3.20 \(\int \frac{(a+b x) (A+B x+C x^2+D x^3)}{(c+d x)^{5/2}} \, dx\)

Optimal. Leaf size=210 \[ \frac{2 \left (a d \left (-B d^2-3 c^2 D+2 c C d\right )-b \left (A d^3-2 B c d^2+3 c^2 C d-4 c^3 D\right )\right )}{d^5 \sqrt{c+d x}}+\frac{2 (b c-a d) \left (A d^3-B c d^2+c^2 C d+c^3 (-D)\right )}{3 d^5 (c+d x)^{3/2}}+\frac{2 \sqrt{c+d x} \left (a d (C d-3 c D)-b \left (-B d^2-6 c^2 D+3 c C d\right )\right )}{d^5}+\frac{2 (c+d x)^{3/2} (a d D-4 b c D+b C d)}{3 d^5}+\frac{2 b D (c+d x)^{5/2}}{5 d^5} \]

[Out]

(2*(b*c - a*d)*(c^2*C*d - B*c*d^2 + A*d^3 - c^3*D))/(3*d^5*(c + d*x)^(3/2)) + (2*(a*d*(2*c*C*d - B*d^2 - 3*c^2
*D) - b*(3*c^2*C*d - 2*B*c*d^2 + A*d^3 - 4*c^3*D)))/(d^5*Sqrt[c + d*x]) + (2*(a*d*(C*d - 3*c*D) - b*(3*c*C*d -
B*d^2 - 6*c^2*D))*Sqrt[c + d*x])/d^5 + (2*(b*C*d - 4*b*c*D + a*d*D)*(c + d*x)^(3/2))/(3*d^5) + (2*b*D*(c + d*
x)^(5/2))/(5*d^5)

________________________________________________________________________________________

Rubi [A]  time = 0.172371, antiderivative size = 210, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.033, Rules used = {1620} \[ \frac{2 \left (a d \left (-B d^2-3 c^2 D+2 c C d\right )-b \left (A d^3-2 B c d^2+3 c^2 C d-4 c^3 D\right )\right )}{d^5 \sqrt{c+d x}}+\frac{2 (b c-a d) \left (A d^3-B c d^2+c^2 C d+c^3 (-D)\right )}{3 d^5 (c+d x)^{3/2}}+\frac{2 \sqrt{c+d x} \left (a d (C d-3 c D)-b \left (-B d^2-6 c^2 D+3 c C d\right )\right )}{d^5}+\frac{2 (c+d x)^{3/2} (a d D-4 b c D+b C d)}{3 d^5}+\frac{2 b D (c+d x)^{5/2}}{5 d^5} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)*(A + B*x + C*x^2 + D*x^3))/(c + d*x)^(5/2),x]

[Out]

(2*(b*c - a*d)*(c^2*C*d - B*c*d^2 + A*d^3 - c^3*D))/(3*d^5*(c + d*x)^(3/2)) + (2*(a*d*(2*c*C*d - B*d^2 - 3*c^2
*D) - b*(3*c^2*C*d - 2*B*c*d^2 + A*d^3 - 4*c^3*D)))/(d^5*Sqrt[c + d*x]) + (2*(a*d*(C*d - 3*c*D) - b*(3*c*C*d -
B*d^2 - 6*c^2*D))*Sqrt[c + d*x])/d^5 + (2*(b*C*d - 4*b*c*D + a*d*D)*(c + d*x)^(3/2))/(3*d^5) + (2*b*D*(c + d*
x)^(5/2))/(5*d^5)

Rule 1620

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[Px*(a + b*x)
^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2]) &&
GtQ[Expon[Px, x], 2]

Rubi steps

\begin{align*} \int \frac{(a+b x) \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{5/2}} \, dx &=\int \left (\frac{(-b c+a d) \left (c^2 C d-B c d^2+A d^3-c^3 D\right )}{d^4 (c+d x)^{5/2}}+\frac{-a d \left (2 c C d-B d^2-3 c^2 D\right )+b \left (3 c^2 C d-2 B c d^2+A d^3-4 c^3 D\right )}{d^4 (c+d x)^{3/2}}+\frac{a d (C d-3 c D)-b \left (3 c C d-B d^2-6 c^2 D\right )}{d^4 \sqrt{c+d x}}+\frac{(b C d-4 b c D+a d D) \sqrt{c+d x}}{d^4}+\frac{b D (c+d x)^{3/2}}{d^4}\right ) \, dx\\ &=\frac{2 (b c-a d) \left (c^2 C d-B c d^2+A d^3-c^3 D\right )}{3 d^5 (c+d x)^{3/2}}+\frac{2 \left (a d \left (2 c C d-B d^2-3 c^2 D\right )-b \left (3 c^2 C d-2 B c d^2+A d^3-4 c^3 D\right )\right )}{d^5 \sqrt{c+d x}}+\frac{2 \left (a d (C d-3 c D)-b \left (3 c C d-B d^2-6 c^2 D\right )\right ) \sqrt{c+d x}}{d^5}+\frac{2 (b C d-4 b c D+a d D) (c+d x)^{3/2}}{3 d^5}+\frac{2 b D (c+d x)^{5/2}}{5 d^5}\\ \end{align*}

Mathematica [A]  time = 0.294512, size = 177, normalized size = 0.84 \[ \frac{2 \left (b \left (-2 c d^3 \left (5 A+x \left (-30 B+15 C x+4 D x^2\right )\right )+d^4 x \left (x \left (15 B+5 C x+3 D x^2\right )-15 A\right )+8 c^2 d^2 (5 B+3 x (2 D x-5 C))+c^3 (192 d D x-80 C d)+128 c^4 D\right )-5 a d \left (d^3 \left (A+3 B x+x^2 (-(3 C+D x))\right )+2 c d^2 (B+3 x (D x-2 C))-8 c^2 d (C-3 D x)+16 c^3 D\right )\right )}{15 d^5 (c+d x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)*(A + B*x + C*x^2 + D*x^3))/(c + d*x)^(5/2),x]

[Out]

(2*(-5*a*d*(16*c^3*D - 8*c^2*d*(C - 3*D*x) + 2*c*d^2*(B + 3*x*(-2*C + D*x)) + d^3*(A + 3*B*x - x^2*(3*C + D*x)
)) + b*(128*c^4*D + c^3*(-80*C*d + 192*d*D*x) + 8*c^2*d^2*(5*B + 3*x*(-5*C + 2*D*x)) + d^4*x*(-15*A + x*(15*B
+ 5*C*x + 3*D*x^2)) - 2*c*d^3*(5*A + x*(-30*B + 15*C*x + 4*D*x^2)))))/(15*d^5*(c + d*x)^(3/2))

________________________________________________________________________________________

Maple [A]  time = 0.004, size = 241, normalized size = 1.2 \begin{align*} -{\frac{-6\,Db{x}^{4}{d}^{4}-10\,Cb{d}^{4}{x}^{3}-10\,Da{d}^{4}{x}^{3}+16\,Dbc{d}^{3}{x}^{3}-30\,Bb{d}^{4}{x}^{2}-30\,Ca{d}^{4}{x}^{2}+60\,Cbc{d}^{3}{x}^{2}+60\,Dac{d}^{3}{x}^{2}-96\,Db{c}^{2}{d}^{2}{x}^{2}+30\,Ab{d}^{4}x+30\,Ba{d}^{4}x-120\,Bbc{d}^{3}x-120\,Cac{d}^{3}x+240\,Cb{c}^{2}{d}^{2}x+240\,Da{c}^{2}{d}^{2}x-384\,Db{c}^{3}dx+10\,Aa{d}^{4}+20\,Abc{d}^{3}+20\,Bac{d}^{3}-80\,Bb{c}^{2}{d}^{2}-80\,Ca{c}^{2}{d}^{2}+160\,Cb{c}^{3}d+160\,Da{c}^{3}d-256\,Db{c}^{4}}{15\,{d}^{5}} \left ( dx+c \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^(5/2),x)

[Out]

-2/15/(d*x+c)^(3/2)*(-3*D*b*d^4*x^4-5*C*b*d^4*x^3-5*D*a*d^4*x^3+8*D*b*c*d^3*x^3-15*B*b*d^4*x^2-15*C*a*d^4*x^2+
30*C*b*c*d^3*x^2+30*D*a*c*d^3*x^2-48*D*b*c^2*d^2*x^2+15*A*b*d^4*x+15*B*a*d^4*x-60*B*b*c*d^3*x-60*C*a*c*d^3*x+1
20*C*b*c^2*d^2*x+120*D*a*c^2*d^2*x-192*D*b*c^3*d*x+5*A*a*d^4+10*A*b*c*d^3+10*B*a*c*d^3-40*B*b*c^2*d^2-40*C*a*c
^2*d^2+80*C*b*c^3*d+80*D*a*c^3*d-128*D*b*c^4)/d^5

________________________________________________________________________________________

Maxima [A]  time = 1.18824, size = 275, normalized size = 1.31 \begin{align*} \frac{2 \,{\left (\frac{3 \,{\left (d x + c\right )}^{\frac{5}{2}} D b - 5 \,{\left (4 \, D b c -{\left (D a + C b\right )} d\right )}{\left (d x + c\right )}^{\frac{3}{2}} + 15 \,{\left (6 \, D b c^{2} - 3 \,{\left (D a + C b\right )} c d +{\left (C a + B b\right )} d^{2}\right )} \sqrt{d x + c}}{d^{4}} - \frac{5 \,{\left (D b c^{4} + A a d^{4} -{\left (D a + C b\right )} c^{3} d +{\left (C a + B b\right )} c^{2} d^{2} -{\left (B a + A b\right )} c d^{3} - 3 \,{\left (4 \, D b c^{3} - 3 \,{\left (D a + C b\right )} c^{2} d + 2 \,{\left (C a + B b\right )} c d^{2} -{\left (B a + A b\right )} d^{3}\right )}{\left (d x + c\right )}\right )}}{{\left (d x + c\right )}^{\frac{3}{2}} d^{4}}\right )}}{15 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^(5/2),x, algorithm="maxima")

[Out]

2/15*((3*(d*x + c)^(5/2)*D*b - 5*(4*D*b*c - (D*a + C*b)*d)*(d*x + c)^(3/2) + 15*(6*D*b*c^2 - 3*(D*a + C*b)*c*d
+ (C*a + B*b)*d^2)*sqrt(d*x + c))/d^4 - 5*(D*b*c^4 + A*a*d^4 - (D*a + C*b)*c^3*d + (C*a + B*b)*c^2*d^2 - (B*a
+ A*b)*c*d^3 - 3*(4*D*b*c^3 - 3*(D*a + C*b)*c^2*d + 2*(C*a + B*b)*c*d^2 - (B*a + A*b)*d^3)*(d*x + c))/((d*x +
c)^(3/2)*d^4))/d

________________________________________________________________________________________

Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^(5/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

________________________________________________________________________________________

Sympy [A]  time = 60.0426, size = 230, normalized size = 1.1 \begin{align*} \frac{2 D b \left (c + d x\right )^{\frac{5}{2}}}{5 d^{5}} + \frac{\left (c + d x\right )^{\frac{3}{2}} \left (2 C b d + 2 D a d - 8 D b c\right )}{3 d^{5}} + \frac{\sqrt{c + d x} \left (2 B b d^{2} + 2 C a d^{2} - 6 C b c d - 6 D a c d + 12 D b c^{2}\right )}{d^{5}} - \frac{2 \left (A b d^{3} + B a d^{3} - 2 B b c d^{2} - 2 C a c d^{2} + 3 C b c^{2} d + 3 D a c^{2} d - 4 D b c^{3}\right )}{d^{5} \sqrt{c + d x}} + \frac{2 \left (a d - b c\right ) \left (- A d^{3} + B c d^{2} - C c^{2} d + D c^{3}\right )}{3 d^{5} \left (c + d x\right )^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(D*x**3+C*x**2+B*x+A)/(d*x+c)**(5/2),x)

[Out]

2*D*b*(c + d*x)**(5/2)/(5*d**5) + (c + d*x)**(3/2)*(2*C*b*d + 2*D*a*d - 8*D*b*c)/(3*d**5) + sqrt(c + d*x)*(2*B
*b*d**2 + 2*C*a*d**2 - 6*C*b*c*d - 6*D*a*c*d + 12*D*b*c**2)/d**5 - 2*(A*b*d**3 + B*a*d**3 - 2*B*b*c*d**2 - 2*C
*a*c*d**2 + 3*C*b*c**2*d + 3*D*a*c**2*d - 4*D*b*c**3)/(d**5*sqrt(c + d*x)) + 2*(a*d - b*c)*(-A*d**3 + B*c*d**2
- C*c**2*d + D*c**3)/(3*d**5*(c + d*x)**(3/2))

________________________________________________________________________________________

Giac [A]  time = 2.27812, size = 408, normalized size = 1.94 \begin{align*} \frac{2 \,{\left (12 \,{\left (d x + c\right )} D b c^{3} - D b c^{4} - 9 \,{\left (d x + c\right )} D a c^{2} d - 9 \,{\left (d x + c\right )} C b c^{2} d + D a c^{3} d + C b c^{3} d + 6 \,{\left (d x + c\right )} C a c d^{2} + 6 \,{\left (d x + c\right )} B b c d^{2} - C a c^{2} d^{2} - B b c^{2} d^{2} - 3 \,{\left (d x + c\right )} B a d^{3} - 3 \,{\left (d x + c\right )} A b d^{3} + B a c d^{3} + A b c d^{3} - A a d^{4}\right )}}{3 \,{\left (d x + c\right )}^{\frac{3}{2}} d^{5}} + \frac{2 \,{\left (3 \,{\left (d x + c\right )}^{\frac{5}{2}} D b d^{20} - 20 \,{\left (d x + c\right )}^{\frac{3}{2}} D b c d^{20} + 90 \, \sqrt{d x + c} D b c^{2} d^{20} + 5 \,{\left (d x + c\right )}^{\frac{3}{2}} D a d^{21} + 5 \,{\left (d x + c\right )}^{\frac{3}{2}} C b d^{21} - 45 \, \sqrt{d x + c} D a c d^{21} - 45 \, \sqrt{d x + c} C b c d^{21} + 15 \, \sqrt{d x + c} C a d^{22} + 15 \, \sqrt{d x + c} B b d^{22}\right )}}{15 \, d^{25}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^(5/2),x, algorithm="giac")

[Out]

2/3*(12*(d*x + c)*D*b*c^3 - D*b*c^4 - 9*(d*x + c)*D*a*c^2*d - 9*(d*x + c)*C*b*c^2*d + D*a*c^3*d + C*b*c^3*d +
6*(d*x + c)*C*a*c*d^2 + 6*(d*x + c)*B*b*c*d^2 - C*a*c^2*d^2 - B*b*c^2*d^2 - 3*(d*x + c)*B*a*d^3 - 3*(d*x + c)*
A*b*d^3 + B*a*c*d^3 + A*b*c*d^3 - A*a*d^4)/((d*x + c)^(3/2)*d^5) + 2/15*(3*(d*x + c)^(5/2)*D*b*d^20 - 20*(d*x
+ c)^(3/2)*D*b*c*d^20 + 90*sqrt(d*x + c)*D*b*c^2*d^20 + 5*(d*x + c)^(3/2)*D*a*d^21 + 5*(d*x + c)^(3/2)*C*b*d^2
1 - 45*sqrt(d*x + c)*D*a*c*d^21 - 45*sqrt(d*x + c)*C*b*c*d^21 + 15*sqrt(d*x + c)*C*a*d^22 + 15*sqrt(d*x + c)*B
*b*d^22)/d^25