3.4.58 \(\int \frac {(b x+c x^2)^3}{(d+e x)^{3/2}} \, dx\) [358]

Optimal. Leaf size=242 \[ -\frac {2 d^3 (c d-b e)^3}{e^7 \sqrt {d+e x}}-\frac {6 d^2 (c d-b e)^2 (2 c d-b e) \sqrt {d+e x}}{e^7}+\frac {2 d (c d-b e) \left (5 c^2 d^2-5 b c d e+b^2 e^2\right ) (d+e x)^{3/2}}{e^7}-\frac {2 (2 c d-b e) \left (10 c^2 d^2-10 b c d e+b^2 e^2\right ) (d+e x)^{5/2}}{5 e^7}+\frac {6 c \left (5 c^2 d^2-5 b c d e+b^2 e^2\right ) (d+e x)^{7/2}}{7 e^7}-\frac {2 c^2 (2 c d-b e) (d+e x)^{9/2}}{3 e^7}+\frac {2 c^3 (d+e x)^{11/2}}{11 e^7} \]

[Out]

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

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Rubi [A]
time = 0.07, antiderivative size = 242, 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)^{7/2} \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{7 e^7}-\frac {2 (d+e x)^{5/2} (2 c d-b e) \left (b^2 e^2-10 b c d e+10 c^2 d^2\right )}{5 e^7}+\frac {2 d (d+e x)^{3/2} (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{e^7}-\frac {2 c^2 (d+e x)^{9/2} (2 c d-b e)}{3 e^7}-\frac {2 d^3 (c d-b e)^3}{e^7 \sqrt {d+e x}}-\frac {6 d^2 \sqrt {d+e x} (c d-b e)^2 (2 c d-b e)}{e^7}+\frac {2 c^3 (d+e x)^{11/2}}{11 e^7} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]
time = 0.11, size = 231, normalized size = 0.95 \begin {gather*} \frac {2 \left (231 b^3 e^3 \left (16 d^3+8 d^2 e x-2 d e^2 x^2+e^3 x^3\right )+99 b^2 c e^2 \left (-128 d^4-64 d^3 e x+16 d^2 e^2 x^2-8 d e^3 x^3+5 e^4 x^4\right )+55 b c^2 e \left (256 d^5+128 d^4 e x-32 d^3 e^2 x^2+16 d^2 e^3 x^3-10 d e^4 x^4+7 e^5 x^5\right )-5 c^3 \left (1024 d^6+512 d^5 e x-128 d^4 e^2 x^2+64 d^3 e^3 x^3-40 d^2 e^4 x^4+28 d e^5 x^5-21 e^6 x^6\right )\right )}{1155 e^7 \sqrt {d+e x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(231*b^3*e^3*(16*d^3 + 8*d^2*e*x - 2*d*e^2*x^2 + e^3*x^3) + 99*b^2*c*e^2*(-128*d^4 - 64*d^3*e*x + 16*d^2*e^
2*x^2 - 8*d*e^3*x^3 + 5*e^4*x^4) + 55*b*c^2*e*(256*d^5 + 128*d^4*e*x - 32*d^3*e^2*x^2 + 16*d^2*e^3*x^3 - 10*d*
e^4*x^4 + 7*e^5*x^5) - 5*c^3*(1024*d^6 + 512*d^5*e*x - 128*d^4*e^2*x^2 + 64*d^3*e^3*x^3 - 40*d^2*e^4*x^4 + 28*
d*e^5*x^5 - 21*e^6*x^6)))/(1155*e^7*Sqrt[d + e*x])

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Maple [A]
time = 0.42, size = 339, normalized size = 1.40

method	result	size
risch	\(\frac {2 \left (105 c^{3} x^{5} e^{5}+385 b \,c^{2} e^{5} x^{4}-245 c^{3} d \,e^{4} x^{4}+495 b^{2} c \,e^{5} x^{3}-935 b \,c^{2} d \,e^{4} x^{3}+445 c^{3} d^{2} e^{3} x^{3}+231 b^{3} e^{5} x^{2}-1287 b^{2} c d \,e^{4} x^{2}+1815 b \,c^{2} d^{2} e^{3} x^{2}-765 c^{3} d^{3} e^{2} x^{2}-693 b^{3} d \,e^{4} x +2871 b^{2} c \,d^{2} e^{3} x -3575 b \,c^{2} d^{3} e^{2} x +1405 c^{3} d^{4} e x +2541 b^{3} d^{2} e^{3}-9207 b^{2} c \,d^{3} e^{2}+10615 b \,c^{2} d^{4} e -3965 c^{3} d^{5}\right ) \sqrt {e x +d}}{1155 e^{7}}+\frac {2 d^{3} \left (b^{3} e^{3}-3 b^{2} d \,e^{2} c +3 b \,c^{2} d^{2} e -c^{3} d^{3}\right )}{e^{7} \sqrt {e x +d}}\)	\(280\)
gosper	\(\frac {\frac {2}{11} c^{3} x^{6} e^{6}+\frac {2}{3} b \,c^{2} e^{6} x^{5}-\frac {8}{33} c^{3} d \,e^{5} x^{5}+\frac {6}{7} b^{2} c \,e^{6} x^{4}-\frac {20}{21} b \,c^{2} d \,e^{5} x^{4}+\frac {80}{231} c^{3} d^{2} e^{4} x^{4}+\frac {2}{5} b^{3} e^{6} x^{3}-\frac {48}{35} b^{2} c d \,e^{5} x^{3}+\frac {32}{21} b \,c^{2} d^{2} e^{4} x^{3}-\frac {128}{231} c^{3} d^{3} e^{3} x^{3}-\frac {4}{5} b^{3} d \,e^{5} x^{2}+\frac {96}{35} b^{2} c \,d^{2} e^{4} x^{2}-\frac {64}{21} b \,c^{2} d^{3} e^{3} x^{2}+\frac {256}{231} c^{3} d^{4} e^{2} x^{2}+\frac {16}{5} b^{3} d^{2} e^{4} x -\frac {384}{35} b^{2} c \,d^{3} e^{3} x +\frac {256}{21} b \,c^{2} d^{4} e^{2} x -\frac {1024}{231} c^{3} d^{5} e x +\frac {32}{5} b^{3} d^{3} e^{3}-\frac {768}{35} b^{2} c \,d^{4} e^{2}+\frac {512}{21} b \,c^{2} d^{5} e -\frac {2048}{231} c^{3} d^{6}}{e^{7} \sqrt {e x +d}}\)	\(286\)
trager	\(\frac {\frac {2}{11} c^{3} x^{6} e^{6}+\frac {2}{3} b \,c^{2} e^{6} x^{5}-\frac {8}{33} c^{3} d \,e^{5} x^{5}+\frac {6}{7} b^{2} c \,e^{6} x^{4}-\frac {20}{21} b \,c^{2} d \,e^{5} x^{4}+\frac {80}{231} c^{3} d^{2} e^{4} x^{4}+\frac {2}{5} b^{3} e^{6} x^{3}-\frac {48}{35} b^{2} c d \,e^{5} x^{3}+\frac {32}{21} b \,c^{2} d^{2} e^{4} x^{3}-\frac {128}{231} c^{3} d^{3} e^{3} x^{3}-\frac {4}{5} b^{3} d \,e^{5} x^{2}+\frac {96}{35} b^{2} c \,d^{2} e^{4} x^{2}-\frac {64}{21} b \,c^{2} d^{3} e^{3} x^{2}+\frac {256}{231} c^{3} d^{4} e^{2} x^{2}+\frac {16}{5} b^{3} d^{2} e^{4} x -\frac {384}{35} b^{2} c \,d^{3} e^{3} x +\frac {256}{21} b \,c^{2} d^{4} e^{2} x -\frac {1024}{231} c^{3} d^{5} e x +\frac {32}{5} b^{3} d^{3} e^{3}-\frac {768}{35} b^{2} c \,d^{4} e^{2}+\frac {512}{21} b \,c^{2} d^{5} e -\frac {2048}{231} c^{3} d^{6}}{e^{7} \sqrt {e x +d}}\)	\(286\)
derivativedivides	\(\frac {\frac {2 c^{3} \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 b \,c^{2} e \left (e x +d \right )^{\frac {9}{2}}}{3}-\frac {4 c^{3} d \left (e x +d \right )^{\frac {9}{2}}}{3}+\frac {6 b^{2} c \,e^{2} \left (e x +d \right )^{\frac {7}{2}}}{7}-\frac {30 b \,c^{2} d e \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {30 c^{3} d^{2} \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 b^{3} e^{3} \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {24 b^{2} c d \,e^{2} \left (e x +d \right )^{\frac {5}{2}}}{5}+12 b \,c^{2} d^{2} e \left (e x +d \right )^{\frac {5}{2}}-8 c^{3} d^{3} \left (e x +d \right )^{\frac {5}{2}}-2 b^{3} d \,e^{3} \left (e x +d \right )^{\frac {3}{2}}+12 b^{2} c \,d^{2} e^{2} \left (e x +d \right )^{\frac {3}{2}}-20 b \,c^{2} d^{3} e \left (e x +d \right )^{\frac {3}{2}}+10 c^{3} d^{4} \left (e x +d \right )^{\frac {3}{2}}+6 b^{3} d^{2} e^{3} \sqrt {e x +d}-24 b^{2} c \,d^{3} e^{2} \sqrt {e x +d}+30 b \,c^{2} d^{4} e \sqrt {e x +d}-12 c^{3} d^{5} \sqrt {e x +d}+\frac {2 d^{3} \left (b^{3} e^{3}-3 b^{2} d \,e^{2} c +3 b \,c^{2} d^{2} e -c^{3} d^{3}\right )}{\sqrt {e x +d}}}{e^{7}}\)	\(339\)
default	\(\frac {\frac {2 c^{3} \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 b \,c^{2} e \left (e x +d \right )^{\frac {9}{2}}}{3}-\frac {4 c^{3} d \left (e x +d \right )^{\frac {9}{2}}}{3}+\frac {6 b^{2} c \,e^{2} \left (e x +d \right )^{\frac {7}{2}}}{7}-\frac {30 b \,c^{2} d e \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {30 c^{3} d^{2} \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 b^{3} e^{3} \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {24 b^{2} c d \,e^{2} \left (e x +d \right )^{\frac {5}{2}}}{5}+12 b \,c^{2} d^{2} e \left (e x +d \right )^{\frac {5}{2}}-8 c^{3} d^{3} \left (e x +d \right )^{\frac {5}{2}}-2 b^{3} d \,e^{3} \left (e x +d \right )^{\frac {3}{2}}+12 b^{2} c \,d^{2} e^{2} \left (e x +d \right )^{\frac {3}{2}}-20 b \,c^{2} d^{3} e \left (e x +d \right )^{\frac {3}{2}}+10 c^{3} d^{4} \left (e x +d \right )^{\frac {3}{2}}+6 b^{3} d^{2} e^{3} \sqrt {e x +d}-24 b^{2} c \,d^{3} e^{2} \sqrt {e x +d}+30 b \,c^{2} d^{4} e \sqrt {e x +d}-12 c^{3} d^{5} \sqrt {e x +d}+\frac {2 d^{3} \left (b^{3} e^{3}-3 b^{2} d \,e^{2} c +3 b \,c^{2} d^{2} e -c^{3} d^{3}\right )}{\sqrt {e x +d}}}{e^{7}}\)	\(339\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/e^7*(1/11*c^3*(e*x+d)^(11/2)+1/3*b*c^2*e*(e*x+d)^(9/2)-2/3*c^3*d*(e*x+d)^(9/2)+3/7*b^2*c*e^2*(e*x+d)^(7/2)-1
5/7*b*c^2*d*e*(e*x+d)^(7/2)+15/7*c^3*d^2*(e*x+d)^(7/2)+1/5*b^3*e^3*(e*x+d)^(5/2)-12/5*b^2*c*d*e^2*(e*x+d)^(5/2
)+6*b*c^2*d^2*e*(e*x+d)^(5/2)-4*c^3*d^3*(e*x+d)^(5/2)-b^3*d*e^3*(e*x+d)^(3/2)+6*b^2*c*d^2*e^2*(e*x+d)^(3/2)-10
*b*c^2*d^3*e*(e*x+d)^(3/2)+5*c^3*d^4*(e*x+d)^(3/2)+3*b^3*d^2*e^3*(e*x+d)^(1/2)-12*b^2*c*d^3*e^2*(e*x+d)^(1/2)+
15*b*c^2*d^4*e*(e*x+d)^(1/2)-6*c^3*d^5*(e*x+d)^(1/2)+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)
^(1/2))

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Maxima [A]
time = 0.31, size = 280, normalized size = 1.16 \begin {gather*} \frac {2}{1155} \, {\left ({\left (105 \, {\left (x e + d\right )}^{\frac {11}{2}} c^{3} - 385 \, {\left (2 \, c^{3} d - b c^{2} e\right )} {\left (x e + d\right )}^{\frac {9}{2}} + 495 \, {\left (5 \, c^{3} d^{2} - 5 \, b c^{2} d e + b^{2} c e^{2}\right )} {\left (x e + d\right )}^{\frac {7}{2}} - 231 \, {\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 {5}{2}} + 1155 \, {\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 (x e + d\right )}^{\frac {3}{2}} - 3465 \, {\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 )} \sqrt {x e + d}\right )} e^{\left (-6\right )} - \frac {1155 \, {\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 )} e^{\left (-6\right )}}{\sqrt {x e + d}}\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)^(3/2),x, algorithm="maxima")

[Out]

2/1155*((105*(x*e + d)^(11/2)*c^3 - 385*(2*c^3*d - b*c^2*e)*(x*e + d)^(9/2) + 495*(5*c^3*d^2 - 5*b*c^2*d*e + b
^2*c*e^2)*(x*e + d)^(7/2) - 231*(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)^(5/2) + 115
5*(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)*(x*e + d)^(3/2) - 3465*(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)*sqrt(x*e + d))*e^(-6) - 1155*(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^(-6)/sqrt(x*e + d))*e^(-1)

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Fricas [A]
time = 1.41, size = 265, normalized size = 1.10 \begin {gather*} -\frac {2 \, {\left (5120 \, c^{3} d^{6} - {\left (105 \, c^{3} x^{6} + 385 \, b c^{2} x^{5} + 495 \, b^{2} c x^{4} + 231 \, b^{3} x^{3}\right )} e^{6} + 2 \, {\left (70 \, c^{3} d x^{5} + 275 \, b c^{2} d x^{4} + 396 \, b^{2} c d x^{3} + 231 \, b^{3} d x^{2}\right )} e^{5} - 8 \, {\left (25 \, c^{3} d^{2} x^{4} + 110 \, b c^{2} d^{2} x^{3} + 198 \, b^{2} c d^{2} x^{2} + 231 \, b^{3} d^{2} x\right )} e^{4} + 16 \, {\left (20 \, c^{3} d^{3} x^{3} + 110 \, b c^{2} d^{3} x^{2} + 396 \, b^{2} c d^{3} x - 231 \, b^{3} d^{3}\right )} e^{3} - 128 \, {\left (5 \, c^{3} d^{4} x^{2} + 55 \, b c^{2} d^{4} x - 99 \, b^{2} c d^{4}\right )} e^{2} + 1280 \, {\left (2 \, c^{3} d^{5} x - 11 \, b c^{2} d^{5}\right )} e\right )} \sqrt {x e + d}}{1155 \, {\left (x e^{8} + d 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)^(3/2),x, algorithm="fricas")

[Out]

-2/1155*(5120*c^3*d^6 - (105*c^3*x^6 + 385*b*c^2*x^5 + 495*b^2*c*x^4 + 231*b^3*x^3)*e^6 + 2*(70*c^3*d*x^5 + 27
5*b*c^2*d*x^4 + 396*b^2*c*d*x^3 + 231*b^3*d*x^2)*e^5 - 8*(25*c^3*d^2*x^4 + 110*b*c^2*d^2*x^3 + 198*b^2*c*d^2*x
^2 + 231*b^3*d^2*x)*e^4 + 16*(20*c^3*d^3*x^3 + 110*b*c^2*d^3*x^2 + 396*b^2*c*d^3*x - 231*b^3*d^3)*e^3 - 128*(5
*c^3*d^4*x^2 + 55*b*c^2*d^4*x - 99*b^2*c*d^4)*e^2 + 1280*(2*c^3*d^5*x - 11*b*c^2*d^5)*e)*sqrt(x*e + d)/(x*e^8
+ d*e^7)

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Sympy [A]
time = 16.49, size = 284, normalized size = 1.17 \begin {gather*} \frac {2 c^{3} \left (d + e x\right )^{\frac {11}{2}}}{11 e^{7}} + \frac {2 d^{3} \left (b e - c d\right )^{3}}{e^{7} \sqrt {d + e x}} + \frac {\left (d + e x\right )^{\frac {9}{2}} \cdot \left (6 b c^{2} e - 12 c^{3} d\right )}{9 e^{7}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \cdot \left (6 b^{2} c e^{2} - 30 b c^{2} d e + 30 c^{3} d^{2}\right )}{7 e^{7}} + \frac {\left (d + e x\right )^{\frac {5}{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 )}{5 e^{7}} + \frac {\left (d + e x\right )^{\frac {3}{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 )}{3 e^{7}} + \frac {\sqrt {d + e x} \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 )}{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)**(3/2),x)

[Out]

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

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Giac [A]
time = 1.02, size = 371, normalized size = 1.53 \begin {gather*} \frac {2}{1155} \, {\left (105 \, {\left (x e + d\right )}^{\frac {11}{2}} c^{3} e^{70} - 770 \, {\left (x e + d\right )}^{\frac {9}{2}} c^{3} d e^{70} + 2475 \, {\left (x e + d\right )}^{\frac {7}{2}} c^{3} d^{2} e^{70} - 4620 \, {\left (x e + d\right )}^{\frac {5}{2}} c^{3} d^{3} e^{70} + 5775 \, {\left (x e + d\right )}^{\frac {3}{2}} c^{3} d^{4} e^{70} - 6930 \, \sqrt {x e + d} c^{3} d^{5} e^{70} + 385 \, {\left (x e + d\right )}^{\frac {9}{2}} b c^{2} e^{71} - 2475 \, {\left (x e + d\right )}^{\frac {7}{2}} b c^{2} d e^{71} + 6930 \, {\left (x e + d\right )}^{\frac {5}{2}} b c^{2} d^{2} e^{71} - 11550 \, {\left (x e + d\right )}^{\frac {3}{2}} b c^{2} d^{3} e^{71} + 17325 \, \sqrt {x e + d} b c^{2} d^{4} e^{71} + 495 \, {\left (x e + d\right )}^{\frac {7}{2}} b^{2} c e^{72} - 2772 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{2} c d e^{72} + 6930 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{2} c d^{2} e^{72} - 13860 \, \sqrt {x e + d} b^{2} c d^{3} e^{72} + 231 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{3} e^{73} - 1155 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{3} d e^{73} + 3465 \, \sqrt {x e + d} b^{3} d^{2} e^{73}\right )} e^{\left (-77\right )} - \frac {2 \, {\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 )} e^{\left (-7\right )}}{\sqrt {x e + d}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/1155*(105*(x*e + d)^(11/2)*c^3*e^70 - 770*(x*e + d)^(9/2)*c^3*d*e^70 + 2475*(x*e + d)^(7/2)*c^3*d^2*e^70 - 4
620*(x*e + d)^(5/2)*c^3*d^3*e^70 + 5775*(x*e + d)^(3/2)*c^3*d^4*e^70 - 6930*sqrt(x*e + d)*c^3*d^5*e^70 + 385*(
x*e + d)^(9/2)*b*c^2*e^71 - 2475*(x*e + d)^(7/2)*b*c^2*d*e^71 + 6930*(x*e + d)^(5/2)*b*c^2*d^2*e^71 - 11550*(x
*e + d)^(3/2)*b*c^2*d^3*e^71 + 17325*sqrt(x*e + d)*b*c^2*d^4*e^71 + 495*(x*e + d)^(7/2)*b^2*c*e^72 - 2772*(x*e
+ d)^(5/2)*b^2*c*d*e^72 + 6930*(x*e + d)^(3/2)*b^2*c*d^2*e^72 - 13860*sqrt(x*e + d)*b^2*c*d^3*e^72 + 231*(x*e
+ d)^(5/2)*b^3*e^73 - 1155*(x*e + d)^(3/2)*b^3*d*e^73 + 3465*sqrt(x*e + d)*b^3*d^2*e^73)*e^(-77) - 2*(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^(-7)/sqrt(x*e + d)

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Mupad [B]
time = 0.21, size = 268, normalized size = 1.11 \begin {gather*} \frac {{\left (d+e\,x\right )}^{5/2}\,\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 )}{5\,e^7}-\frac {-2\,b^3\,d^3\,e^3+6\,b^2\,c\,d^4\,e^2-6\,b\,c^2\,d^5\,e+2\,c^3\,d^6}{e^7\,\sqrt {d+e\,x}}+\frac {2\,c^3\,{\left (d+e\,x\right )}^{11/2}}{11\,e^7}-\frac {\left (12\,c^3\,d-6\,b\,c^2\,e\right )\,{\left (d+e\,x\right )}^{9/2}}{9\,e^7}+\frac {{\left (d+e\,x\right )}^{7/2}\,\left (6\,b^2\,c\,e^2-30\,b\,c^2\,d\,e+30\,c^3\,d^2\right )}{7\,e^7}+\frac {{\left (d+e\,x\right )}^{3/2}\,\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 )}{3\,e^7}+\frac {6\,d^2\,{\left (b\,e-c\,d\right )}^2\,\left (b\,e-2\,c\,d\right )\,\sqrt {d+e\,x}}{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)^(3/2),x)

[Out]

((d + e*x)^(5/2)*(2*b^3*e^3 - 40*c^3*d^3 + 60*b*c^2*d^2*e - 24*b^2*c*d*e^2))/(5*e^7) - (2*c^3*d^6 - 2*b^3*d^3*
e^3 + 6*b^2*c*d^4*e^2 - 6*b*c^2*d^5*e)/(e^7*(d + e*x)^(1/2)) + (2*c^3*(d + e*x)^(11/2))/(11*e^7) - ((12*c^3*d
- 6*b*c^2*e)*(d + e*x)^(9/2))/(9*e^7) + ((d + e*x)^(7/2)*(30*c^3*d^2 + 6*b^2*c*e^2 - 30*b*c^2*d*e))/(7*e^7) +
((d + e*x)^(3/2)*(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))/(3*e^7) + (6*d^2*(b*e - c*d)^
2*(b*e - 2*c*d)*(d + e*x)^(1/2))/e^7

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