^{2} d^{2} e -c^{3} d^{3}\right )}{3 \left (e x +d \right )^{\frac {3}{2}}}}{e^{7}}\)	\(319\)
default	\(\frac {\frac {2 c^{3} \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {6 b \,c^{2} e \left (e x +d \right )^{\frac {7}{2}}}{7}-\frac {12 c^{3} d \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {6 b^{2} c \,e^{2} \left (e x +d \right )^{\frac {5}{2}}}{5}-6 b \,c^{2} d e \left (e x +d \right )^{\frac {5}{2}}+6 c^{3} d^{2} \left (e x +d \right )^{\frac {5}{2}}+\frac {2 b^{3} e^{3} \left (e x +d \right )^{\frac {3}{2}}}{3}-8 b^{2} c d \,e^{2} \left (e x +d \right )^{\frac {3}{2}}+20 b \,c^{2} d^{2} e \left (e x +d \right )^{\frac {3}{2}}-\frac {40 c^{3} d^{3} \left (e x +d \right )^{\frac {3}{2}}}{3}-6 b^{3} d \,e^{3} \sqrt {e x +d}+36 b^{2} c \,d^{2} e^{2} \sqrt {e x +d}-60 b \,c^{2} d^{3} e \sqrt {e x +d}+30 c^{3} d^{4} \sqrt {e x +d}-\frac {6 d^{2} \left (b^{3} e^{3}-4 b^{2} d \,e^{2} c +5 b \,c^{2} d^{2} e -2 c^{3} d^{3}\right )}{\sqrt {e x +d}}+\frac {2 d^{3} \left (b^{3} e^{3}-3 b^{2} d \,e^{2} c +3 b \,c^{2} d^{2} e -c^{3} d^{3}\right )}{3 \left (e x +d \right )^{\frac {3}{2}}}}{e^{7}}\)	\(319\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+b*x)^3/(e*x+d)^(5/2),x,method=_RETURNVERBOSE)

[Out]

2/e^7*(1/9*c^3*(e*x+d)^(9/2)+3/7*b*c^2*e*(e*x+d)^(7/2)-6/7*c^3*d*(e*x+d)^(7/2)+3/5*b^2*c*e^2*(e*x+d)^(5/2)-3*b
*c^2*d*e*(e*x+d)^(5/2)+3*c^3*d^2*(e*x+d)^(5/2)+1/3*b^3*e^3*(e*x+d)^(3/2)-4*b^2*c*d*e^2*(e*x+d)^(3/2)+10*b*c^2*
d^2*e*(e*x+d)^(3/2)-20/3*c^3*d^3*(e*x+d)^(3/2)-3*b^3*d*e^3*(e*x+d)^(1/2)+18*b^2*c*d^2*e^2*(e*x+d)^(1/2)-30*b*c
^2*d^3*e*(e*x+d)^(1/2)+15*c^3*d^4*(e*x+d)^(1/2)-3*d^2*(b^3*e^3-4*b^2*c*d*e^2+5*b*c^2*d^2*e-2*c^3*d^3)/(e*x+d)^
(1/2)+1/3*d^3*(b^3*e^3-3*b^2*c*d*e^2+3*b*c^2*d^2*e-c^3*d^3)/(e*x+d)^(3/2))

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Maxima [A]
time = 0.28, size = 278, normalized size = 1.14 \begin {gather*} \frac {2}{315} \, {\left ({\left (35 \, {\left (x e + d\right )}^{\frac {9}{2}} c^{3} - 135 \, {\left (2 \, c^{3} d - b c^{2} e\right )} {\left (x e + d\right )}^{\frac {7}{2}} + 189 \, {\left (5 \, c^{3} d^{2} - 5 \, b c^{2} d e + b^{2} c e^{2}\right )} {\left (x e + d\right )}^{\frac {5}{2}} - 105 \, {\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 (x e + d\right )}^{\frac {3}{2}} + 945 \, {\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 )} \sqrt {x e + d}\right )} e^{\left (-6\right )} - \frac {105 \, {\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} - 9 \, {\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 (x e + d\right )}\right )} e^{\left (-6\right )}}{{\left (x e + d\right )}^{\frac {3}{2}}}\right )} e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/315*((35*(x*e + d)^(9/2)*c^3 - 135*(2*c^3*d - b*c^2*e)*(x*e + d)^(7/2) + 189*(5*c^3*d^2 - 5*b*c^2*d*e + b^2*
c*e^2)*(x*e + d)^(5/2) - 105*(20*c^3*d^3 - 30*b*c^2*d^2*e + 12*b^2*c*d*e^2 - b^3*e^3)*(x*e + d)^(3/2) + 945*(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)*sqrt(x*e + d))*e^(-6) - 105*(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 - 9*(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)*(x*e + d))*e^(
-6)/(x*e + d)^(3/2))*e^(-1)

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Fricas [A]
time = 1.57, size = 274, normalized size = 1.12 \begin {gather*} \frac {2 \, {\left (5120 \, c^{3} d^{6} + {\left (35 \, c^{3} x^{6} + 135 \, b c^{2} x^{5} + 189 \, b^{2} c x^{4} + 105 \, b^{3} x^{3}\right )} e^{6} - 6 \, {\left (10 \, c^{3} d x^{5} + 45 \, b c^{2} d x^{4} + 84 \, b^{2} c d x^{3} + 105 \, b^{3} d x^{2}\right )} e^{5} + 24 \, {\left (5 \, c^{3} d^{2} x^{4} + 30 \, b c^{2} d^{2} x^{3} + 126 \, b^{2} c d^{2} x^{2} - 105 \, b^{3} d^{2} x\right )} e^{4} - 16 \, {\left (20 \, c^{3} d^{3} x^{3} + 270 \, b c^{2} d^{3} x^{2} - 756 \, b^{2} c d^{3} x + 105 \, b^{3} d^{3}\right )} e^{3} + 384 \, {\left (5 \, c^{3} d^{4} x^{2} - 45 \, b c^{2} d^{4} x + 21 \, b^{2} c d^{4}\right )} e^{2} + 3840 \, {\left (2 \, c^{3} d^{5} x - 3 \, b c^{2} d^{5}\right )} e\right )} \sqrt {x e + d}}{315 \, {\left (x^{2} e^{9} + 2 \, d x e^{8} + d^{2} e^{7}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/315*(5120*c^3*d^6 + (35*c^3*x^6 + 135*b*c^2*x^5 + 189*b^2*c*x^4 + 105*b^3*x^3)*e^6 - 6*(10*c^3*d*x^5 + 45*b*
c^2*d*x^4 + 84*b^2*c*d*x^3 + 105*b^3*d*x^2)*e^5 + 24*(5*c^3*d^2*x^4 + 30*b*c^2*d^2*x^3 + 126*b^2*c*d^2*x^2 - 1
05*b^3*d^2*x)*e^4 - 16*(20*c^3*d^3*x^3 + 270*b*c^2*d^3*x^2 - 756*b^2*c*d^3*x + 105*b^3*d^3)*e^3 + 384*(5*c^3*d
^4*x^2 - 45*b*c^2*d^4*x + 21*b^2*c*d^4)*e^2 + 3840*(2*c^3*d^5*x - 3*b*c^2*d^5)*e)*sqrt(x*e + d)/(x^2*e^9 + 2*d
*x*e^8 + d^2*e^7)

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Sympy [A]
time = 17.57, size = 260, normalized size = 1.07 \begin {gather*} \frac {2 c^{3} \left (d + e x\right )^{\frac {9}{2}}}{9 e^{7}} + \frac {2 d^{3} \left (b e - c d\right )^{3}}{3 e^{7} \left (d + e x\right )^{\frac {3}{2}}} - \frac {6 d^{2} \left (b e - 2 c d\right ) \left (b e - c d\right )^{2}}{e^{7} \sqrt {d + e x}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \cdot \left (6 b c^{2} e - 12 c^{3} d\right )}{7 e^{7}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \cdot \left (6 b^{2} c e^{2} - 30 b c^{2} d e + 30 c^{3} d^{2}\right )}{5 e^{7}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \cdot \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 )}{3 e^{7}} + \frac {\sqrt {d + e x} \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 )}{e^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+b*x)**3/(e*x+d)**(5/2),x)

[Out]

2*c**3*(d + e*x)**(9/2)/(9*e**7) + 2*d**3*(b*e - c*d)**3/(3*e**7*(d + e*x)**(3/2)) - 6*d**2*(b*e - 2*c*d)*(b*e
- c*d)**2/(e**7*sqrt(d + e*x)) + (d + e*x)**(7/2)*(6*b*c**2*e - 12*c**3*d)/(7*e**7) + (d + e*x)**(5/2)*(6*b**
2*c*e**2 - 30*b*c**2*d*e + 30*c**3*d**2)/(5*e**7) + (d + e*x)**(3/2)*(2*b**3*e**3 - 24*b**2*c*d*e**2 + 60*b*c*
*2*d**2*e - 40*c**3*d**3)/(3*e**7) + sqrt(d + e*x)*(-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)/e**7

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Giac [A]
time = 1.32, size = 361, normalized size = 1.48 \begin {gather*} \frac {2}{315} \, {\left (35 \, {\left (x e + d\right )}^{\frac {9}{2}} c^{3} e^{56} - 270 \, {\left (x e + d\right )}^{\frac {7}{2}} c^{3} d e^{56} + 945 \, {\left (x e + d\right )}^{\frac {5}{2}} c^{3} d^{2} e^{56} - 2100 \, {\left (x e + d\right )}^{\frac {3}{2}} c^{3} d^{3} e^{56} + 4725 \, \sqrt {x e + d} c^{3} d^{4} e^{56} + 135 \, {\left (x e + d\right )}^{\frac {7}{2}} b c^{2} e^{57} - 945 \, {\left (x e + d\right )}^{\frac {5}{2}} b c^{2} d e^{57} + 3150 \, {\left (x e + d\right )}^{\frac {3}{2}} b c^{2} d^{2} e^{57} - 9450 \, \sqrt {x e + d} b c^{2} d^{3} e^{57} + 189 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{2} c e^{58} - 1260 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{2} c d e^{58} + 5670 \, \sqrt {x e + d} b^{2} c d^{2} e^{58} + 105 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{3} e^{59} - 945 \, \sqrt {x e + d} b^{3} d e^{59}\right )} e^{\left (-63\right )} + \frac {2 \, {\left (18 \, {\left (x e + d\right )} c^{3} d^{5} - c^{3} d^{6} - 45 \, {\left (x e + d\right )} b c^{2} d^{4} e + 3 \, b c^{2} d^{5} e + 36 \, {\left (x e + d\right )} b^{2} c d^{3} e^{2} - 3 \, b^{2} c d^{4} e^{2} - 9 \, {\left (x e + d\right )} b^{3} d^{2} e^{3} + b^{3} d^{3} e^{3}\right )} e^{\left (-7\right )}}{3 \, {\left (x e + d\right )}^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/315*(35*(x*e + d)^(9/2)*c^3*e^56 - 270*(x*e + d)^(7/2)*c^3*d*e^56 + 945*(x*e + d)^(5/2)*c^3*d^2*e^56 - 2100*
(x*e + d)^(3/2)*c^3*d^3*e^56 + 4725*sqrt(x*e + d)*c^3*d^4*e^56 + 135*(x*e + d)^(7/2)*b*c^2*e^57 - 945*(x*e + d
)^(5/2)*b*c^2*d*e^57 + 3150*(x*e + d)^(3/2)*b*c^2*d^2*e^57 - 9450*sqrt(x*e + d)*b*c^2*d^3*e^57 + 189*(x*e + d)
^(5/2)*b^2*c*e^58 - 1260*(x*e + d)^(3/2)*b^2*c*d*e^58 + 5670*sqrt(x*e + d)*b^2*c*d^2*e^58 + 105*(x*e + d)^(3/2
)*b^3*e^59 - 945*sqrt(x*e + d)*b^3*d*e^59)*e^(-63) + 2/3*(18*(x*e + d)*c^3*d^5 - c^3*d^6 - 45*(x*e + d)*b*c^2*
d^4*e + 3*b*c^2*d^5*e + 36*(x*e + d)*b^2*c*d^3*e^2 - 3*b^2*c*d^4*e^2 - 9*(x*e + d)*b^3*d^2*e^3 + b^3*d^3*e^3)*
e^(-7)/(x*e + d)^(3/2)

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Mupad [B]
time = 0.07, size = 281, normalized size = 1.15 \begin {gather*} \frac {{\left (d+e\,x\right )}^{3/2}\,\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 )}{3\,e^7}+\frac {\left (d+e\,x\right )\,\left (-6\,b^3\,d^2\,e^3+24\,b^2\,c\,d^3\,e^2-30\,b\,c^2\,d^4\,e+12\,c^3\,d^5\right )-\frac {2\,c^3\,d^6}{3}+\frac {2\,b^3\,d^3\,e^3}{3}-2\,b^2\,c\,d^4\,e^2+2\,b\,c^2\,d^5\,e}{e^7\,{\left (d+e\,x\right )}^{3/2}}+\frac {2\,c^3\,{\left (d+e\,x\right )}^{9/2}}{9\,e^7}-\frac {\left (12\,c^3\,d-6\,b\,c^2\,e\right )\,{\left (d+e\,x\right )}^{7/2}}{7\,e^7}+\frac {{\left (d+e\,x\right )}^{5/2}\,\left (6\,b^2\,c\,e^2-30\,b\,c^2\,d\,e+30\,c^3\,d^2\right )}{5\,e^7}+\frac {\sqrt {d+e\,x}\,\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 )}{e^7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((d + e*x)^(3/2)*(2*b^3*e^3 - 40*c^3*d^3 + 60*b*c^2*d^2*e - 24*b^2*c*d*e^2))/(3*e^7) + ((d + e*x)*(12*c^3*d^5
- 6*b^3*d^2*e^3 + 24*b^2*c*d^3*e^2 - 30*b*c^2*d^4*e) - (2*c^3*d^6)/3 + (2*b^3*d^3*e^3)/3 - 2*b^2*c*d^4*e^2 + 2
*b*c^2*d^5*e)/(e^7*(d + e*x)^(3/2)) + (2*c^3*(d + e*x)^(9/2))/(9*e^7) - ((12*c^3*d - 6*b*c^2*e)*(d + e*x)^(7/2
))/(7*e^7) + ((d + e*x)^(5/2)*(30*c^3*d^2 + 6*b^2*c*e^2 - 30*b*c^2*d*e))/(5*e^7) + ((d + e*x)^(1/2)*(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))/e^7

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