### 3.360 $$\int \frac{(b x+c x^2)^3}{(d+e x)^{7/2}} \, dx$$

Optimal. Leaf size=240 $\frac{2 c (d+e x)^{3/2} \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{e^7}-\frac{2 \sqrt{d+e x} (2 c d-b e) \left (b^2 e^2-10 b c d e+10 c^2 d^2\right )}{e^7}-\frac{6 d (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{e^7 \sqrt{d+e x}}-\frac{6 c^2 (d+e x)^{5/2} (2 c d-b e)}{5 e^7}+\frac{2 d^2 (c d-b e)^2 (2 c d-b e)}{e^7 (d+e x)^{3/2}}-\frac{2 d^3 (c d-b e)^3}{5 e^7 (d+e x)^{5/2}}+\frac{2 c^3 (d+e x)^{7/2}}{7 e^7}$

[Out]

(-2*d^3*(c*d - b*e)^3)/(5*e^7*(d + e*x)^(5/2)) + (2*d^2*(c*d - b*e)^2*(2*c*d - b*e))/(e^7*(d + e*x)^(3/2)) - (
6*d*(c*d - b*e)*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2))/(e^7*Sqrt[d + e*x]) - (2*(2*c*d - b*e)*(10*c^2*d^2 - 10*b*c
*d*e + b^2*e^2)*Sqrt[d + e*x])/e^7 + (2*c*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2)*(d + e*x)^(3/2))/e^7 - (6*c^2*(2*c
*d - b*e)*(d + e*x)^(5/2))/(5*e^7) + (2*c^3*(d + e*x)^(7/2))/(7*e^7)

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Rubi [A]  time = 0.106346, antiderivative size = 240, normalized size of antiderivative = 1., 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} $\frac{2 c (d+e x)^{3/2} \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{e^7}-\frac{2 \sqrt{d+e x} (2 c d-b e) \left (b^2 e^2-10 b c d e+10 c^2 d^2\right )}{e^7}-\frac{6 d (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{e^7 \sqrt{d+e x}}-\frac{6 c^2 (d+e x)^{5/2} (2 c d-b e)}{5 e^7}+\frac{2 d^2 (c d-b e)^2 (2 c d-b e)}{e^7 (d+e x)^{3/2}}-\frac{2 d^3 (c d-b e)^3}{5 e^7 (d+e x)^{5/2}}+\frac{2 c^3 (d+e x)^{7/2}}{7 e^7}$

Antiderivative was successfully veriﬁed.

[In]

Int[(b*x + c*x^2)^3/(d + e*x)^(7/2),x]

[Out]

(-2*d^3*(c*d - b*e)^3)/(5*e^7*(d + e*x)^(5/2)) + (2*d^2*(c*d - b*e)^2*(2*c*d - b*e))/(e^7*(d + e*x)^(3/2)) - (
6*d*(c*d - b*e)*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2))/(e^7*Sqrt[d + e*x]) - (2*(2*c*d - b*e)*(10*c^2*d^2 - 10*b*c
*d*e + b^2*e^2)*Sqrt[d + e*x])/e^7 + (2*c*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2)*(d + e*x)^(3/2))/e^7 - (6*c^2*(2*c
*d - b*e)*(d + e*x)^(5/2))/(5*e^7) + (2*c^3*(d + e*x)^(7/2))/(7*e^7)

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 )^3}{(d+e x)^{7/2}} \, dx &=\int \left (\frac{d^3 (c d-b e)^3}{e^6 (d+e x)^{7/2}}-\frac{3 d^2 (c d-b e)^2 (2 c d-b e)}{e^6 (d+e x)^{5/2}}+\frac{3 d (c d-b e) \left (5 c^2 d^2-5 b c d e+b^2 e^2\right )}{e^6 (d+e x)^{3/2}}+\frac{(2 c d-b e) \left (-10 c^2 d^2+10 b c d e-b^2 e^2\right )}{e^6 \sqrt{d+e x}}+\frac{3 c \left (5 c^2 d^2-5 b c d e+b^2 e^2\right ) \sqrt{d+e x}}{e^6}-\frac{3 c^2 (2 c d-b e) (d+e x)^{3/2}}{e^6}+\frac{c^3 (d+e x)^{5/2}}{e^6}\right ) \, dx\\ &=-\frac{2 d^3 (c d-b e)^3}{5 e^7 (d+e x)^{5/2}}+\frac{2 d^2 (c d-b e)^2 (2 c d-b e)}{e^7 (d+e x)^{3/2}}-\frac{6 d (c d-b e) \left (5 c^2 d^2-5 b c d e+b^2 e^2\right )}{e^7 \sqrt{d+e x}}-\frac{2 (2 c d-b e) \left (10 c^2 d^2-10 b c d e+b^2 e^2\right ) \sqrt{d+e x}}{e^7}+\frac{2 c \left (5 c^2 d^2-5 b c d e+b^2 e^2\right ) (d+e x)^{3/2}}{e^7}-\frac{6 c^2 (2 c d-b e) (d+e x)^{5/2}}{5 e^7}+\frac{2 c^3 (d+e x)^{7/2}}{7 e^7}\\ \end{align*}

Mathematica [A]  time = 0.139434, size = 206, normalized size = 0.86 $\frac{2 \left (35 c (d+e x)^4 \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )-35 (d+e x)^3 (2 c d-b e) \left (b^2 e^2-10 b c d e+10 c^2 d^2\right )-105 d (d+e x)^2 (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )-21 c^2 (d+e x)^5 (2 c d-b e)+35 d^2 (d+e x) (c d-b e)^2 (2 c d-b e)-7 d^3 (c d-b e)^3+5 c^3 (d+e x)^6\right )}{35 e^7 (d+e x)^{5/2}}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(b*x + c*x^2)^3/(d + e*x)^(7/2),x]

[Out]

(2*(-7*d^3*(c*d - b*e)^3 + 35*d^2*(c*d - b*e)^2*(2*c*d - b*e)*(d + e*x) - 105*d*(c*d - b*e)*(5*c^2*d^2 - 5*b*c
*d*e + b^2*e^2)*(d + e*x)^2 - 35*(2*c*d - b*e)*(10*c^2*d^2 - 10*b*c*d*e + b^2*e^2)*(d + e*x)^3 + 35*c*(5*c^2*d
^2 - 5*b*c*d*e + b^2*e^2)*(d + e*x)^4 - 21*c^2*(2*c*d - b*e)*(d + e*x)^5 + 5*c^3*(d + e*x)^6))/(35*e^7*(d + e*
x)^(5/2))

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Maple [A]  time = 0.049, size = 286, normalized size = 1.2 \begin{align*}{\frac{10\,{c}^{3}{x}^{6}{e}^{6}+42\,b{c}^{2}{e}^{6}{x}^{5}-24\,{c}^{3}d{e}^{5}{x}^{5}+70\,{b}^{2}c{e}^{6}{x}^{4}-140\,b{c}^{2}d{e}^{5}{x}^{4}+80\,{c}^{3}{d}^{2}{e}^{4}{x}^{4}+70\,{b}^{3}{e}^{6}{x}^{3}-560\,{b}^{2}cd{e}^{5}{x}^{3}+1120\,b{c}^{2}{d}^{2}{e}^{4}{x}^{3}-640\,{c}^{3}{d}^{3}{e}^{3}{x}^{3}+420\,{b}^{3}d{e}^{5}{x}^{2}-3360\,{b}^{2}c{d}^{2}{e}^{4}{x}^{2}+6720\,b{c}^{2}{d}^{3}{e}^{3}{x}^{2}-3840\,{c}^{3}{d}^{4}{e}^{2}{x}^{2}+560\,{b}^{3}{d}^{2}{e}^{4}x-4480\,{b}^{2}c{d}^{3}{e}^{3}x+8960\,b{c}^{2}{d}^{4}{e}^{2}x-5120\,{c}^{3}{d}^{5}ex+224\,{b}^{3}{d}^{3}{e}^{3}-1792\,{b}^{2}c{d}^{4}{e}^{2}+3584\,b{c}^{2}{d}^{5}e-2048\,{c}^{3}{d}^{6}}{35\,{e}^{7}} \left ( ex+d \right ) ^{-{\frac{5}{2}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+b*x)^3/(e*x+d)^(7/2),x)

[Out]

2/35*(5*c^3*e^6*x^6+21*b*c^2*e^6*x^5-12*c^3*d*e^5*x^5+35*b^2*c*e^6*x^4-70*b*c^2*d*e^5*x^4+40*c^3*d^2*e^4*x^4+3
5*b^3*e^6*x^3-280*b^2*c*d*e^5*x^3+560*b*c^2*d^2*e^4*x^3-320*c^3*d^3*e^3*x^3+210*b^3*d*e^5*x^2-1680*b^2*c*d^2*e
^4*x^2+3360*b*c^2*d^3*e^3*x^2-1920*c^3*d^4*e^2*x^2+280*b^3*d^2*e^4*x-2240*b^2*c*d^3*e^3*x+4480*b*c^2*d^4*e^2*x
-2560*c^3*d^5*e*x+112*b^3*d^3*e^3-896*b^2*c*d^4*e^2+1792*b*c^2*d^5*e-1024*c^3*d^6)/(e*x+d)^(5/2)/e^7

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Maxima [A]  time = 1.12873, size = 374, normalized size = 1.56 \begin{align*} \frac{2 \,{\left (\frac{5 \,{\left (e x + d\right )}^{\frac{7}{2}} c^{3} - 21 \,{\left (2 \, c^{3} d - b c^{2} e\right )}{\left (e x + d\right )}^{\frac{5}{2}} + 35 \,{\left (5 \, c^{3} d^{2} - 5 \, b c^{2} d e + b^{2} c e^{2}\right )}{\left (e x + d\right )}^{\frac{3}{2}} - 35 \,{\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 )} \sqrt{e x + d}}{e^{6}} - \frac{7 \,{\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} + 15 \,{\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 (e x + d\right )}^{2} - 5 \,{\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 )}{\left (e x + d\right )}\right )}}{{\left (e x + d\right )}^{\frac{5}{2}} e^{6}}\right )}}{35 \, e} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x)^3/(e*x+d)^(7/2),x, algorithm="maxima")

[Out]

2/35*((5*(e*x + d)^(7/2)*c^3 - 21*(2*c^3*d - b*c^2*e)*(e*x + d)^(5/2) + 35*(5*c^3*d^2 - 5*b*c^2*d*e + b^2*c*e^
2)*(e*x + d)^(3/2) - 35*(20*c^3*d^3 - 30*b*c^2*d^2*e + 12*b^2*c*d*e^2 - b^3*e^3)*sqrt(e*x + d))/e^6 - 7*(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 + 15*(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)*(e*x + d)^2 - 5*(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)*(e*x + d))/((e*x + d)^(5/2)*e
^6))/e

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Fricas [A]  time = 2.4311, size = 645, normalized size = 2.69 \begin{align*} \frac{2 \,{\left (5 \, c^{3} e^{6} x^{6} - 1024 \, c^{3} d^{6} + 1792 \, b c^{2} d^{5} e - 896 \, b^{2} c d^{4} e^{2} + 112 \, b^{3} d^{3} e^{3} - 3 \,{\left (4 \, c^{3} d e^{5} - 7 \, b c^{2} e^{6}\right )} x^{5} + 5 \,{\left (8 \, c^{3} d^{2} e^{4} - 14 \, b c^{2} d e^{5} + 7 \, b^{2} c e^{6}\right )} x^{4} - 5 \,{\left (64 \, c^{3} d^{3} e^{3} - 112 \, b c^{2} d^{2} e^{4} + 56 \, b^{2} c d e^{5} - 7 \, b^{3} e^{6}\right )} x^{3} - 30 \,{\left (64 \, c^{3} d^{4} e^{2} - 112 \, b c^{2} d^{3} e^{3} + 56 \, b^{2} c d^{2} e^{4} - 7 \, b^{3} d e^{5}\right )} x^{2} - 40 \,{\left (64 \, c^{3} d^{5} e - 112 \, b c^{2} d^{4} e^{2} + 56 \, b^{2} c d^{3} e^{3} - 7 \, b^{3} d^{2} e^{4}\right )} x\right )} \sqrt{e x + d}}{35 \,{\left (e^{10} x^{3} + 3 \, d e^{9} x^{2} + 3 \, d^{2} e^{8} x + d^{3} e^{7}\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x)^3/(e*x+d)^(7/2),x, algorithm="fricas")

[Out]

2/35*(5*c^3*e^6*x^6 - 1024*c^3*d^6 + 1792*b*c^2*d^5*e - 896*b^2*c*d^4*e^2 + 112*b^3*d^3*e^3 - 3*(4*c^3*d*e^5 -
7*b*c^2*e^6)*x^5 + 5*(8*c^3*d^2*e^4 - 14*b*c^2*d*e^5 + 7*b^2*c*e^6)*x^4 - 5*(64*c^3*d^3*e^3 - 112*b*c^2*d^2*e
^4 + 56*b^2*c*d*e^5 - 7*b^3*e^6)*x^3 - 30*(64*c^3*d^4*e^2 - 112*b*c^2*d^3*e^3 + 56*b^2*c*d^2*e^4 - 7*b^3*d*e^5
)*x^2 - 40*(64*c^3*d^5*e - 112*b*c^2*d^4*e^2 + 56*b^2*c*d^3*e^3 - 7*b^3*d^2*e^4)*x)*sqrt(e*x + d)/(e^10*x^3 +
3*d*e^9*x^2 + 3*d^2*e^8*x + d^3*e^7)

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Sympy [A]  time = 82.5956, size = 248, normalized size = 1.03 \begin{align*} \frac{2 c^{3} \left (d + e x\right )^{\frac{7}{2}}}{7 e^{7}} + \frac{2 d^{3} \left (b e - c d\right )^{3}}{5 e^{7} \left (d + e x\right )^{\frac{5}{2}}} - \frac{2 d^{2} \left (b e - 2 c d\right ) \left (b e - c d\right )^{2}}{e^{7} \left (d + e x\right )^{\frac{3}{2}}} + \frac{6 d \left (b e - c d\right ) \left (b^{2} e^{2} - 5 b c d e + 5 c^{2} d^{2}\right )}{e^{7} \sqrt{d + e x}} + \frac{\left (d + e x\right )^{\frac{5}{2}} \left (6 b c^{2} e - 12 c^{3} d\right )}{5 e^{7}} + \frac{\left (d + e x\right )^{\frac{3}{2}} \left (6 b^{2} c e^{2} - 30 b c^{2} d e + 30 c^{3} d^{2}\right )}{3 e^{7}} + \frac{\sqrt{d + e x} \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 )}{e^{7}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+b*x)**3/(e*x+d)**(7/2),x)

[Out]

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

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Giac [A]  time = 1.30761, size = 485, normalized size = 2.02 \begin{align*} \frac{2}{35} \,{\left (5 \,{\left (x e + d\right )}^{\frac{7}{2}} c^{3} e^{42} - 42 \,{\left (x e + d\right )}^{\frac{5}{2}} c^{3} d e^{42} + 175 \,{\left (x e + d\right )}^{\frac{3}{2}} c^{3} d^{2} e^{42} - 700 \, \sqrt{x e + d} c^{3} d^{3} e^{42} + 21 \,{\left (x e + d\right )}^{\frac{5}{2}} b c^{2} e^{43} - 175 \,{\left (x e + d\right )}^{\frac{3}{2}} b c^{2} d e^{43} + 1050 \, \sqrt{x e + d} b c^{2} d^{2} e^{43} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{2} c e^{44} - 420 \, \sqrt{x e + d} b^{2} c d e^{44} + 35 \, \sqrt{x e + d} b^{3} e^{45}\right )} e^{\left (-49\right )} - \frac{2 \,{\left (75 \,{\left (x e + d\right )}^{2} c^{3} d^{4} - 10 \,{\left (x e + d\right )} c^{3} d^{5} + c^{3} d^{6} - 150 \,{\left (x e + d\right )}^{2} b c^{2} d^{3} e + 25 \,{\left (x e + d\right )} b c^{2} d^{4} e - 3 \, b c^{2} d^{5} e + 90 \,{\left (x e + d\right )}^{2} b^{2} c d^{2} e^{2} - 20 \,{\left (x e + d\right )} b^{2} c d^{3} e^{2} + 3 \, b^{2} c d^{4} e^{2} - 15 \,{\left (x e + d\right )}^{2} b^{3} d e^{3} + 5 \,{\left (x e + d\right )} b^{3} d^{2} e^{3} - b^{3} d^{3} e^{3}\right )} e^{\left (-7\right )}}{5 \,{\left (x e + d\right )}^{\frac{5}{2}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x)^3/(e*x+d)^(7/2),x, algorithm="giac")

[Out]

2/35*(5*(x*e + d)^(7/2)*c^3*e^42 - 42*(x*e + d)^(5/2)*c^3*d*e^42 + 175*(x*e + d)^(3/2)*c^3*d^2*e^42 - 700*sqrt
(x*e + d)*c^3*d^3*e^42 + 21*(x*e + d)^(5/2)*b*c^2*e^43 - 175*(x*e + d)^(3/2)*b*c^2*d*e^43 + 1050*sqrt(x*e + d)
*b*c^2*d^2*e^43 + 35*(x*e + d)^(3/2)*b^2*c*e^44 - 420*sqrt(x*e + d)*b^2*c*d*e^44 + 35*sqrt(x*e + d)*b^3*e^45)*
e^(-49) - 2/5*(75*(x*e + d)^2*c^3*d^4 - 10*(x*e + d)*c^3*d^5 + c^3*d^6 - 150*(x*e + d)^2*b*c^2*d^3*e + 25*(x*e
+ d)*b*c^2*d^4*e - 3*b*c^2*d^5*e + 90*(x*e + d)^2*b^2*c*d^2*e^2 - 20*(x*e + d)*b^2*c*d^3*e^2 + 3*b^2*c*d^4*e^
2 - 15*(x*e + d)^2*b^3*d*e^3 + 5*(x*e + d)*b^3*d^2*e^3 - b^3*d^3*e^3)*e^(-7)/(x*e + d)^(5/2)