3.4.57 \(\int \frac {(b x+c x^2)^3}{\sqrt {d+e x}} \, dx\) [357]

Optimal. Leaf size=244 \[ \frac {2 d^3 (c d-b e)^3 \sqrt {d+e x}}{e^7}-\frac {2 d^2 (c d-b e)^2 (2 c d-b e) (d+e x)^{3/2}}{e^7}+\frac {6 d (c d-b e) \left (5 c^2 d^2-5 b c d e+b^2 e^2\right ) (d+e x)^{5/2}}{5 e^7}-\frac {2 (2 c d-b e) \left (10 c^2 d^2-10 b c d e+b^2 e^2\right ) (d+e x)^{7/2}}{7 e^7}+\frac {2 c \left (5 c^2 d^2-5 b c d e+b^2 e^2\right ) (d+e x)^{9/2}}{3 e^7}-\frac {6 c^2 (2 c d-b e) (d+e x)^{11/2}}{11 e^7}+\frac {2 c^3 (d+e x)^{13/2}}{13 e^7} \]

[Out]

-2*d^2*(-b*e+c*d)^2*(-b*e+2*c*d)*(e*x+d)^(3/2)/e^7+6/5*d*(-b*e+c*d)*(b^2*e^2-5*b*c*d*e+5*c^2*d^2)*(e*x+d)^(5/2
)/e^7-2/7*(-b*e+2*c*d)*(b^2*e^2-10*b*c*d*e+10*c^2*d^2)*(e*x+d)^(7/2)/e^7+2/3*c*(b^2*e^2-5*b*c*d*e+5*c^2*d^2)*(
e*x+d)^(9/2)/e^7-6/11*c^2*(-b*e+2*c*d)*(e*x+d)^(11/2)/e^7+2/13*c^3*(e*x+d)^(13/2)/e^7+2*d^3*(-b*e+c*d)^3*(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 {2 c (d+e x)^{9/2} \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{3 e^7}-\frac {2 (d+e x)^{7/2} (2 c d-b e) \left (b^2 e^2-10 b c d e+10 c^2 d^2\right )}{7 e^7}+\frac {6 d (d+e x)^{5/2} (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{5 e^7}-\frac {6 c^2 (d+e x)^{11/2} (2 c d-b e)}{11 e^7}+\frac {2 d^3 \sqrt {d+e x} (c d-b e)^3}{e^7}-\frac {2 d^2 (d+e x)^{3/2} (c d-b e)^2 (2 c d-b e)}{e^7}+\frac {2 c^3 (d+e x)^{13/2}}{13 e^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(b*x + c*x^2)^3/Sqrt[d + e*x],x]

[Out]

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

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Mathematica [A]
time = 0.11, size = 232, normalized size = 0.95 \begin {gather*} \frac {2 \sqrt {d+e x} \left (429 b^3 e^3 \left (-16 d^3+8 d^2 e x-6 d e^2 x^2+5 e^3 x^3\right )+143 b^2 c e^2 \left (128 d^4-64 d^3 e x+48 d^2 e^2 x^2-40 d e^3 x^3+35 e^4 x^4\right )+65 b c^2 e \left (-256 d^5+128 d^4 e x-96 d^3 e^2 x^2+80 d^2 e^3 x^3-70 d e^4 x^4+63 e^5 x^5\right )+5 c^3 \left (1024 d^6-512 d^5 e x+384 d^4 e^2 x^2-320 d^3 e^3 x^3+280 d^2 e^4 x^4-252 d e^5 x^5+231 e^6 x^6\right )\right )}{15015 e^7} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[d + e*x]*(429*b^3*e^3*(-16*d^3 + 8*d^2*e*x - 6*d*e^2*x^2 + 5*e^3*x^3) + 143*b^2*c*e^2*(128*d^4 - 64*d^
3*e*x + 48*d^2*e^2*x^2 - 40*d*e^3*x^3 + 35*e^4*x^4) + 65*b*c^2*e*(-256*d^5 + 128*d^4*e*x - 96*d^3*e^2*x^2 + 80
*d^2*e^3*x^3 - 70*d*e^4*x^4 + 63*e^5*x^5) + 5*c^3*(1024*d^6 - 512*d^5*e*x + 384*d^4*e^2*x^2 - 320*d^3*e^3*x^3
+ 280*d^2*e^4*x^4 - 252*d*e^5*x^5 + 231*e^6*x^6)))/(15015*e^7)

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Maple [A]
time = 0.48, size = 269, normalized size = 1.10

method	result	size
derivativedivides	\(\frac {\frac {2 c^{3} \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {2 \left (-3 c^{3} d +3 \left (b e -c d \right ) c^{2}\right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (3 c^{3} d^{2}-9 d \left (b e -c d \right ) c^{2}+3 \left (b e -c d \right )^{2} c \right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (-c^{3} d^{3}+9 d^{2} \left (b e -c d \right ) c^{2}-9 d \left (b e -c d \right )^{2} c +\left (b e -c d \right )^{3}\right ) \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (-3 d^{3} \left (b e -c d \right ) c^{2}+9 d^{2} \left (b e -c d \right )^{2} c -3 d \left (b e -c d \right )^{3}\right ) \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {2 \left (-3 d^{3} \left (b e -c d \right )^{2} c +3 d^{2} \left (b e -c d \right )^{3}\right ) \left (e x +d \right )^{\frac {3}{2}}}{3}-2 d^{3} \left (b e -c d \right )^{3} \sqrt {e x +d}}{e^{7}}\)	\(269\)
default	\(\frac {\frac {2 c^{3} \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {2 \left (-3 c^{3} d +3 \left (b e -c d \right ) c^{2}\right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (3 c^{3} d^{2}-9 d \left (b e -c d \right ) c^{2}+3 \left (b e -c d \right )^{2} c \right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (-c^{3} d^{3}+9 d^{2} \left (b e -c d \right ) c^{2}-9 d \left (b e -c d \right )^{2} c +\left (b e -c d \right )^{3}\right ) \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (-3 d^{3} \left (b e -c d \right ) c^{2}+9 d^{2} \left (b e -c d \right )^{2} c -3 d \left (b e -c d \right )^{3}\right ) \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {2 \left (-3 d^{3} \left (b e -c d \right )^{2} c +3 d^{2} \left (b e -c d \right )^{3}\right ) \left (e x +d \right )^{\frac {3}{2}}}{3}-2 d^{3} \left (b e -c d \right )^{3} \sqrt {e x +d}}{e^{7}}\)	\(269\)
gosper	\(-\frac {2 \left (-1155 c^{3} x^{6} e^{6}-4095 b \,c^{2} e^{6} x^{5}+1260 c^{3} d \,e^{5} x^{5}-5005 b^{2} c \,e^{6} x^{4}+4550 b \,c^{2} d \,e^{5} x^{4}-1400 c^{3} d^{2} e^{4} x^{4}-2145 b^{3} e^{6} x^{3}+5720 b^{2} c d \,e^{5} x^{3}-5200 b \,c^{2} d^{2} e^{4} x^{3}+1600 c^{3} d^{3} e^{3} x^{3}+2574 b^{3} d \,e^{5} x^{2}-6864 b^{2} c \,d^{2} e^{4} x^{2}+6240 b \,c^{2} d^{3} e^{3} x^{2}-1920 c^{3} d^{4} e^{2} x^{2}-3432 b^{3} d^{2} e^{4} x +9152 b^{2} c \,d^{3} e^{3} x -8320 b \,c^{2} d^{4} e^{2} x +2560 c^{3} d^{5} e x +6864 b^{3} d^{3} e^{3}-18304 b^{2} c \,d^{4} e^{2}+16640 b \,c^{2} d^{5} e -5120 c^{3} d^{6}\right ) \sqrt {e x +d}}{15015 e^{7}}\)	\(286\)
trager	\(-\frac {2 \left (-1155 c^{3} x^{6} e^{6}-4095 b \,c^{2} e^{6} x^{5}+1260 c^{3} d \,e^{5} x^{5}-5005 b^{2} c \,e^{6} x^{4}+4550 b \,c^{2} d \,e^{5} x^{4}-1400 c^{3} d^{2} e^{4} x^{4}-2145 b^{3} e^{6} x^{3}+5720 b^{2} c d \,e^{5} x^{3}-5200 b \,c^{2} d^{2} e^{4} x^{3}+1600 c^{3} d^{3} e^{3} x^{3}+2574 b^{3} d \,e^{5} x^{2}-6864 b^{2} c \,d^{2} e^{4} x^{2}+6240 b \,c^{2} d^{3} e^{3} x^{2}-1920 c^{3} d^{4} e^{2} x^{2}-3432 b^{3} d^{2} e^{4} x +9152 b^{2} c \,d^{3} e^{3} x -8320 b \,c^{2} d^{4} e^{2} x +2560 c^{3} d^{5} e x +6864 b^{3} d^{3} e^{3}-18304 b^{2} c \,d^{4} e^{2}+16640 b \,c^{2} d^{5} e -5120 c^{3} d^{6}\right ) \sqrt {e x +d}}{15015 e^{7}}\)	\(286\)
risch	\(-\frac {2 \left (-1155 c^{3} x^{6} e^{6}-4095 b \,c^{2} e^{6} x^{5}+1260 c^{3} d \,e^{5} x^{5}-5005 b^{2} c \,e^{6} x^{4}+4550 b \,c^{2} d \,e^{5} x^{4}-1400 c^{3} d^{2} e^{4} x^{4}-2145 b^{3} e^{6} x^{3}+5720 b^{2} c d \,e^{5} x^{3}-5200 b \,c^{2} d^{2} e^{4} x^{3}+1600 c^{3} d^{3} e^{3} x^{3}+2574 b^{3} d \,e^{5} x^{2}-6864 b^{2} c \,d^{2} e^{4} x^{2}+6240 b \,c^{2} d^{3} e^{3} x^{2}-1920 c^{3} d^{4} e^{2} x^{2}-3432 b^{3} d^{2} e^{4} x +9152 b^{2} c \,d^{3} e^{3} x -8320 b \,c^{2} d^{4} e^{2} x +2560 c^{3} d^{5} e x +6864 b^{3} d^{3} e^{3}-18304 b^{2} c \,d^{4} e^{2}+16640 b \,c^{2} d^{5} e -5120 c^{3} d^{6}\right ) \sqrt {e x +d}}{15015 e^{7}}\)	\(286\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.30, size = 305, normalized size = 1.25 \begin {gather*} \frac {2}{15015} \, {\left (429 \, {\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} 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 )} b^{2} c e^{\left (-4\right )} + 65 \, {\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} e^{\left (-5\right )} + 5 \, {\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} e^{\left (-6\right )}\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)^(1/2),x, algorithm="maxima")

[Out]

2/15015*(429*(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*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 + 3
15*sqrt(x*e + d)*d^4)*b^2*c*e^(-4) + 65*(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*e^(-5) + 5*(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*e^(-6))*e^(-1)

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Fricas [A]
time = 1.54, size = 256, normalized size = 1.05 \begin {gather*} \frac {2}{15015} \, {\left (5120 \, c^{3} d^{6} + 5 \, {\left (231 \, c^{3} x^{6} + 819 \, b c^{2} x^{5} + 1001 \, b^{2} c x^{4} + 429 \, b^{3} x^{3}\right )} e^{6} - 2 \, {\left (630 \, c^{3} d x^{5} + 2275 \, b c^{2} d x^{4} + 2860 \, b^{2} c d x^{3} + 1287 \, b^{3} d x^{2}\right )} e^{5} + 8 \, {\left (175 \, c^{3} d^{2} x^{4} + 650 \, b c^{2} d^{2} x^{3} + 858 \, b^{2} c d^{2} x^{2} + 429 \, b^{3} d^{2} x\right )} e^{4} - 16 \, {\left (100 \, c^{3} d^{3} x^{3} + 390 \, b c^{2} d^{3} x^{2} + 572 \, b^{2} c d^{3} x + 429 \, b^{3} d^{3}\right )} e^{3} + 128 \, {\left (15 \, c^{3} d^{4} x^{2} + 65 \, b c^{2} d^{4} x + 143 \, b^{2} c d^{4}\right )} e^{2} - 1280 \, {\left (2 \, c^{3} d^{5} x + 13 \, b c^{2} d^{5}\right )} e\right )} \sqrt {x e + d} e^{\left (-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)^(1/2),x, algorithm="fricas")

[Out]

2/15015*(5120*c^3*d^6 + 5*(231*c^3*x^6 + 819*b*c^2*x^5 + 1001*b^2*c*x^4 + 429*b^3*x^3)*e^6 - 2*(630*c^3*d*x^5
+ 2275*b*c^2*d*x^4 + 2860*b^2*c*d*x^3 + 1287*b^3*d*x^2)*e^5 + 8*(175*c^3*d^2*x^4 + 650*b*c^2*d^2*x^3 + 858*b^2
*c*d^2*x^2 + 429*b^3*d^2*x)*e^4 - 16*(100*c^3*d^3*x^3 + 390*b*c^2*d^3*x^2 + 572*b^2*c*d^3*x + 429*b^3*d^3)*e^3
+ 128*(15*c^3*d^4*x^2 + 65*b*c^2*d^4*x + 143*b^2*c*d^4)*e^2 - 1280*(2*c^3*d^5*x + 13*b*c^2*d^5)*e)*sqrt(x*e +
d)*e^(-7)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 745 vs. \(2 (241) = 482\).
time = 40.16, size = 745, normalized size = 3.05 \begin {gather*} \begin {cases} \frac {- \frac {2 b^{3} d \left (- \frac {d^{3}}{\sqrt {d + e x}} - 3 d^{2} \sqrt {d + e x} + d \left (d + e x\right )^{\frac {3}{2}} - \frac {\left (d + e x\right )^{\frac {5}{2}}}{5}\right )}{e^{3}} - \frac {2 b^{3} \left (\frac {d^{4}}{\sqrt {d + e x}} + 4 d^{3} \sqrt {d + e x} - 2 d^{2} \left (d + e x\right )^{\frac {3}{2}} + \frac {4 d \left (d + e x\right )^{\frac {5}{2}}}{5} - \frac {\left (d + e x\right )^{\frac {7}{2}}}{7}\right )}{e^{3}} - \frac {6 b^{2} c d \left (\frac {d^{4}}{\sqrt {d + e x}} + 4 d^{3} \sqrt {d + e x} - 2 d^{2} \left (d + e x\right )^{\frac {3}{2}} + \frac {4 d \left (d + e x\right )^{\frac {5}{2}}}{5} - \frac {\left (d + e x\right )^{\frac {7}{2}}}{7}\right )}{e^{4}} - \frac {6 b^{2} c \left (- \frac {d^{5}}{\sqrt {d + e x}} - 5 d^{4} \sqrt {d + e x} + \frac {10 d^{3} \left (d + e x\right )^{\frac {3}{2}}}{3} - 2 d^{2} \left (d + e x\right )^{\frac {5}{2}} + \frac {5 d \left (d + e x\right )^{\frac {7}{2}}}{7} - \frac {\left (d + e x\right )^{\frac {9}{2}}}{9}\right )}{e^{4}} - \frac {6 b c^{2} d \left (- \frac {d^{5}}{\sqrt {d + e x}} - 5 d^{4} \sqrt {d + e x} + \frac {10 d^{3} \left (d + e x\right )^{\frac {3}{2}}}{3} - 2 d^{2} \left (d + e x\right )^{\frac {5}{2}} + \frac {5 d \left (d + e x\right )^{\frac {7}{2}}}{7} - \frac {\left (d + e x\right )^{\frac {9}{2}}}{9}\right )}{e^{5}} - \frac {6 b c^{2} \left (\frac {d^{6}}{\sqrt {d + e x}} + 6 d^{5} \sqrt {d + e x} - 5 d^{4} \left (d + e x\right )^{\frac {3}{2}} + 4 d^{3} \left (d + e x\right )^{\frac {5}{2}} - \frac {15 d^{2} \left (d + e x\right )^{\frac {7}{2}}}{7} + \frac {2 d \left (d + e x\right )^{\frac {9}{2}}}{3} - \frac {\left (d + e x\right )^{\frac {11}{2}}}{11}\right )}{e^{5}} - \frac {2 c^{3} d \left (\frac {d^{6}}{\sqrt {d + e x}} + 6 d^{5} \sqrt {d + e x} - 5 d^{4} \left (d + e x\right )^{\frac {3}{2}} + 4 d^{3} \left (d + e x\right )^{\frac {5}{2}} - \frac {15 d^{2} \left (d + e x\right )^{\frac {7}{2}}}{7} + \frac {2 d \left (d + e x\right )^{\frac {9}{2}}}{3} - \frac {\left (d + e x\right )^{\frac {11}{2}}}{11}\right )}{e^{6}} - \frac {2 c^{3} \left (- \frac {d^{7}}{\sqrt {d + e x}} - 7 d^{6} \sqrt {d + e x} + 7 d^{5} \left (d + e x\right )^{\frac {3}{2}} - 7 d^{4} \left (d + e x\right )^{\frac {5}{2}} + 5 d^{3} \left (d + e x\right )^{\frac {7}{2}} - \frac {7 d^{2} \left (d + e x\right )^{\frac {9}{2}}}{3} + \frac {7 d \left (d + e x\right )^{\frac {11}{2}}}{11} - \frac {\left (d + e x\right )^{\frac {13}{2}}}{13}\right )}{e^{6}}}{e} & \text {for}\: e \neq 0 \\\frac {\frac {b^{3} x^{4}}{4} + \frac {3 b^{2} c x^{5}}{5} + \frac {b c^{2} x^{6}}{2} + \frac {c^{3} x^{7}}{7}}{\sqrt {d}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise(((-2*b**3*d*(-d**3/sqrt(d + e*x) - 3*d**2*sqrt(d + e*x) + d*(d + e*x)**(3/2) - (d + e*x)**(5/2)/5)/e
**3 - 2*b**3*(d**4/sqrt(d + e*x) + 4*d**3*sqrt(d + e*x) - 2*d**2*(d + e*x)**(3/2) + 4*d*(d + e*x)**(5/2)/5 - (
d + e*x)**(7/2)/7)/e**3 - 6*b**2*c*d*(d**4/sqrt(d + e*x) + 4*d**3*sqrt(d + e*x) - 2*d**2*(d + e*x)**(3/2) + 4*
d*(d + e*x)**(5/2)/5 - (d + e*x)**(7/2)/7)/e**4 - 6*b**2*c*(-d**5/sqrt(d + e*x) - 5*d**4*sqrt(d + e*x) + 10*d*
*3*(d + e*x)**(3/2)/3 - 2*d**2*(d + e*x)**(5/2) + 5*d*(d + e*x)**(7/2)/7 - (d + e*x)**(9/2)/9)/e**4 - 6*b*c**2
*d*(-d**5/sqrt(d + e*x) - 5*d**4*sqrt(d + e*x) + 10*d**3*(d + e*x)**(3/2)/3 - 2*d**2*(d + e*x)**(5/2) + 5*d*(d
+ e*x)**(7/2)/7 - (d + e*x)**(9/2)/9)/e**5 - 6*b*c**2*(d**6/sqrt(d + e*x) + 6*d**5*sqrt(d + e*x) - 5*d**4*(d
+ e*x)**(3/2) + 4*d**3*(d + e*x)**(5/2) - 15*d**2*(d + e*x)**(7/2)/7 + 2*d*(d + e*x)**(9/2)/3 - (d + e*x)**(11
/2)/11)/e**5 - 2*c**3*d*(d**6/sqrt(d + e*x) + 6*d**5*sqrt(d + e*x) - 5*d**4*(d + e*x)**(3/2) + 4*d**3*(d + e*x
)**(5/2) - 15*d**2*(d + e*x)**(7/2)/7 + 2*d*(d + e*x)**(9/2)/3 - (d + e*x)**(11/2)/11)/e**6 - 2*c**3*(-d**7/sq
rt(d + e*x) - 7*d**6*sqrt(d + e*x) + 7*d**5*(d + e*x)**(3/2) - 7*d**4*(d + e*x)**(5/2) + 5*d**3*(d + e*x)**(7/
2) - 7*d**2*(d + e*x)**(9/2)/3 + 7*d*(d + e*x)**(11/2)/11 - (d + e*x)**(13/2)/13)/e**6)/e, Ne(e, 0)), ((b**3*x
**4/4 + 3*b**2*c*x**5/5 + b*c**2*x**6/2 + c**3*x**7/7)/sqrt(d), True))

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Giac [A]
time = 1.13, size = 305, normalized size = 1.25 \begin {gather*} \frac {2}{15015} \, {\left (429 \, {\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} 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 )} b^{2} c e^{\left (-4\right )} + 65 \, {\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} e^{\left (-5\right )} + 5 \, {\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} e^{\left (-6\right )}\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)^(1/2),x, algorithm="giac")

[Out]

2/15015*(429*(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*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 + 3
15*sqrt(x*e + d)*d^4)*b^2*c*e^(-4) + 65*(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*e^(-5) + 5*(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*e^(-6))*e^(-1)

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Mupad [B]
time = 0.06, size = 239, normalized size = 0.98 \begin {gather*} \frac {{\left (d+e\,x\right )}^{7/2}\,\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 )}{7\,e^7}+\frac {2\,c^3\,{\left (d+e\,x\right )}^{13/2}}{13\,e^7}-\frac {\left (12\,c^3\,d-6\,b\,c^2\,e\right )\,{\left (d+e\,x\right )}^{11/2}}{11\,e^7}+\frac {{\left (d+e\,x\right )}^{9/2}\,\left (6\,b^2\,c\,e^2-30\,b\,c^2\,d\,e+30\,c^3\,d^2\right )}{9\,e^7}+\frac {{\left (d+e\,x\right )}^{5/2}\,\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 )}{5\,e^7}-\frac {2\,d^3\,{\left (b\,e-c\,d\right )}^3\,\sqrt {d+e\,x}}{e^7}+\frac {2\,d^2\,{\left (b\,e-c\,d\right )}^2\,\left (b\,e-2\,c\,d\right )\,{\left (d+e\,x\right )}^{3/2}}{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)^(1/2),x)

[Out]

((d + e*x)^(7/2)*(2*b^3*e^3 - 40*c^3*d^3 + 60*b*c^2*d^2*e - 24*b^2*c*d*e^2))/(7*e^7) + (2*c^3*(d + e*x)^(13/2)
)/(13*e^7) - ((12*c^3*d - 6*b*c^2*e)*(d + e*x)^(11/2))/(11*e^7) + ((d + e*x)^(9/2)*(30*c^3*d^2 + 6*b^2*c*e^2 -
30*b*c^2*d*e))/(9*e^7) + ((d + e*x)^(5/2)*(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))/(5*
e^7) - (2*d^3*(b*e - c*d)^3*(d + e*x)^(1/2))/e^7 + (2*d^2*(b*e - c*d)^2*(b*e - 2*c*d)*(d + e*x)^(3/2))/e^7

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