3.4.50 \(\int \frac {(b x+c x^2)^2}{(d+e x)^{3/2}} \, dx\) [350]

Optimal. Leaf size=143 \[ -\frac {2 d^2 (c d-b e)^2}{e^5 \sqrt {d+e x}}-\frac {4 d (c d-b e) (2 c d-b e) \sqrt {d+e x}}{e^5}+\frac {2 \left (6 c^2 d^2-6 b c d e+b^2 e^2\right ) (d+e x)^{3/2}}{3 e^5}-\frac {4 c (2 c d-b e) (d+e x)^{5/2}}{5 e^5}+\frac {2 c^2 (d+e x)^{7/2}}{7 e^5} \]

[Out]

2/3*(b^2*e^2-6*b*c*d*e+6*c^2*d^2)*(e*x+d)^(3/2)/e^5-4/5*c*(-b*e+2*c*d)*(e*x+d)^(5/2)/e^5+2/7*c^2*(e*x+d)^(7/2)
/e^5-2*d^2*(-b*e+c*d)^2/e^5/(e*x+d)^(1/2)-4*d*(-b*e+c*d)*(-b*e+2*c*d)*(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 (d+e x)^{3/2} \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )}{3 e^5}-\frac {2 d^2 (c d-b e)^2}{e^5 \sqrt {d+e x}}-\frac {4 c (d+e x)^{5/2} (2 c d-b e)}{5 e^5}-\frac {4 d \sqrt {d+e x} (c d-b e) (2 c d-b e)}{e^5}+\frac {2 c^2 (d+e x)^{7/2}}{7 e^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]
time = 0.07, size = 123, normalized size = 0.86 \begin {gather*} \frac {70 b^2 e^2 \left (-8 d^2-4 d e x+e^2 x^2\right )+84 b c e \left (16 d^3+8 d^2 e x-2 d e^2 x^2+e^3 x^3\right )-6 c^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 )}{105 e^5 \sqrt {d+e x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(70*b^2*e^2*(-8*d^2 - 4*d*e*x + e^2*x^2) + 84*b*c*e*(16*d^3 + 8*d^2*e*x - 2*d*e^2*x^2 + e^3*x^3) - 6*c^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))/(105*e^5*Sqrt[d + e*x])

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Maple [A]
time = 0.45, size = 166, normalized size = 1.16

method	result	size
risch	\(-\frac {2 \left (-15 c^{2} x^{3} e^{3}-42 b c \,e^{3} x^{2}+39 c^{2} d \,e^{2} x^{2}-35 e^{3} x \,b^{2}+126 b c d \,e^{2} x -87 c^{2} d^{2} e x +175 b^{2} d \,e^{2}-462 b c \,d^{2} e +279 c^{2} d^{3}\right ) \sqrt {e x +d}}{105 e^{5}}-\frac {2 d^{2} \left (b^{2} e^{2}-2 b c d e +d^{2} c^{2}\right )}{e^{5} \sqrt {e x +d}}\)	\(137\)
gosper	\(-\frac {2 \left (-15 c^{2} x^{4} e^{4}-42 b c \,e^{4} x^{3}+24 c^{2} d \,e^{3} x^{3}-35 b^{2} e^{4} x^{2}+84 b c d \,e^{3} x^{2}-48 c^{2} d^{2} e^{2} x^{2}+140 b^{2} d \,e^{3} x -336 b c \,d^{2} e^{2} x +192 c^{2} d^{3} e x +280 d^{2} e^{2} b^{2}-672 b c \,d^{3} e +384 c^{2} d^{4}\right )}{105 \sqrt {e x +d}\, e^{5}}\)	\(141\)
trager	\(-\frac {2 \left (-15 c^{2} x^{4} e^{4}-42 b c \,e^{4} x^{3}+24 c^{2} d \,e^{3} x^{3}-35 b^{2} e^{4} x^{2}+84 b c d \,e^{3} x^{2}-48 c^{2} d^{2} e^{2} x^{2}+140 b^{2} d \,e^{3} x -336 b c \,d^{2} e^{2} x +192 c^{2} d^{3} e x +280 d^{2} e^{2} b^{2}-672 b c \,d^{3} e +384 c^{2} d^{4}\right )}{105 \sqrt {e x +d}\, e^{5}}\)	\(141\)
derivativedivides	\(\frac {\frac {2 c^{2} \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {4 b c e \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {8 c^{2} d \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {2 b^{2} e^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}-4 b c d e \left (e x +d \right )^{\frac {3}{2}}+4 c^{2} d^{2} \left (e x +d \right )^{\frac {3}{2}}-4 b^{2} d \,e^{2} \sqrt {e x +d}+12 b c \,d^{2} e \sqrt {e x +d}-8 c^{2} d^{3} \sqrt {e x +d}-\frac {2 d^{2} \left (b^{2} e^{2}-2 b c d e +d^{2} c^{2}\right )}{\sqrt {e x +d}}}{e^{5}}\)	\(166\)
default	\(\frac {\frac {2 c^{2} \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {4 b c e \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {8 c^{2} d \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {2 b^{2} e^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}-4 b c d e \left (e x +d \right )^{\frac {3}{2}}+4 c^{2} d^{2} \left (e x +d \right )^{\frac {3}{2}}-4 b^{2} d \,e^{2} \sqrt {e x +d}+12 b c \,d^{2} e \sqrt {e x +d}-8 c^{2} d^{3} \sqrt {e x +d}-\frac {2 d^{2} \left (b^{2} e^{2}-2 b c d e +d^{2} c^{2}\right )}{\sqrt {e x +d}}}{e^{5}}\)	\(166\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/e^5*(1/7*c^2*(e*x+d)^(7/2)+2/5*b*c*e*(e*x+d)^(5/2)-4/5*c^2*d*(e*x+d)^(5/2)+1/3*b^2*e^2*(e*x+d)^(3/2)-2*b*c*d
*e*(e*x+d)^(3/2)+2*c^2*d^2*(e*x+d)^(3/2)-2*b^2*d*e^2*(e*x+d)^(1/2)+6*b*c*d^2*e*(e*x+d)^(1/2)-4*c^2*d^3*(e*x+d)
^(1/2)-d^2*(b^2*e^2-2*b*c*d*e+c^2*d^2)/(e*x+d)^(1/2))

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Maxima [A]
time = 0.30, size = 150, normalized size = 1.05 \begin {gather*} \frac {2}{105} \, {\left ({\left (15 \, {\left (x e + d\right )}^{\frac {7}{2}} c^{2} - 42 \, {\left (2 \, c^{2} d - b c e\right )} {\left (x e + d\right )}^{\frac {5}{2}} + 35 \, {\left (6 \, c^{2} d^{2} - 6 \, b c d e + b^{2} e^{2}\right )} {\left (x e + d\right )}^{\frac {3}{2}} - 210 \, {\left (2 \, c^{2} d^{3} - 3 \, b c d^{2} e + b^{2} d e^{2}\right )} \sqrt {x e + d}\right )} e^{\left (-4\right )} - \frac {105 \, {\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2}\right )} e^{\left (-4\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)^2/(e*x+d)^(3/2),x, algorithm="maxima")

[Out]

2/105*((15*(x*e + d)^(7/2)*c^2 - 42*(2*c^2*d - b*c*e)*(x*e + d)^(5/2) + 35*(6*c^2*d^2 - 6*b*c*d*e + b^2*e^2)*(
x*e + d)^(3/2) - 210*(2*c^2*d^3 - 3*b*c*d^2*e + b^2*d*e^2)*sqrt(x*e + d))*e^(-4) - 105*(c^2*d^4 - 2*b*c*d^3*e
+ b^2*d^2*e^2)*e^(-4)/sqrt(x*e + d))*e^(-1)

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Fricas [A]
time = 1.54, size = 140, normalized size = 0.98 \begin {gather*} -\frac {2 \, {\left (384 \, c^{2} d^{4} - {\left (15 \, c^{2} x^{4} + 42 \, b c x^{3} + 35 \, b^{2} x^{2}\right )} e^{4} + 4 \, {\left (6 \, c^{2} d x^{3} + 21 \, b c d x^{2} + 35 \, b^{2} d x\right )} e^{3} - 8 \, {\left (6 \, c^{2} d^{2} x^{2} + 42 \, b c d^{2} x - 35 \, b^{2} d^{2}\right )} e^{2} + 96 \, {\left (2 \, c^{2} d^{3} x - 7 \, b c d^{3}\right )} e\right )} \sqrt {x e + d}}{105 \, {\left (x e^{6} + d 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)^(3/2),x, algorithm="fricas")

[Out]

-2/105*(384*c^2*d^4 - (15*c^2*x^4 + 42*b*c*x^3 + 35*b^2*x^2)*e^4 + 4*(6*c^2*d*x^3 + 21*b*c*d*x^2 + 35*b^2*d*x)
*e^3 - 8*(6*c^2*d^2*x^2 + 42*b*c*d^2*x - 35*b^2*d^2)*e^2 + 96*(2*c^2*d^3*x - 7*b*c*d^3)*e)*sqrt(x*e + d)/(x*e^
6 + d*e^5)

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Sympy [A]
time = 8.85, size = 150, normalized size = 1.05 \begin {gather*} \frac {2 c^{2} \left (d + e x\right )^{\frac {7}{2}}}{7 e^{5}} - \frac {2 d^{2} \left (b e - c d\right )^{2}}{e^{5} \sqrt {d + e x}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \cdot \left (4 b c e - 8 c^{2} d\right )}{5 e^{5}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \cdot \left (2 b^{2} e^{2} - 12 b c d e + 12 c^{2} d^{2}\right )}{3 e^{5}} + \frac {\sqrt {d + e x} \left (- 4 b^{2} d e^{2} + 12 b c d^{2} e - 8 c^{2} d^{3}\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)**(3/2),x)

[Out]

2*c**2*(d + e*x)**(7/2)/(7*e**5) - 2*d**2*(b*e - c*d)**2/(e**5*sqrt(d + e*x)) + (d + e*x)**(5/2)*(4*b*c*e - 8*
c**2*d)/(5*e**5) + (d + e*x)**(3/2)*(2*b**2*e**2 - 12*b*c*d*e + 12*c**2*d**2)/(3*e**5) + sqrt(d + e*x)*(-4*b**
2*d*e**2 + 12*b*c*d**2*e - 8*c**2*d**3)/e**5

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Giac [A]
time = 1.66, size = 188, normalized size = 1.31 \begin {gather*} \frac {2}{105} \, {\left (15 \, {\left (x e + d\right )}^{\frac {7}{2}} c^{2} e^{30} - 84 \, {\left (x e + d\right )}^{\frac {5}{2}} c^{2} d e^{30} + 210 \, {\left (x e + d\right )}^{\frac {3}{2}} c^{2} d^{2} e^{30} - 420 \, \sqrt {x e + d} c^{2} d^{3} e^{30} + 42 \, {\left (x e + d\right )}^{\frac {5}{2}} b c e^{31} - 210 \, {\left (x e + d\right )}^{\frac {3}{2}} b c d e^{31} + 630 \, \sqrt {x e + d} b c d^{2} e^{31} + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{2} e^{32} - 210 \, \sqrt {x e + d} b^{2} d e^{32}\right )} e^{\left (-35\right )} - \frac {2 \, {\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2}\right )} e^{\left (-5\right )}}{\sqrt {x e + d}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/105*(15*(x*e + d)^(7/2)*c^2*e^30 - 84*(x*e + d)^(5/2)*c^2*d*e^30 + 210*(x*e + d)^(3/2)*c^2*d^2*e^30 - 420*sq
rt(x*e + d)*c^2*d^3*e^30 + 42*(x*e + d)^(5/2)*b*c*e^31 - 210*(x*e + d)^(3/2)*b*c*d*e^31 + 630*sqrt(x*e + d)*b*
c*d^2*e^31 + 35*(x*e + d)^(3/2)*b^2*e^32 - 210*sqrt(x*e + d)*b^2*d*e^32)*e^(-35) - 2*(c^2*d^4 - 2*b*c*d^3*e +
b^2*d^2*e^2)*e^(-5)/sqrt(x*e + d)

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Mupad [B]
time = 0.19, size = 153, normalized size = 1.07 \begin {gather*} \frac {2\,c^2\,{\left (d+e\,x\right )}^{7/2}}{7\,e^5}-\frac {\sqrt {d+e\,x}\,\left (4\,b^2\,d\,e^2-12\,b\,c\,d^2\,e+8\,c^2\,d^3\right )}{e^5}+\frac {{\left (d+e\,x\right )}^{3/2}\,\left (2\,b^2\,e^2-12\,b\,c\,d\,e+12\,c^2\,d^2\right )}{3\,e^5}-\frac {2\,b^2\,d^2\,e^2-4\,b\,c\,d^3\,e+2\,c^2\,d^4}{e^5\,\sqrt {d+e\,x}}-\frac {\left (8\,c^2\,d-4\,b\,c\,e\right )\,{\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)^(7/2))/(7*e^5) - ((d + e*x)^(1/2)*(8*c^2*d^3 + 4*b^2*d*e^2 - 12*b*c*d^2*e))/e^5 + ((d + e*x)^
(3/2)*(2*b^2*e^2 + 12*c^2*d^2 - 12*b*c*d*e))/(3*e^5) - (2*c^2*d^4 + 2*b^2*d^2*e^2 - 4*b*c*d^3*e)/(e^5*(d + e*x
)^(1/2)) - ((8*c^2*d - 4*b*c*e)*(d + e*x)^(5/2))/(5*e^5)

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