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

Optimal. Leaf size=210 \[ -\frac{2 \sqrt{c+d x} \left (a d \left (-B d^2-3 c^2 D+2 c C d\right )-b \left (A d^3-2 B c d^2+3 c^2 C d-4 c^3 D\right )\right )}{d^5}+\frac{2 (b c-a d) \left (A d^3-B c d^2+c^2 C d+c^3 (-D)\right )}{d^5 \sqrt{c+d x}}+\frac{2 (c+d x)^{3/2} \left (a d (C d-3 c D)-b \left (-B d^2-6 c^2 D+3 c C d\right )\right )}{3 d^5}+\frac{2 (c+d x)^{5/2} (a d D-4 b c D+b C d)}{5 d^5}+\frac{2 b D (c+d x)^{7/2}}{7 d^5} \]

[Out]

(2*(b*c - a*d)*(c^2*C*d - B*c*d^2 + A*d^3 - c^3*D))/(d^5*Sqrt[c + d*x]) - (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))*Sqrt[c + d*x])/d^5 + (2*(a*d*(C*d - 3*c*D) - b*(3*c*C*d - B*d^2
- 6*c^2*D))*(c + d*x)^(3/2))/(3*d^5) + (2*(b*C*d - 4*b*c*D + a*d*D)*(c + d*x)^(5/2))/(5*d^5) + (2*b*D*(c + d*
x)^(7/2))/(7*d^5)

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Rubi [A]  time = 0.166234, 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} \[ -\frac{2 \sqrt{c+d x} \left (a d \left (-B d^2-3 c^2 D+2 c C d\right )-b \left (A d^3-2 B c d^2+3 c^2 C d-4 c^3 D\right )\right )}{d^5}+\frac{2 (b c-a d) \left (A d^3-B c d^2+c^2 C d+c^3 (-D)\right )}{d^5 \sqrt{c+d x}}+\frac{2 (c+d x)^{3/2} \left (a d (C d-3 c D)-b \left (-B d^2-6 c^2 D+3 c C d\right )\right )}{3 d^5}+\frac{2 (c+d x)^{5/2} (a d D-4 b c D+b C d)}{5 d^5}+\frac{2 b D (c+d x)^{7/2}}{7 d^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(b*c - a*d)*(c^2*C*d - B*c*d^2 + A*d^3 - c^3*D))/(d^5*Sqrt[c + d*x]) - (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))*Sqrt[c + d*x])/d^5 + (2*(a*d*(C*d - 3*c*D) - b*(3*c*C*d - B*d^2
- 6*c^2*D))*(c + d*x)^(3/2))/(3*d^5) + (2*(b*C*d - 4*b*c*D + a*d*D)*(c + d*x)^(5/2))/(5*d^5) + (2*b*D*(c + d*
x)^(7/2))/(7*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)^{3/2}} \, 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)^{3/2}}+\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 \sqrt{c+d x}}+\frac{\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^4}+\frac{(b C d-4 b c D+a d D) (c+d x)^{3/2}}{d^4}+\frac{b D (c+d x)^{5/2}}{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 )}{d^5 \sqrt{c+d x}}-\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 ) \sqrt{c+d x}}{d^5}+\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 ) (c+d x)^{3/2}}{3 d^5}+\frac{2 (b C d-4 b c D+a d D) (c+d x)^{5/2}}{5 d^5}+\frac{2 b D (c+d x)^{7/2}}{7 d^5}\\ \end{align*}

Mathematica [A]  time = 0.304956, size = 188, normalized size = 0.9 \[ \frac{14 a d \left (d^3 \left (x \left (15 B+5 C x+3 D x^2\right )-15 A\right )+2 c d^2 (15 B-x (10 C+3 D x))-8 c^2 d (5 C-3 D x)+48 c^3 D\right )+b \left (4 c d^3 (105 A-x (70 B+3 x (7 C+4 D x)))+2 d^4 x (105 A+x (35 B+3 x (7 C+5 D x)))+16 c^2 d^2 (3 x (7 C+2 D x)-35 B)+96 c^3 d (7 C-4 D x)-768 c^4 D\right )}{105 d^5 \sqrt{c+d x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(14*a*d*(48*c^3*D - 8*c^2*d*(5*C - 3*D*x) + 2*c*d^2*(15*B - x*(10*C + 3*D*x)) + d^3*(-15*A + x*(15*B + 5*C*x +
3*D*x^2))) + b*(-768*c^4*D + 96*c^3*d*(7*C - 4*D*x) + 16*c^2*d^2*(-35*B + 3*x*(7*C + 2*D*x)) + 4*c*d^3*(105*A
- x*(70*B + 3*x*(7*C + 4*D*x))) + 2*d^4*x*(105*A + x*(35*B + 3*x*(7*C + 5*D*x)))))/(105*d^5*Sqrt[c + d*x])

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Maple [A]  time = 0.005, size = 241, normalized size = 1.2 \begin{align*} -{\frac{-30\,Db{x}^{4}{d}^{4}-42\,Cb{d}^{4}{x}^{3}-42\,Da{d}^{4}{x}^{3}+48\,Dbc{d}^{3}{x}^{3}-70\,Bb{d}^{4}{x}^{2}-70\,Ca{d}^{4}{x}^{2}+84\,Cbc{d}^{3}{x}^{2}+84\,Dac{d}^{3}{x}^{2}-96\,Db{c}^{2}{d}^{2}{x}^{2}-210\,Ab{d}^{4}x-210\,Ba{d}^{4}x+280\,Bbc{d}^{3}x+280\,Cac{d}^{3}x-336\,Cb{c}^{2}{d}^{2}x-336\,Da{c}^{2}{d}^{2}x+384\,Db{c}^{3}dx+210\,Aa{d}^{4}-420\,Abc{d}^{3}-420\,Bac{d}^{3}+560\,Bb{c}^{2}{d}^{2}+560\,Ca{c}^{2}{d}^{2}-672\,Cb{c}^{3}d-672\,Da{c}^{3}d+768\,Db{c}^{4}}{105\,{d}^{5}}{\frac{1}{\sqrt{dx+c}}}} \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)^(3/2),x)

[Out]

-2/105/(d*x+c)^(1/2)*(-15*D*b*d^4*x^4-21*C*b*d^4*x^3-21*D*a*d^4*x^3+24*D*b*c*d^3*x^3-35*B*b*d^4*x^2-35*C*a*d^4
*x^2+42*C*b*c*d^3*x^2+42*D*a*c*d^3*x^2-48*D*b*c^2*d^2*x^2-105*A*b*d^4*x-105*B*a*d^4*x+140*B*b*c*d^3*x+140*C*a*
c*d^3*x-168*C*b*c^2*d^2*x-168*D*a*c^2*d^2*x+192*D*b*c^3*d*x+105*A*a*d^4-210*A*b*c*d^3-210*B*a*c*d^3+280*B*b*c^
2*d^2+280*C*a*c^2*d^2-336*C*b*c^3*d-336*D*a*c^3*d+384*D*b*c^4)/d^5

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Maxima [A]  time = 1.77846, size = 278, normalized size = 1.32 \begin{align*} \frac{2 \,{\left (\frac{15 \,{\left (d x + c\right )}^{\frac{7}{2}} D b - 21 \,{\left (4 \, D b c -{\left (D a + C b\right )} d\right )}{\left (d x + c\right )}^{\frac{5}{2}} + 35 \,{\left (6 \, D b c^{2} - 3 \,{\left (D a + C b\right )} c d +{\left (C a + B b\right )} d^{2}\right )}{\left (d x + c\right )}^{\frac{3}{2}} - 105 \,{\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 )} \sqrt{d x + c}}{d^{4}} - \frac{105 \,{\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}\right )}}{\sqrt{d x + c} d^{4}}\right )}}{105 \, 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)^(3/2),x, algorithm="maxima")

[Out]

2/105*((15*(d*x + c)^(7/2)*D*b - 21*(4*D*b*c - (D*a + C*b)*d)*(d*x + c)^(5/2) + 35*(6*D*b*c^2 - 3*(D*a + C*b)*
c*d + (C*a + B*b)*d^2)*(d*x + c)^(3/2) - 105*(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)*sqrt(d*x + c))/d^4 - 105*(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)/(sqrt(d*x + c)*d^4))/d

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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)^(3/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [A]  time = 43.6359, size = 230, normalized size = 1.1 \begin{align*} \frac{2 D b \left (c + d x\right )^{\frac{7}{2}}}{7 d^{5}} + \frac{\left (c + d x\right )^{\frac{5}{2}} \left (2 C b d + 2 D a d - 8 D b c\right )}{5 d^{5}} + \frac{\left (c + d x\right )^{\frac{3}{2}} \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 )}{3 d^{5}} + \frac{\sqrt{c + d x} \left (2 A b d^{3} + 2 B a d^{3} - 4 B b c d^{2} - 4 C a c d^{2} + 6 C b c^{2} d + 6 D a c^{2} d - 8 D b c^{3}\right )}{d^{5}} + \frac{2 \left (a d - b c\right ) \left (- A d^{3} + B c d^{2} - C c^{2} d + D c^{3}\right )}{d^{5} \sqrt{c + d x}} \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)**(3/2),x)

[Out]

2*D*b*(c + d*x)**(7/2)/(7*d**5) + (c + d*x)**(5/2)*(2*C*b*d + 2*D*a*d - 8*D*b*c)/(5*d**5) + (c + d*x)**(3/2)*(
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)/(3*d**5) + sqrt(c + d*x)*(2*A*b*d**3 + 2*B*a*d*
*3 - 4*B*b*c*d**2 - 4*C*a*c*d**2 + 6*C*b*c**2*d + 6*D*a*c**2*d - 8*D*b*c**3)/d**5 + 2*(a*d - b*c)*(-A*d**3 + B
*c*d**2 - C*c**2*d + D*c**3)/(d**5*sqrt(c + d*x))

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Giac [A]  time = 2.4333, size = 436, normalized size = 2.08 \begin{align*} -\frac{2 \,{\left (D b c^{4} - D a c^{3} d - C b c^{3} d + C a c^{2} d^{2} + B b c^{2} d^{2} - B a c d^{3} - A b c d^{3} + A a d^{4}\right )}}{\sqrt{d x + c} d^{5}} + \frac{2 \,{\left (15 \,{\left (d x + c\right )}^{\frac{7}{2}} D b d^{30} - 84 \,{\left (d x + c\right )}^{\frac{5}{2}} D b c d^{30} + 210 \,{\left (d x + c\right )}^{\frac{3}{2}} D b c^{2} d^{30} - 420 \, \sqrt{d x + c} D b c^{3} d^{30} + 21 \,{\left (d x + c\right )}^{\frac{5}{2}} D a d^{31} + 21 \,{\left (d x + c\right )}^{\frac{5}{2}} C b d^{31} - 105 \,{\left (d x + c\right )}^{\frac{3}{2}} D a c d^{31} - 105 \,{\left (d x + c\right )}^{\frac{3}{2}} C b c d^{31} + 315 \, \sqrt{d x + c} D a c^{2} d^{31} + 315 \, \sqrt{d x + c} C b c^{2} d^{31} + 35 \,{\left (d x + c\right )}^{\frac{3}{2}} C a d^{32} + 35 \,{\left (d x + c\right )}^{\frac{3}{2}} B b d^{32} - 210 \, \sqrt{d x + c} C a c d^{32} - 210 \, \sqrt{d x + c} B b c d^{32} + 105 \, \sqrt{d x + c} B a d^{33} + 105 \, \sqrt{d x + c} A b d^{33}\right )}}{105 \, d^{35}} \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)^(3/2),x, algorithm="giac")

[Out]

-2*(D*b*c^4 - D*a*c^3*d - C*b*c^3*d + C*a*c^2*d^2 + B*b*c^2*d^2 - B*a*c*d^3 - A*b*c*d^3 + A*a*d^4)/(sqrt(d*x +
c)*d^5) + 2/105*(15*(d*x + c)^(7/2)*D*b*d^30 - 84*(d*x + c)^(5/2)*D*b*c*d^30 + 210*(d*x + c)^(3/2)*D*b*c^2*d^
30 - 420*sqrt(d*x + c)*D*b*c^3*d^30 + 21*(d*x + c)^(5/2)*D*a*d^31 + 21*(d*x + c)^(5/2)*C*b*d^31 - 105*(d*x + c
)^(3/2)*D*a*c*d^31 - 105*(d*x + c)^(3/2)*C*b*c*d^31 + 315*sqrt(d*x + c)*D*a*c^2*d^31 + 315*sqrt(d*x + c)*C*b*c
^2*d^31 + 35*(d*x + c)^(3/2)*C*a*d^32 + 35*(d*x + c)^(3/2)*B*b*d^32 - 210*sqrt(d*x + c)*C*a*c*d^32 - 210*sqrt(
d*x + c)*B*b*c*d^32 + 105*sqrt(d*x + c)*B*a*d^33 + 105*sqrt(d*x + c)*A*b*d^33)/d^35