3.4.51 \(\int \frac {(b x+c x^2)^2}{(d+e x)^{5/2}} \, dx\) [351]

Optimal. Leaf size=143 \[ -\frac {2 d^2 (c d-b e)^2}{3 e^5 (d+e x)^{3/2}}+\frac {4 d (c d-b e) (2 c d-b e)}{e^5 \sqrt {d+e x}}+\frac {2 \left (6 c^2 d^2-6 b c d e+b^2 e^2\right ) \sqrt {d+e x}}{e^5}-\frac {4 c (2 c d-b e) (d+e x)^{3/2}}{3 e^5}+\frac {2 c^2 (d+e x)^{5/2}}{5 e^5} \]

[Out]

-2/3*d^2*(-b*e+c*d)^2/e^5/(e*x+d)^(3/2)-4/3*c*(-b*e+2*c*d)*(e*x+d)^(3/2)/e^5+2/5*c^2*(e*x+d)^(5/2)/e^5+4*d*(-b
*e+c*d)*(-b*e+2*c*d)/e^5/(e*x+d)^(1/2)+2*(b^2*e^2-6*b*c*d*e+6*c^2*d^2)*(e*x+d)^(1/2)/e^5

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Rubi [A]
time = 0.04, antiderivative size = 143, 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 \sqrt {d+e x} \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )}{e^5}-\frac {2 d^2 (c d-b e)^2}{3 e^5 (d+e x)^{3/2}}-\frac {4 c (d+e x)^{3/2} (2 c d-b e)}{3 e^5}+\frac {4 d (c d-b e) (2 c d-b e)}{e^5 \sqrt {d+e x}}+\frac {2 c^2 (d+e x)^{5/2}}{5 e^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*d^2*(c*d - b*e)^2)/(3*e^5*(d + e*x)^(3/2)) + (4*d*(c*d - b*e)*(2*c*d - b*e))/(e^5*Sqrt[d + e*x]) + (2*(6*c
^2*d^2 - 6*b*c*d*e + b^2*e^2)*Sqrt[d + e*x])/e^5 - (4*c*(2*c*d - b*e)*(d + e*x)^(3/2))/(3*e^5) + (2*c^2*(d + e
*x)^(5/2))/(5*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 \frac {\left (b x+c x^2\right )^2}{(d+e x)^{5/2}} \, dx &=\int \left (\frac {d^2 (c d-b e)^2}{e^4 (d+e x)^{5/2}}+\frac {2 d (c d-b e) (-2 c d+b e)}{e^4 (d+e x)^{3/2}}+\frac {6 c^2 d^2-6 b c d e+b^2 e^2}{e^4 \sqrt {d+e x}}-\frac {2 c (2 c d-b e) \sqrt {d+e x}}{e^4}+\frac {c^2 (d+e x)^{3/2}}{e^4}\right ) \, dx\\ &=-\frac {2 d^2 (c d-b e)^2}{3 e^5 (d+e x)^{3/2}}+\frac {4 d (c d-b e) (2 c d-b e)}{e^5 \sqrt {d+e x}}+\frac {2 \left (6 c^2 d^2-6 b c d e+b^2 e^2\right ) \sqrt {d+e x}}{e^5}-\frac {4 c (2 c d-b e) (d+e x)^{3/2}}{3 e^5}+\frac {2 c^2 (d+e x)^{5/2}}{5 e^5}\\ \end {align*}

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Mathematica [A]
time = 0.08, size = 123, normalized size = 0.86 \begin {gather*} \frac {2 \left (5 b^2 e^2 \left (8 d^2+12 d e x+3 e^2 x^2\right )+10 b c e \left (-16 d^3-24 d^2 e x-6 d e^2 x^2+e^3 x^3\right )+c^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 )\right )}{15 e^5 (d+e x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(5*b^2*e^2*(8*d^2 + 12*d*e*x + 3*e^2*x^2) + 10*b*c*e*(-16*d^3 - 24*d^2*e*x - 6*d*e^2*x^2 + e^3*x^3) + c^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)))/(15*e^5*(d + e*x)^(3/2))

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Maple [A]
time = 0.46, size = 151, normalized size = 1.06

method	result	size
risch	\(\frac {2 \left (3 c^{2} e^{2} x^{2}+10 b c \,e^{2} x -14 c^{2} d x e +15 b^{2} e^{2}-80 b c d e +73 d^{2} c^{2}\right ) \sqrt {e x +d}}{15 e^{5}}+\frac {2 \left (6 b \,e^{2} x -12 c d e x +5 b d e -11 c \,d^{2}\right ) d \left (b e -c d \right )}{3 e^{5} \left (e x +d \right )^{\frac {3}{2}}}\)	\(110\)
gosper	\(\frac {\frac {2}{5} c^{2} x^{4} e^{4}+\frac {4}{3} b c \,e^{4} x^{3}-\frac {16}{15} c^{2} d \,e^{3} x^{3}+2 b^{2} e^{4} x^{2}-8 b c d \,e^{3} x^{2}+\frac {32}{5} c^{2} d^{2} e^{2} x^{2}+8 b^{2} d \,e^{3} x -32 b c \,d^{2} e^{2} x +\frac {128}{5} c^{2} d^{3} e x +\frac {16}{3} d^{2} e^{2} b^{2}-\frac {64}{3} b c \,d^{3} e +\frac {256}{15} c^{2} d^{4}}{e^{5} \left (e x +d \right )^{\frac {3}{2}}}\)	\(141\)
trager	\(\frac {\frac {2}{5} c^{2} x^{4} e^{4}+\frac {4}{3} b c \,e^{4} x^{3}-\frac {16}{15} c^{2} d \,e^{3} x^{3}+2 b^{2} e^{4} x^{2}-8 b c d \,e^{3} x^{2}+\frac {32}{5} c^{2} d^{2} e^{2} x^{2}+8 b^{2} d \,e^{3} x -32 b c \,d^{2} e^{2} x +\frac {128}{5} c^{2} d^{3} e x +\frac {16}{3} d^{2} e^{2} b^{2}-\frac {64}{3} b c \,d^{3} e +\frac {256}{15} c^{2} d^{4}}{e^{5} \left (e x +d \right )^{\frac {3}{2}}}\)	\(141\)
derivativedivides	\(\frac {\frac {2 c^{2} \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {4 b c e \left (e x +d \right )^{\frac {3}{2}}}{3}-\frac {8 c^{2} d \left (e x +d \right )^{\frac {3}{2}}}{3}+2 b^{2} e^{2} \sqrt {e x +d}-12 b c d e \sqrt {e x +d}+12 c^{2} d^{2} \sqrt {e x +d}-\frac {2 d^{2} \left (b^{2} e^{2}-2 b c d e +d^{2} c^{2}\right )}{3 \left (e x +d \right )^{\frac {3}{2}}}+\frac {4 d \left (b^{2} e^{2}-3 b c d e +2 d^{2} c^{2}\right )}{\sqrt {e x +d}}}{e^{5}}\)	\(151\)
default	\(\frac {\frac {2 c^{2} \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {4 b c e \left (e x +d \right )^{\frac {3}{2}}}{3}-\frac {8 c^{2} d \left (e x +d \right )^{\frac {3}{2}}}{3}+2 b^{2} e^{2} \sqrt {e x +d}-12 b c d e \sqrt {e x +d}+12 c^{2} d^{2} \sqrt {e x +d}-\frac {2 d^{2} \left (b^{2} e^{2}-2 b c d e +d^{2} c^{2}\right )}{3 \left (e x +d \right )^{\frac {3}{2}}}+\frac {4 d \left (b^{2} e^{2}-3 b c d e +2 d^{2} c^{2}\right )}{\sqrt {e x +d}}}{e^{5}}\)	\(151\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/e^5*(1/5*c^2*(e*x+d)^(5/2)+2/3*b*c*e*(e*x+d)^(3/2)-4/3*c^2*d*(e*x+d)^(3/2)+b^2*e^2*(e*x+d)^(1/2)-6*b*c*d*e*(
e*x+d)^(1/2)+6*c^2*d^2*(e*x+d)^(1/2)-1/3*d^2*(b^2*e^2-2*b*c*d*e+c^2*d^2)/(e*x+d)^(3/2)+2*d*(b^2*e^2-3*b*c*d*e+
2*c^2*d^2)/(e*x+d)^(1/2))

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Maxima [A]
time = 0.32, size = 148, normalized size = 1.03 \begin {gather*} \frac {2}{15} \, {\left ({\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} c^{2} - 10 \, {\left (2 \, c^{2} d - b c e\right )} {\left (x e + d\right )}^{\frac {3}{2}} + 15 \, {\left (6 \, c^{2} d^{2} - 6 \, b c d e + b^{2} e^{2}\right )} \sqrt {x e + d}\right )} e^{\left (-4\right )} - \frac {5 \, {\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2} - 6 \, {\left (2 \, c^{2} d^{3} - 3 \, b c d^{2} e + b^{2} d e^{2}\right )} {\left (x e + d\right )}\right )} e^{\left (-4\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)^2/(e*x+d)^(5/2),x, algorithm="maxima")

[Out]

2/15*((3*(x*e + d)^(5/2)*c^2 - 10*(2*c^2*d - b*c*e)*(x*e + d)^(3/2) + 15*(6*c^2*d^2 - 6*b*c*d*e + b^2*e^2)*sqr
t(x*e + d))*e^(-4) - 5*(c^2*d^4 - 2*b*c*d^3*e + b^2*d^2*e^2 - 6*(2*c^2*d^3 - 3*b*c*d^2*e + b^2*d*e^2)*(x*e + d
))*e^(-4)/(x*e + d)^(3/2))*e^(-1)

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Fricas [A]
time = 1.45, size = 149, normalized size = 1.04 \begin {gather*} \frac {2 \, {\left (128 \, c^{2} d^{4} + {\left (3 \, c^{2} x^{4} + 10 \, b c x^{3} + 15 \, b^{2} x^{2}\right )} e^{4} - 4 \, {\left (2 \, c^{2} d x^{3} + 15 \, b c d x^{2} - 15 \, b^{2} d x\right )} e^{3} + 8 \, {\left (6 \, c^{2} d^{2} x^{2} - 30 \, b c d^{2} x + 5 \, b^{2} d^{2}\right )} e^{2} + 32 \, {\left (6 \, c^{2} d^{3} x - 5 \, b c d^{3}\right )} e\right )} \sqrt {x e + d}}{15 \, {\left (x^{2} e^{7} + 2 \, d x e^{6} + d^{2} e^{5}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15*(128*c^2*d^4 + (3*c^2*x^4 + 10*b*c*x^3 + 15*b^2*x^2)*e^4 - 4*(2*c^2*d*x^3 + 15*b*c*d*x^2 - 15*b^2*d*x)*e^
3 + 8*(6*c^2*d^2*x^2 - 30*b*c*d^2*x + 5*b^2*d^2)*e^2 + 32*(6*c^2*d^3*x - 5*b*c*d^3)*e)*sqrt(x*e + d)/(x^2*e^7
+ 2*d*x*e^6 + d^2*e^5)

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Sympy [A]
time = 13.46, size = 139, normalized size = 0.97 \begin {gather*} \frac {2 c^{2} \left (d + e x\right )^{\frac {5}{2}}}{5 e^{5}} - \frac {2 d^{2} \left (b e - c d\right )^{2}}{3 e^{5} \left (d + e x\right )^{\frac {3}{2}}} + \frac {4 d \left (b e - 2 c d\right ) \left (b e - c d\right )}{e^{5} \sqrt {d + e x}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \cdot \left (4 b c e - 8 c^{2} d\right )}{3 e^{5}} + \frac {\sqrt {d + e x} \left (2 b^{2} e^{2} - 12 b c d e + 12 c^{2} d^{2}\right )}{e^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*c**2*(d + e*x)**(5/2)/(5*e**5) - 2*d**2*(b*e - c*d)**2/(3*e**5*(d + e*x)**(3/2)) + 4*d*(b*e - 2*c*d)*(b*e -
c*d)/(e**5*sqrt(d + e*x)) + (d + e*x)**(3/2)*(4*b*c*e - 8*c**2*d)/(3*e**5) + sqrt(d + e*x)*(2*b**2*e**2 - 12*b
*c*d*e + 12*c**2*d**2)/e**5

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Giac [A]
time = 1.28, size = 182, normalized size = 1.27 \begin {gather*} \frac {2}{15} \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} c^{2} e^{20} - 20 \, {\left (x e + d\right )}^{\frac {3}{2}} c^{2} d e^{20} + 90 \, \sqrt {x e + d} c^{2} d^{2} e^{20} + 10 \, {\left (x e + d\right )}^{\frac {3}{2}} b c e^{21} - 90 \, \sqrt {x e + d} b c d e^{21} + 15 \, \sqrt {x e + d} b^{2} e^{22}\right )} e^{\left (-25\right )} + \frac {2 \, {\left (12 \, {\left (x e + d\right )} c^{2} d^{3} - c^{2} d^{4} - 18 \, {\left (x e + d\right )} b c d^{2} e + 2 \, b c d^{3} e + 6 \, {\left (x e + d\right )} b^{2} d e^{2} - b^{2} d^{2} e^{2}\right )} e^{\left (-5\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)^2/(e*x+d)^(5/2),x, algorithm="giac")

[Out]

2/15*(3*(x*e + d)^(5/2)*c^2*e^20 - 20*(x*e + d)^(3/2)*c^2*d*e^20 + 90*sqrt(x*e + d)*c^2*d^2*e^20 + 10*(x*e + d
)^(3/2)*b*c*e^21 - 90*sqrt(x*e + d)*b*c*d*e^21 + 15*sqrt(x*e + d)*b^2*e^22)*e^(-25) + 2/3*(12*(x*e + d)*c^2*d^
3 - c^2*d^4 - 18*(x*e + d)*b*c*d^2*e + 2*b*c*d^3*e + 6*(x*e + d)*b^2*d*e^2 - b^2*d^2*e^2)*e^(-5)/(x*e + d)^(3/
2)

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Mupad [B]
time = 0.08, size = 145, normalized size = 1.01 \begin {gather*} \frac {2\,c^2\,{\left (d+e\,x\right )}^{5/2}}{5\,e^5}+\frac {\sqrt {d+e\,x}\,\left (2\,b^2\,e^2-12\,b\,c\,d\,e+12\,c^2\,d^2\right )}{e^5}-\frac {\left (8\,c^2\,d-4\,b\,c\,e\right )\,{\left (d+e\,x\right )}^{3/2}}{3\,e^5}+\frac {\left (d+e\,x\right )\,\left (4\,b^2\,d\,e^2-12\,b\,c\,d^2\,e+8\,c^2\,d^3\right )-\frac {2\,c^2\,d^4}{3}-\frac {2\,b^2\,d^2\,e^2}{3}+\frac {4\,b\,c\,d^3\,e}{3}}{e^5\,{\left (d+e\,x\right )}^{3/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*c^2*(d + e*x)^(5/2))/(5*e^5) + ((d + e*x)^(1/2)*(2*b^2*e^2 + 12*c^2*d^2 - 12*b*c*d*e))/e^5 - ((8*c^2*d - 4*
b*c*e)*(d + e*x)^(3/2))/(3*e^5) + ((d + e*x)*(8*c^2*d^3 + 4*b^2*d*e^2 - 12*b*c*d^2*e) - (2*c^2*d^4)/3 - (2*b^2
*d^2*e^2)/3 + (4*b*c*d^3*e)/3)/(e^5*(d + e*x)^(3/2))

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