3.23.88 \(\int \frac {(a+b x+c x^2)^3}{(d+e x)^{7/2}} \, dx\) [2288]

Optimal. Leaf size=278 \[ -\frac {2 \left (c d^2-b d e+a e^2\right )^3}{5 e^7 (d+e x)^{5/2}}+\frac {2 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^2}{e^7 (d+e x)^{3/2}}-\frac {6 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )}{e^7 \sqrt {d+e x}}-\frac {2 (2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right ) \sqrt {d+e x}}{e^7}+\frac {2 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^{3/2}}{e^7}-\frac {6 c^2 (2 c d-b e) (d+e x)^{5/2}}{5 e^7}+\frac {2 c^3 (d+e x)^{7/2}}{7 e^7} \]

[Out]

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

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Rubi [A]
time = 0.08, antiderivative size = 278, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {712} \begin {gather*} \frac {2 c (d+e x)^{3/2} \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^7}-\frac {2 \sqrt {d+e x} (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{e^7}-\frac {6 \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^7 \sqrt {d+e x}}+\frac {2 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{e^7 (d+e x)^{3/2}}-\frac {2 \left (a e^2-b d e+c d^2\right )^3}{5 e^7 (d+e x)^{5/2}}-\frac {6 c^2 (d+e x)^{5/2} (2 c d-b e)}{5 e^7}+\frac {2 c^3 (d+e x)^{7/2}}{7 e^7} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]
time = 0.29, size = 391, normalized size = 1.41 \begin {gather*} -\frac {2 \left (c^3 \left (1024 d^6+2560 d^5 e x+1920 d^4 e^2 x^2+320 d^3 e^3 x^3-40 d^2 e^4 x^4+12 d e^5 x^5-5 e^6 x^6\right )+7 e^3 \left (a^3 e^3+a^2 b e^2 (2 d+5 e x)+a b^2 e \left (8 d^2+20 d e x+15 e^2 x^2\right )-b^3 \left (16 d^3+40 d^2 e x+30 d e^2 x^2+5 e^3 x^3\right )\right )+7 c e^2 \left (a^2 e^2 \left (8 d^2+20 d e x+15 e^2 x^2\right )-6 a b e \left (16 d^3+40 d^2 e x+30 d e^2 x^2+5 e^3 x^3\right )+b^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 )-7 c^2 e \left (a e \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 )+b \left (256 d^5+640 d^4 e x+480 d^3 e^2 x^2+80 d^2 e^3 x^3-10 d e^4 x^4+3 e^5 x^5\right )\right )\right )}{35 e^7 (d+e x)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(c^3*(1024*d^6 + 2560*d^5*e*x + 1920*d^4*e^2*x^2 + 320*d^3*e^3*x^3 - 40*d^2*e^4*x^4 + 12*d*e^5*x^5 - 5*e^6
*x^6) + 7*e^3*(a^3*e^3 + a^2*b*e^2*(2*d + 5*e*x) + a*b^2*e*(8*d^2 + 20*d*e*x + 15*e^2*x^2) - b^3*(16*d^3 + 40*
d^2*e*x + 30*d*e^2*x^2 + 5*e^3*x^3)) + 7*c*e^2*(a^2*e^2*(8*d^2 + 20*d*e*x + 15*e^2*x^2) - 6*a*b*e*(16*d^3 + 40
*d^2*e*x + 30*d*e^2*x^2 + 5*e^3*x^3) + b^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
)) - 7*c^2*e*(a*e*(-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) + b*(256*d^5 + 640*d^4
*e*x + 480*d^3*e^2*x^2 + 80*d^2*e^3*x^3 - 10*d*e^4*x^4 + 3*e^5*x^5))))/(35*e^7*(d + e*x)^(5/2))

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Maple [A]
time = 0.76, size = 506, normalized size = 1.82 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/e^7*(1/7*c^3*(e*x+d)^(7/2)+3/5*b*c^2*e*(e*x+d)^(5/2)-6/5*c^3*d*(e*x+d)^(5/2)+a*c^2*e^2*(e*x+d)^(3/2)+b^2*c*e
^2*(e*x+d)^(3/2)-5*b*c^2*d*e*(e*x+d)^(3/2)+5*c^3*d^2*(e*x+d)^(3/2)+6*a*b*c*e^3*(e*x+d)^(1/2)-12*a*c^2*d*e^2*(e
*x+d)^(1/2)+b^3*e^3*(e*x+d)^(1/2)-12*b^2*c*d*e^2*(e*x+d)^(1/2)+30*b*c^2*d^2*e*(e*x+d)^(1/2)-20*c^3*d^3*(e*x+d)
^(1/2)-(3*a^2*c*e^4+3*a*b^2*e^4-18*a*b*c*d*e^3+18*a*c^2*d^2*e^2-3*b^3*d*e^3+18*b^2*c*d^2*e^2-30*b*c^2*d^3*e+15
*c^3*d^4)/(e*x+d)^(1/2)-1/3*(3*a^2*b*e^5-6*a^2*c*d*e^4-6*a*b^2*d*e^4+18*a*b*c*d^2*e^3-12*a*c^2*d^3*e^2+3*b^3*d
^2*e^3-12*b^2*c*d^3*e^2+15*b*c^2*d^4*e-6*c^3*d^5)/(e*x+d)^(3/2)-1/5*(a^3*e^6-3*a^2*b*d*e^5+3*a^2*c*d^2*e^4+3*a
*b^2*d^2*e^4-6*a*b*c*d^3*e^3+3*a*c^2*d^4*e^2-b^3*d^3*e^3+3*b^2*c*d^4*e^2-3*b*c^2*d^5*e+c^3*d^6)/(e*x+d)^(5/2))

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Maxima [A]
time = 0.29, size = 430, normalized size = 1.55 \begin {gather*} \frac {2}{35} \, {\left ({\left (5 \, {\left (x e + d\right )}^{\frac {7}{2}} c^{3} - 21 \, {\left (2 \, c^{3} d - b c^{2} e\right )} {\left (x e + d\right )}^{\frac {5}{2}} + 35 \, {\left (5 \, c^{3} d^{2} - 5 \, b c^{2} d e + b^{2} c e^{2} + a c^{2} e^{2}\right )} {\left (x e + d\right )}^{\frac {3}{2}} - 35 \, {\left (20 \, c^{3} d^{3} - 30 \, b c^{2} d^{2} e - b^{3} e^{3} - 6 \, a b c e^{3} + 12 \, {\left (b^{2} c e^{2} + a c^{2} e^{2}\right )} d\right )} \sqrt {x e + d}\right )} e^{\left (-6\right )} - \frac {7 \, {\left (c^{3} d^{6} - 3 \, b c^{2} d^{5} e + 3 \, {\left (b^{2} c e^{2} + a c^{2} e^{2}\right )} d^{4} - 3 \, a^{2} b d e^{5} - {\left (b^{3} e^{3} + 6 \, a b c e^{3}\right )} d^{3} + a^{3} e^{6} + 15 \, {\left (5 \, c^{3} d^{4} - 10 \, b c^{2} d^{3} e + a b^{2} e^{4} + a^{2} c e^{4} + 6 \, {\left (b^{2} c e^{2} + a c^{2} e^{2}\right )} d^{2} - {\left (b^{3} e^{3} + 6 \, a b c e^{3}\right )} d\right )} {\left (x e + d\right )}^{2} + 3 \, {\left (a b^{2} e^{4} + a^{2} c e^{4}\right )} d^{2} - 5 \, {\left (2 \, c^{3} d^{5} - 5 \, b c^{2} d^{4} e + 4 \, {\left (b^{2} c e^{2} + a c^{2} e^{2}\right )} d^{3} - a^{2} b e^{5} - {\left (b^{3} e^{3} + 6 \, a b c e^{3}\right )} d^{2} + 2 \, {\left (a b^{2} e^{4} + a^{2} c e^{4}\right )} d\right )} {\left (x e + d\right )}\right )} e^{\left (-6\right )}}{{\left (x e + d\right )}^{\frac {5}{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+a)^3/(e*x+d)^(7/2),x, algorithm="maxima")

[Out]

2/35*((5*(x*e + d)^(7/2)*c^3 - 21*(2*c^3*d - b*c^2*e)*(x*e + d)^(5/2) + 35*(5*c^3*d^2 - 5*b*c^2*d*e + b^2*c*e^
2 + a*c^2*e^2)*(x*e + d)^(3/2) - 35*(20*c^3*d^3 - 30*b*c^2*d^2*e - b^3*e^3 - 6*a*b*c*e^3 + 12*(b^2*c*e^2 + a*c
^2*e^2)*d)*sqrt(x*e + d))*e^(-6) - 7*(c^3*d^6 - 3*b*c^2*d^5*e + 3*(b^2*c*e^2 + a*c^2*e^2)*d^4 - 3*a^2*b*d*e^5
- (b^3*e^3 + 6*a*b*c*e^3)*d^3 + a^3*e^6 + 15*(5*c^3*d^4 - 10*b*c^2*d^3*e + a*b^2*e^4 + a^2*c*e^4 + 6*(b^2*c*e^
2 + a*c^2*e^2)*d^2 - (b^3*e^3 + 6*a*b*c*e^3)*d)*(x*e + d)^2 + 3*(a*b^2*e^4 + a^2*c*e^4)*d^2 - 5*(2*c^3*d^5 - 5
*b*c^2*d^4*e + 4*(b^2*c*e^2 + a*c^2*e^2)*d^3 - a^2*b*e^5 - (b^3*e^3 + 6*a*b*c*e^3)*d^2 + 2*(a*b^2*e^4 + a^2*c*
e^4)*d)*(x*e + d))*e^(-6)/(x*e + d)^(5/2))*e^(-1)

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Fricas [A]
time = 2.69, size = 410, normalized size = 1.47 \begin {gather*} -\frac {2 \, {\left (1024 \, c^{3} d^{6} - {\left (5 \, c^{3} x^{6} + 21 \, b c^{2} x^{5} + 35 \, {\left (b^{2} c + a c^{2}\right )} x^{4} - 35 \, a^{2} b x + 35 \, {\left (b^{3} + 6 \, a b c\right )} x^{3} - 7 \, a^{3} - 105 \, {\left (a b^{2} + a^{2} c\right )} x^{2}\right )} e^{6} + 2 \, {\left (6 \, c^{3} d x^{5} + 35 \, b c^{2} d x^{4} + 140 \, {\left (b^{2} c + a c^{2}\right )} d x^{3} + 7 \, a^{2} b d - 105 \, {\left (b^{3} + 6 \, a b c\right )} d x^{2} + 70 \, {\left (a b^{2} + a^{2} c\right )} d x\right )} e^{5} - 8 \, {\left (5 \, c^{3} d^{2} x^{4} + 70 \, b c^{2} d^{2} x^{3} - 210 \, {\left (b^{2} c + a c^{2}\right )} d^{2} x^{2} + 35 \, {\left (b^{3} + 6 \, a b c\right )} d^{2} x - 7 \, {\left (a b^{2} + a^{2} c\right )} d^{2}\right )} e^{4} + 16 \, {\left (20 \, c^{3} d^{3} x^{3} - 210 \, b c^{2} d^{3} x^{2} + 140 \, {\left (b^{2} c + a c^{2}\right )} d^{3} x - 7 \, {\left (b^{3} + 6 \, a b c\right )} d^{3}\right )} e^{3} + 128 \, {\left (15 \, c^{3} d^{4} x^{2} - 35 \, b c^{2} d^{4} x + 7 \, {\left (b^{2} c + a c^{2}\right )} d^{4}\right )} e^{2} + 256 \, {\left (10 \, c^{3} d^{5} x - 7 \, b c^{2} d^{5}\right )} e\right )} \sqrt {x e + d}}{35 \, {\left (x^{3} e^{10} + 3 \, d x^{2} e^{9} + 3 \, d^{2} x e^{8} + d^{3} e^{7}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/35*(1024*c^3*d^6 - (5*c^3*x^6 + 21*b*c^2*x^5 + 35*(b^2*c + a*c^2)*x^4 - 35*a^2*b*x + 35*(b^3 + 6*a*b*c)*x^3
- 7*a^3 - 105*(a*b^2 + a^2*c)*x^2)*e^6 + 2*(6*c^3*d*x^5 + 35*b*c^2*d*x^4 + 140*(b^2*c + a*c^2)*d*x^3 + 7*a^2*
b*d - 105*(b^3 + 6*a*b*c)*d*x^2 + 70*(a*b^2 + a^2*c)*d*x)*e^5 - 8*(5*c^3*d^2*x^4 + 70*b*c^2*d^2*x^3 - 210*(b^2
*c + a*c^2)*d^2*x^2 + 35*(b^3 + 6*a*b*c)*d^2*x - 7*(a*b^2 + a^2*c)*d^2)*e^4 + 16*(20*c^3*d^3*x^3 - 210*b*c^2*d
^3*x^2 + 140*(b^2*c + a*c^2)*d^3*x - 7*(b^3 + 6*a*b*c)*d^3)*e^3 + 128*(15*c^3*d^4*x^2 - 35*b*c^2*d^4*x + 7*(b^
2*c + a*c^2)*d^4)*e^2 + 256*(10*c^3*d^5*x - 7*b*c^2*d^5)*e)*sqrt(x*e + d)/(x^3*e^10 + 3*d*x^2*e^9 + 3*d^2*x*e^
8 + d^3*e^7)

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Sympy [A]
time = 114.93, size = 304, normalized size = 1.09 \begin {gather*} \frac {2 c^{3} \left (d + e x\right )^{\frac {7}{2}}}{7 e^{7}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \cdot \left (6 b c^{2} e - 12 c^{3} d\right )}{5 e^{7}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \cdot \left (6 a c^{2} e^{2} + 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 (12 a b c e^{3} - 24 a c^{2} d e^{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}\right )}{e^{7}} - \frac {6 \left (a e^{2} - b d e + c d^{2}\right ) \left (a c e^{2} + b^{2} e^{2} - 5 b c d e + 5 c^{2} d^{2}\right )}{e^{7} \sqrt {d + e x}} - \frac {2 \left (b e - 2 c d\right ) \left (a e^{2} - b d e + c d^{2}\right )^{2}}{e^{7} \left (d + e x\right )^{\frac {3}{2}}} - \frac {2 \left (a e^{2} - b d e + c d^{2}\right )^{3}}{5 e^{7} \left (d + e x\right )^{\frac {5}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 609 vs. \(2 (264) = 528\).
time = 1.09, size = 609, normalized size = 2.19 \begin {gather*} \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} + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} a c^{2} e^{44} - 420 \, \sqrt {x e + d} b^{2} c d e^{44} - 420 \, \sqrt {x e + d} a c^{2} d e^{44} + 35 \, \sqrt {x e + d} b^{3} e^{45} + 210 \, \sqrt {x e + d} a b c 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} + 90 \, {\left (x e + d\right )}^{2} a c^{2} d^{2} e^{2} - 20 \, {\left (x e + d\right )} b^{2} c d^{3} e^{2} - 20 \, {\left (x e + d\right )} a c^{2} d^{3} e^{2} + 3 \, b^{2} c d^{4} e^{2} + 3 \, a c^{2} d^{4} e^{2} - 15 \, {\left (x e + d\right )}^{2} b^{3} d e^{3} - 90 \, {\left (x e + d\right )}^{2} a b c d e^{3} + 5 \, {\left (x e + d\right )} b^{3} d^{2} e^{3} + 30 \, {\left (x e + d\right )} a b c d^{2} e^{3} - b^{3} d^{3} e^{3} - 6 \, a b c d^{3} e^{3} + 15 \, {\left (x e + d\right )}^{2} a b^{2} e^{4} + 15 \, {\left (x e + d\right )}^{2} a^{2} c e^{4} - 10 \, {\left (x e + d\right )} a b^{2} d e^{4} - 10 \, {\left (x e + d\right )} a^{2} c d e^{4} + 3 \, a b^{2} d^{2} e^{4} + 3 \, a^{2} c d^{2} e^{4} + 5 \, {\left (x e + d\right )} a^{2} b e^{5} - 3 \, a^{2} b d e^{5} + a^{3} e^{6}\right )} e^{\left (-7\right )}}{5 \, {\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+a)^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 + 35*(x*e + d)^(3/2)*a*c^2*e^44 - 420*sqrt(x*e + d)*b^2*c*d*e^
44 - 420*sqrt(x*e + d)*a*c^2*d*e^44 + 35*sqrt(x*e + d)*b^3*e^45 + 210*sqrt(x*e + d)*a*b*c*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 + 90*(x*e + d)^2*a*c^2*d^2*e^2 - 20*(x*e + d)*b^2*c*d^3*e^2
- 20*(x*e + d)*a*c^2*d^3*e^2 + 3*b^2*c*d^4*e^2 + 3*a*c^2*d^4*e^2 - 15*(x*e + d)^2*b^3*d*e^3 - 90*(x*e + d)^2*
a*b*c*d*e^3 + 5*(x*e + d)*b^3*d^2*e^3 + 30*(x*e + d)*a*b*c*d^2*e^3 - b^3*d^3*e^3 - 6*a*b*c*d^3*e^3 + 15*(x*e +
d)^2*a*b^2*e^4 + 15*(x*e + d)^2*a^2*c*e^4 - 10*(x*e + d)*a*b^2*d*e^4 - 10*(x*e + d)*a^2*c*d*e^4 + 3*a*b^2*d^2
*e^4 + 3*a^2*c*d^2*e^4 + 5*(x*e + d)*a^2*b*e^5 - 3*a^2*b*d*e^5 + a^3*e^6)*e^(-7)/(x*e + d)^(5/2)

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Mupad [B]
time = 0.89, size = 445, normalized size = 1.60 \begin {gather*} \frac {2\,c^3\,{\left (d+e\,x\right )}^{7/2}}{7\,e^7}-\frac {\left (12\,c^3\,d-6\,b\,c^2\,e\right )\,{\left (d+e\,x\right )}^{5/2}}{5\,e^7}+\frac {{\left (d+e\,x\right )}^{3/2}\,\left (6\,b^2\,c\,e^2-30\,b\,c^2\,d\,e+30\,c^3\,d^2+6\,a\,c^2\,e^2\right )}{3\,e^7}-\frac {{\left (d+e\,x\right )}^2\,\left (6\,a^2\,c\,e^4+6\,a\,b^2\,e^4-36\,a\,b\,c\,d\,e^3+36\,a\,c^2\,d^2\,e^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\right )-\left (d+e\,x\right )\,\left (-2\,a^2\,b\,e^5+4\,a^2\,c\,d\,e^4+4\,a\,b^2\,d\,e^4-12\,a\,b\,c\,d^2\,e^3+8\,a\,c^2\,d^3\,e^2-2\,b^3\,d^2\,e^3+8\,b^2\,c\,d^3\,e^2-10\,b\,c^2\,d^4\,e+4\,c^3\,d^5\right )+\frac {2\,a^3\,e^6}{5}+\frac {2\,c^3\,d^6}{5}-\frac {2\,b^3\,d^3\,e^3}{5}+\frac {6\,a\,b^2\,d^2\,e^4}{5}+\frac {6\,a\,c^2\,d^4\,e^2}{5}+\frac {6\,a^2\,c\,d^2\,e^4}{5}+\frac {6\,b^2\,c\,d^4\,e^2}{5}-\frac {6\,a^2\,b\,d\,e^5}{5}-\frac {6\,b\,c^2\,d^5\,e}{5}-\frac {12\,a\,b\,c\,d^3\,e^3}{5}}{e^7\,{\left (d+e\,x\right )}^{5/2}}+\frac {2\,\left (b\,e-2\,c\,d\right )\,\sqrt {d+e\,x}\,\left (b^2\,e^2-10\,b\,c\,d\,e+10\,c^2\,d^2+6\,a\,c\,e^2\right )}{e^7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*c^3*(d + e*x)^(7/2))/(7*e^7) - ((12*c^3*d - 6*b*c^2*e)*(d + e*x)^(5/2))/(5*e^7) + ((d + e*x)^(3/2)*(30*c^3*
d^2 + 6*a*c^2*e^2 + 6*b^2*c*e^2 - 30*b*c^2*d*e))/(3*e^7) - ((d + e*x)^2*(30*c^3*d^4 + 6*a*b^2*e^4 + 6*a^2*c*e^
4 - 6*b^3*d*e^3 + 36*a*c^2*d^2*e^2 + 36*b^2*c*d^2*e^2 - 60*b*c^2*d^3*e - 36*a*b*c*d*e^3) - (d + e*x)*(4*c^3*d^
5 - 2*a^2*b*e^5 - 2*b^3*d^2*e^3 + 8*a*c^2*d^3*e^2 + 8*b^2*c*d^3*e^2 + 4*a*b^2*d*e^4 + 4*a^2*c*d*e^4 - 10*b*c^2
*d^4*e - 12*a*b*c*d^2*e^3) + (2*a^3*e^6)/5 + (2*c^3*d^6)/5 - (2*b^3*d^3*e^3)/5 + (6*a*b^2*d^2*e^4)/5 + (6*a*c^
2*d^4*e^2)/5 + (6*a^2*c*d^2*e^4)/5 + (6*b^2*c*d^4*e^2)/5 - (6*a^2*b*d*e^5)/5 - (6*b*c^2*d^5*e)/5 - (12*a*b*c*d
^3*e^3)/5)/(e^7*(d + e*x)^(5/2)) + (2*(b*e - 2*c*d)*(d + e*x)^(1/2)*(b^2*e^2 + 10*c^2*d^2 + 6*a*c*e^2 - 10*b*c
*d*e))/e^7

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