3.4.30 \(\int \frac {(b x+c x^2)^2}{(d+e x)^{7/2}} \, dx\)

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

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Rubi [A]  time = 0.06, 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 = {698} \begin {gather*} -\frac {2 \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )}{e^5 \sqrt {d+e x}}-\frac {2 d^2 (c d-b e)^2}{5 e^5 (d+e x)^{5/2}}-\frac {4 c \sqrt {d+e x} (2 c d-b e)}{e^5}+\frac {4 d (c d-b e) (2 c d-b e)}{3 e^5 (d+e x)^{3/2}}+\frac {2 c^2 (d+e x)^{3/2}}{3 e^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(b*x + c*x^2)^2/(d + e*x)^(7/2),x]

[Out]

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

Rule 698

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

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Mathematica [A]  time = 0.06, size = 123, normalized size = 0.86 \begin {gather*} -\frac {2 \left (b^2 e^2 \left (8 d^2+20 d e x+15 e^2 x^2\right )-6 b c e \left (16 d^3+40 d^2 e x+30 d e^2 x^2+5 e^3 x^3\right )+c^2 \left (128 d^4+320 d^3 e x+240 d^2 e^2 x^2+40 d e^3 x^3-5 e^4 x^4\right )\right )}{15 e^5 (d+e x)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(b*x + c*x^2)^2/(d + e*x)^(7/2),x]

[Out]

(-2*(b^2*e^2*(8*d^2 + 20*d*e*x + 15*e^2*x^2) - 6*b*c*e*(16*d^3 + 40*d^2*e*x + 30*d*e^2*x^2 + 5*e^3*x^3) + c^2*
(128*d^4 + 320*d^3*e*x + 240*d^2*e^2*x^2 + 40*d*e^3*x^3 - 5*e^4*x^4)))/(15*e^5*(d + e*x)^(5/2))

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IntegrateAlgebraic [A]  time = 0.07, size = 164, normalized size = 1.15 \begin {gather*} \frac {2 \left (-3 b^2 d^2 e^2+10 b^2 d e^2 (d+e x)-15 b^2 e^2 (d+e x)^2+6 b c d^3 e-30 b c d^2 e (d+e x)+90 b c d e (d+e x)^2+30 b c e (d+e x)^3-3 c^2 d^4+20 c^2 d^3 (d+e x)-90 c^2 d^2 (d+e x)^2-60 c^2 d (d+e x)^3+5 c^2 (d+e x)^4\right )}{15 e^5 (d+e x)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(b*x + c*x^2)^2/(d + e*x)^(7/2),x]

[Out]

(2*(-3*c^2*d^4 + 6*b*c*d^3*e - 3*b^2*d^2*e^2 + 20*c^2*d^3*(d + e*x) - 30*b*c*d^2*e*(d + e*x) + 10*b^2*d*e^2*(d
+ e*x) - 90*c^2*d^2*(d + e*x)^2 + 90*b*c*d*e*(d + e*x)^2 - 15*b^2*e^2*(d + e*x)^2 - 60*c^2*d*(d + e*x)^3 + 30
*b*c*e*(d + e*x)^3 + 5*c^2*(d + e*x)^4))/(15*e^5*(d + e*x)^(5/2))

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fricas [A]  time = 0.40, size = 168, normalized size = 1.17 \begin {gather*} \frac {2 \, {\left (5 \, c^{2} e^{4} x^{4} - 128 \, c^{2} d^{4} + 96 \, b c d^{3} e - 8 \, b^{2} d^{2} e^{2} - 10 \, {\left (4 \, c^{2} d e^{3} - 3 \, b c e^{4}\right )} x^{3} - 15 \, {\left (16 \, c^{2} d^{2} e^{2} - 12 \, b c d e^{3} + b^{2} e^{4}\right )} x^{2} - 20 \, {\left (16 \, c^{2} d^{3} e - 12 \, b c d^{2} e^{2} + b^{2} d e^{3}\right )} x\right )} \sqrt {e x + d}}{15 \, {\left (e^{8} x^{3} + 3 \, d e^{7} x^{2} + 3 \, d^{2} e^{6} x + d^{3} 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)^(7/2),x, algorithm="fricas")

[Out]

2/15*(5*c^2*e^4*x^4 - 128*c^2*d^4 + 96*b*c*d^3*e - 8*b^2*d^2*e^2 - 10*(4*c^2*d*e^3 - 3*b*c*e^4)*x^3 - 15*(16*c
^2*d^2*e^2 - 12*b*c*d*e^3 + b^2*e^4)*x^2 - 20*(16*c^2*d^3*e - 12*b*c*d^2*e^2 + b^2*d*e^3)*x)*sqrt(e*x + d)/(e^
8*x^3 + 3*d*e^7*x^2 + 3*d^2*e^6*x + d^3*e^5)

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giac [A]  time = 0.18, size = 179, normalized size = 1.25 \begin {gather*} \frac {2}{3} \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} c^{2} e^{10} - 12 \, \sqrt {x e + d} c^{2} d e^{10} + 6 \, \sqrt {x e + d} b c e^{11}\right )} e^{\left (-15\right )} - \frac {2 \, {\left (90 \, {\left (x e + d\right )}^{2} c^{2} d^{2} - 20 \, {\left (x e + d\right )} c^{2} d^{3} + 3 \, c^{2} d^{4} - 90 \, {\left (x e + d\right )}^{2} b c d e + 30 \, {\left (x e + d\right )} b c d^{2} e - 6 \, b c d^{3} e + 15 \, {\left (x e + d\right )}^{2} b^{2} e^{2} - 10 \, {\left (x e + d\right )} b^{2} d e^{2} + 3 \, b^{2} d^{2} e^{2}\right )} e^{\left (-5\right )}}{15 \, {\left (x e + d\right )}^{\frac {5}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x)^2/(e*x+d)^(7/2),x, algorithm="giac")

[Out]

2/3*((x*e + d)^(3/2)*c^2*e^10 - 12*sqrt(x*e + d)*c^2*d*e^10 + 6*sqrt(x*e + d)*b*c*e^11)*e^(-15) - 2/15*(90*(x*
e + d)^2*c^2*d^2 - 20*(x*e + d)*c^2*d^3 + 3*c^2*d^4 - 90*(x*e + d)^2*b*c*d*e + 30*(x*e + d)*b*c*d^2*e - 6*b*c*
d^3*e + 15*(x*e + d)^2*b^2*e^2 - 10*(x*e + d)*b^2*d*e^2 + 3*b^2*d^2*e^2)*e^(-5)/(x*e + d)^(5/2)

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maple [A]  time = 0.05, size = 141, normalized size = 0.99 \begin {gather*} -\frac {2 \left (-5 c^{2} x^{4} e^{4}-30 b c \,e^{4} x^{3}+40 c^{2} d \,e^{3} x^{3}+15 b^{2} e^{4} x^{2}-180 b c d \,e^{3} x^{2}+240 c^{2} d^{2} e^{2} x^{2}+20 b^{2} d \,e^{3} x -240 b c \,d^{2} e^{2} x +320 c^{2} d^{3} e x +8 b^{2} d^{2} e^{2}-96 b c \,d^{3} e +128 c^{2} d^{4}\right )}{15 \left (e x +d \right )^{\frac {5}{2}} e^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+b*x)^2/(e*x+d)^(7/2),x)

[Out]

-2/15*(-5*c^2*e^4*x^4-30*b*c*e^4*x^3+40*c^2*d*e^3*x^3+15*b^2*e^4*x^2-180*b*c*d*e^3*x^2+240*c^2*d^2*e^2*x^2+20*
b^2*d*e^3*x-240*b*c*d^2*e^2*x+320*c^2*d^3*e*x+8*b^2*d^2*e^2-96*b*c*d^3*e+128*c^2*d^4)/(e*x+d)^(5/2)/e^5

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maxima [A]  time = 1.33, size = 147, normalized size = 1.03 \begin {gather*} \frac {2 \, {\left (\frac {5 \, {\left ({\left (e x + d\right )}^{\frac {3}{2}} c^{2} - 6 \, {\left (2 \, c^{2} d - b c e\right )} \sqrt {e x + d}\right )}}{e^{4}} - \frac {3 \, c^{2} d^{4} - 6 \, b c d^{3} e + 3 \, b^{2} d^{2} e^{2} + 15 \, {\left (6 \, c^{2} d^{2} - 6 \, b c d e + b^{2} e^{2}\right )} {\left (e x + d\right )}^{2} - 10 \, {\left (2 \, c^{2} d^{3} - 3 \, b c d^{2} e + b^{2} d e^{2}\right )} {\left (e x + d\right )}}{{\left (e x + d\right )}^{\frac {5}{2}} e^{4}}\right )}}{15 \, e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x)^2/(e*x+d)^(7/2),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.24, size = 140, normalized size = 0.98 \begin {gather*} -\frac {2\,\left (8\,b^2\,d^2\,e^2+20\,b^2\,d\,e^3\,x+15\,b^2\,e^4\,x^2-96\,b\,c\,d^3\,e-240\,b\,c\,d^2\,e^2\,x-180\,b\,c\,d\,e^3\,x^2-30\,b\,c\,e^4\,x^3+128\,c^2\,d^4+320\,c^2\,d^3\,e\,x+240\,c^2\,d^2\,e^2\,x^2+40\,c^2\,d\,e^3\,x^3-5\,c^2\,e^4\,x^4\right )}{15\,e^5\,{\left (d+e\,x\right )}^{5/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x + c*x^2)^2/(d + e*x)^(7/2),x)

[Out]

-(2*(128*c^2*d^4 + 8*b^2*d^2*e^2 + 15*b^2*e^4*x^2 - 5*c^2*e^4*x^4 + 40*c^2*d*e^3*x^3 - 96*b*c*d^3*e + 240*c^2*
d^2*e^2*x^2 - 30*b*c*e^4*x^3 + 20*b^2*d*e^3*x + 320*c^2*d^3*e*x - 240*b*c*d^2*e^2*x - 180*b*c*d*e^3*x^2))/(15*
e^5*(d + e*x)^(5/2))

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sympy [A]  time = 3.73, size = 787, normalized size = 5.50 \begin {gather*} \begin {cases} - \frac {16 b^{2} d^{2} e^{2}}{15 d^{2} e^{5} \sqrt {d + e x} + 30 d e^{6} x \sqrt {d + e x} + 15 e^{7} x^{2} \sqrt {d + e x}} - \frac {40 b^{2} d e^{3} x}{15 d^{2} e^{5} \sqrt {d + e x} + 30 d e^{6} x \sqrt {d + e x} + 15 e^{7} x^{2} \sqrt {d + e x}} - \frac {30 b^{2} e^{4} x^{2}}{15 d^{2} e^{5} \sqrt {d + e x} + 30 d e^{6} x \sqrt {d + e x} + 15 e^{7} x^{2} \sqrt {d + e x}} + \frac {192 b c d^{3} e}{15 d^{2} e^{5} \sqrt {d + e x} + 30 d e^{6} x \sqrt {d + e x} + 15 e^{7} x^{2} \sqrt {d + e x}} + \frac {480 b c d^{2} e^{2} x}{15 d^{2} e^{5} \sqrt {d + e x} + 30 d e^{6} x \sqrt {d + e x} + 15 e^{7} x^{2} \sqrt {d + e x}} + \frac {360 b c d e^{3} x^{2}}{15 d^{2} e^{5} \sqrt {d + e x} + 30 d e^{6} x \sqrt {d + e x} + 15 e^{7} x^{2} \sqrt {d + e x}} + \frac {60 b c e^{4} x^{3}}{15 d^{2} e^{5} \sqrt {d + e x} + 30 d e^{6} x \sqrt {d + e x} + 15 e^{7} x^{2} \sqrt {d + e x}} - \frac {256 c^{2} d^{4}}{15 d^{2} e^{5} \sqrt {d + e x} + 30 d e^{6} x \sqrt {d + e x} + 15 e^{7} x^{2} \sqrt {d + e x}} - \frac {640 c^{2} d^{3} e x}{15 d^{2} e^{5} \sqrt {d + e x} + 30 d e^{6} x \sqrt {d + e x} + 15 e^{7} x^{2} \sqrt {d + e x}} - \frac {480 c^{2} d^{2} e^{2} x^{2}}{15 d^{2} e^{5} \sqrt {d + e x} + 30 d e^{6} x \sqrt {d + e x} + 15 e^{7} x^{2} \sqrt {d + e x}} - \frac {80 c^{2} d e^{3} x^{3}}{15 d^{2} e^{5} \sqrt {d + e x} + 30 d e^{6} x \sqrt {d + e x} + 15 e^{7} x^{2} \sqrt {d + e x}} + \frac {10 c^{2} e^{4} x^{4}}{15 d^{2} e^{5} \sqrt {d + e x} + 30 d e^{6} x \sqrt {d + e x} + 15 e^{7} x^{2} \sqrt {d + e x}} & \text {for}\: e \neq 0 \\\frac {\frac {b^{2} x^{3}}{3} + \frac {b c x^{4}}{2} + \frac {c^{2} x^{5}}{5}}{d^{\frac {7}{2}}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+b*x)**2/(e*x+d)**(7/2),x)

[Out]

Piecewise((-16*b**2*d**2*e**2/(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d +
e*x)) - 40*b**2*d*e**3*x/(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x))
- 30*b**2*e**4*x**2/(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) + 1
92*b*c*d**3*e/(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) + 480*b*c*
d**2*e**2*x/(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) + 360*b*c*d*
e**3*x**2/(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) + 60*b*c*e**4*
x**3/(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) - 256*c**2*d**4/(15
*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) - 640*c**2*d**3*e*x/(15*d**
2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) - 480*c**2*d**2*e**2*x**2/(15*d
**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) - 80*c**2*d*e**3*x**3/(15*d**
2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) + 10*c**2*e**4*x**4/(15*d**2*e*
*5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)), Ne(e, 0)), ((b**2*x**3/3 + b*c*x**
4/2 + c**2*x**5/5)/d**(7/2), True))

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