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

Optimal. Leaf size=212 \[ -\frac{2 (c+d x)^{3/2} \left (a d \left (-B d^2-3 c^2 D+2 c C d\right )-b \left (A d^3-2 B c d^2+3 c^2 C d-4 c^3 D\right )\right )}{3 d^5}-\frac{2 \sqrt{c+d x} (b c-a d) \left (A d^3-B c d^2+c^2 C d+c^3 (-D)\right )}{d^5}+\frac{2 (c+d x)^{5/2} \left (a d (C d-3 c D)-b \left (-B d^2-6 c^2 D+3 c C d\right )\right )}{5 d^5}+\frac{2 (c+d x)^{7/2} (a d D-4 b c D+b C d)}{7 d^5}+\frac{2 b D (c+d x)^{9/2}}{9 d^5} \]

[Out]

(-2*(b*c - a*d)*(c^2*C*d - B*c*d^2 + A*d^3 - c^3*D)*Sqrt[c + d*x])/d^5 - (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))*(c + d*x)^(3/2))/(3*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)^(5/2))/(5*d^5) + (2*(b*C*d - 4*b*c*D + a*d*D)*(c + d*x)^(7/2))/(7*d^5) + (2*b*D*(c
+ d*x)^(9/2))/(9*d^5)

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Rubi [A]  time = 0.173042, antiderivative size = 212, 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} \[ -\frac{2 (c+d x)^{3/2} \left (a d \left (-B d^2-3 c^2 D+2 c C d\right )-b \left (A d^3-2 B c d^2+3 c^2 C d-4 c^3 D\right )\right )}{3 d^5}-\frac{2 \sqrt{c+d x} (b c-a d) \left (A d^3-B c d^2+c^2 C d+c^3 (-D)\right )}{d^5}+\frac{2 (c+d x)^{5/2} \left (a d (C d-3 c D)-b \left (-B d^2-6 c^2 D+3 c C d\right )\right )}{5 d^5}+\frac{2 (c+d x)^{7/2} (a d D-4 b c D+b C d)}{7 d^5}+\frac{2 b D (c+d x)^{9/2}}{9 d^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(b*c - a*d)*(c^2*C*d - B*c*d^2 + A*d^3 - c^3*D)*Sqrt[c + d*x])/d^5 - (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))*(c + d*x)^(3/2))/(3*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)^(5/2))/(5*d^5) + (2*(b*C*d - 4*b*c*D + a*d*D)*(c + d*x)^(7/2))/(7*d^5) + (2*b*D*(c
+ d*x)^(9/2))/(9*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 )}{\sqrt{c+d x}} \, 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 \sqrt{c+d x}}+\frac{\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^4}+\frac{\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}}{d^4}+\frac{(b C d-4 b c D+a d D) (c+d x)^{5/2}}{d^4}+\frac{b D (c+d x)^{7/2}}{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 ) \sqrt{c+d x}}{d^5}-\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 ) (c+d x)^{3/2}}{3 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)^{5/2}}{5 d^5}+\frac{2 (b C d-4 b c D+a d D) (c+d x)^{7/2}}{7 d^5}+\frac{2 b D (c+d x)^{9/2}}{9 d^5}\\ \end{align*}

Mathematica [A]  time = 0.300073, size = 184, normalized size = 0.87 \[ \frac{2 \sqrt{c+d x} \left (3 a d \left (d^3 (105 A+x (35 B+3 x (7 C+5 D x)))-2 c d^2 (35 B+x (14 C+9 D x))+8 c^2 d (7 C+3 D x)-48 c^3 D\right )+b \left (-2 c d^3 (105 A+x (42 B+x (27 C+20 D x)))+d^4 x (105 A+x (63 B+5 x (9 C+7 D x)))+24 c^2 d^2 (7 B+x (3 C+2 D x))-16 c^3 d (9 C+4 D x)+128 c^4 D\right )\right )}{315 d^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[c + d*x]*(3*a*d*(-48*c^3*D + 8*c^2*d*(7*C + 3*D*x) - 2*c*d^2*(35*B + x*(14*C + 9*D*x)) + d^3*(105*A +
x*(35*B + 3*x*(7*C + 5*D*x)))) + b*(128*c^4*D - 16*c^3*d*(9*C + 4*D*x) + 24*c^2*d^2*(7*B + x*(3*C + 2*D*x)) +
d^4*x*(105*A + x*(63*B + 5*x*(9*C + 7*D*x))) - 2*c*d^3*(105*A + x*(42*B + x*(27*C + 20*D*x))))))/(315*d^5)

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Maple [A]  time = 0.004, size = 241, normalized size = 1.1 \begin{align*}{\frac{70\,Db{x}^{4}{d}^{4}+90\,Cb{d}^{4}{x}^{3}+90\,Da{d}^{4}{x}^{3}-80\,Dbc{d}^{3}{x}^{3}+126\,Bb{d}^{4}{x}^{2}+126\,Ca{d}^{4}{x}^{2}-108\,Cbc{d}^{3}{x}^{2}-108\,Dac{d}^{3}{x}^{2}+96\,Db{c}^{2}{d}^{2}{x}^{2}+210\,Ab{d}^{4}x+210\,Ba{d}^{4}x-168\,Bbc{d}^{3}x-168\,Cac{d}^{3}x+144\,Cb{c}^{2}{d}^{2}x+144\,Da{c}^{2}{d}^{2}x-128\,Db{c}^{3}dx+630\,Aa{d}^{4}-420\,Abc{d}^{3}-420\,Bac{d}^{3}+336\,Bb{c}^{2}{d}^{2}+336\,Ca{c}^{2}{d}^{2}-288\,Cb{c}^{3}d-288\,Da{c}^{3}d+256\,Db{c}^{4}}{315\,{d}^{5}}\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)^(1/2),x)

[Out]

2/315*(d*x+c)^(1/2)*(35*D*b*d^4*x^4+45*C*b*d^4*x^3+45*D*a*d^4*x^3-40*D*b*c*d^3*x^3+63*B*b*d^4*x^2+63*C*a*d^4*x
^2-54*C*b*c*d^3*x^2-54*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-84*B*b*c*d^3*x-84*C*a*c*d^
3*x+72*C*b*c^2*d^2*x+72*D*a*c^2*d^2*x-64*D*b*c^3*d*x+315*A*a*d^4-210*A*b*c*d^3-210*B*a*c*d^3+168*B*b*c^2*d^2+1
68*C*a*c^2*d^2-144*C*b*c^3*d-144*D*a*c^3*d+128*D*b*c^4)/d^5

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Maxima [A]  time = 2.58009, size = 267, normalized size = 1.26 \begin{align*} \frac{2 \,{\left (35 \,{\left (d x + c\right )}^{\frac{9}{2}} D b - 45 \,{\left (4 \, D b c -{\left (D a + C b\right )} d\right )}{\left (d x + c\right )}^{\frac{7}{2}} + 63 \,{\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{5}{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 )}{\left (d x + c\right )}^{\frac{3}{2}} + 315 \,{\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}\right )}}{315 \, d^{5}} \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)^(1/2),x, algorithm="maxima")

[Out]

2/315*(35*(d*x + c)^(9/2)*D*b - 45*(4*D*b*c - (D*a + C*b)*d)*(d*x + c)^(7/2) + 63*(6*D*b*c^2 - 3*(D*a + C*b)*c
*d + (C*a + B*b)*d^2)*(d*x + c)^(5/2) - 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)*(d*x + c)^(3/2) + 315*(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^5

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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)^(1/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [A]  time = 70.9248, size = 848, normalized size = 4. \begin{align*} \text{result too large to display} \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)**(1/2),x)

[Out]

Piecewise((-(2*A*a*c/sqrt(c + d*x) + 2*A*a*(-c/sqrt(c + d*x) - sqrt(c + d*x)) + 2*A*b*c*(-c/sqrt(c + d*x) - sq
rt(c + d*x))/d + 2*A*b*(c**2/sqrt(c + d*x) + 2*c*sqrt(c + d*x) - (c + d*x)**(3/2)/3)/d + 2*B*a*c*(-c/sqrt(c +
d*x) - sqrt(c + d*x))/d + 2*B*a*(c**2/sqrt(c + d*x) + 2*c*sqrt(c + d*x) - (c + d*x)**(3/2)/3)/d + 2*B*b*c*(c**
2/sqrt(c + d*x) + 2*c*sqrt(c + d*x) - (c + d*x)**(3/2)/3)/d**2 + 2*B*b*(-c**3/sqrt(c + d*x) - 3*c**2*sqrt(c +
d*x) + c*(c + d*x)**(3/2) - (c + d*x)**(5/2)/5)/d**2 + 2*C*a*c*(c**2/sqrt(c + d*x) + 2*c*sqrt(c + d*x) - (c +
d*x)**(3/2)/3)/d**2 + 2*C*a*(-c**3/sqrt(c + d*x) - 3*c**2*sqrt(c + d*x) + c*(c + d*x)**(3/2) - (c + d*x)**(5/2
)/5)/d**2 + 2*C*b*c*(-c**3/sqrt(c + d*x) - 3*c**2*sqrt(c + d*x) + c*(c + d*x)**(3/2) - (c + d*x)**(5/2)/5)/d**
3 + 2*C*b*(c**4/sqrt(c + d*x) + 4*c**3*sqrt(c + d*x) - 2*c**2*(c + d*x)**(3/2) + 4*c*(c + d*x)**(5/2)/5 - (c +
d*x)**(7/2)/7)/d**3 + 2*D*a*c*(-c**3/sqrt(c + d*x) - 3*c**2*sqrt(c + d*x) + c*(c + d*x)**(3/2) - (c + d*x)**(
5/2)/5)/d**3 + 2*D*a*(c**4/sqrt(c + d*x) + 4*c**3*sqrt(c + d*x) - 2*c**2*(c + d*x)**(3/2) + 4*c*(c + d*x)**(5/
2)/5 - (c + d*x)**(7/2)/7)/d**3 + 2*D*b*c*(c**4/sqrt(c + d*x) + 4*c**3*sqrt(c + d*x) - 2*c**2*(c + d*x)**(3/2)
+ 4*c*(c + d*x)**(5/2)/5 - (c + d*x)**(7/2)/7)/d**4 + 2*D*b*(-c**5/sqrt(c + d*x) - 5*c**4*sqrt(c + d*x) + 10*
c**3*(c + d*x)**(3/2)/3 - 2*c**2*(c + d*x)**(5/2) + 5*c*(c + d*x)**(7/2)/7 - (c + d*x)**(9/2)/9)/d**4)/d, Ne(d
, 0)), ((A*a*x + D*b*x**5/5 + x**4*(C*b + D*a)/4 + x**3*(B*b + C*a)/3 + x**2*(A*b + B*a)/2)/sqrt(c), True))

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Giac [A]  time = 2.0287, size = 417, normalized size = 1.97 \begin{align*} \frac{2 \,{\left (315 \, \sqrt{d x + c} A a + \frac{105 \,{\left ({\left (d x + c\right )}^{\frac{3}{2}} - 3 \, \sqrt{d x + c} c\right )} B a}{d} + \frac{105 \,{\left ({\left (d x + c\right )}^{\frac{3}{2}} - 3 \, \sqrt{d x + c} c\right )} A b}{d} + \frac{21 \,{\left (3 \,{\left (d x + c\right )}^{\frac{5}{2}} - 10 \,{\left (d x + c\right )}^{\frac{3}{2}} c + 15 \, \sqrt{d x + c} c^{2}\right )} C a}{d^{2}} + \frac{21 \,{\left (3 \,{\left (d x + c\right )}^{\frac{5}{2}} - 10 \,{\left (d x + c\right )}^{\frac{3}{2}} c + 15 \, \sqrt{d x + c} c^{2}\right )} B b}{d^{2}} + \frac{9 \,{\left (5 \,{\left (d x + c\right )}^{\frac{7}{2}} - 21 \,{\left (d x + c\right )}^{\frac{5}{2}} c + 35 \,{\left (d x + c\right )}^{\frac{3}{2}} c^{2} - 35 \, \sqrt{d x + c} c^{3}\right )} D a}{d^{3}} + \frac{9 \,{\left (5 \,{\left (d x + c\right )}^{\frac{7}{2}} - 21 \,{\left (d x + c\right )}^{\frac{5}{2}} c + 35 \,{\left (d x + c\right )}^{\frac{3}{2}} c^{2} - 35 \, \sqrt{d x + c} c^{3}\right )} C b}{d^{3}} + \frac{{\left (35 \,{\left (d x + c\right )}^{\frac{9}{2}} - 180 \,{\left (d x + c\right )}^{\frac{7}{2}} c + 378 \,{\left (d x + c\right )}^{\frac{5}{2}} c^{2} - 420 \,{\left (d x + c\right )}^{\frac{3}{2}} c^{3} + 315 \, \sqrt{d x + c} c^{4}\right )} D b}{d^{4}}\right )}}{315 \, 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)^(1/2),x, algorithm="giac")

[Out]

2/315*(315*sqrt(d*x + c)*A*a + 105*((d*x + c)^(3/2) - 3*sqrt(d*x + c)*c)*B*a/d + 105*((d*x + c)^(3/2) - 3*sqrt
(d*x + c)*c)*A*b/d + 21*(3*(d*x + c)^(5/2) - 10*(d*x + c)^(3/2)*c + 15*sqrt(d*x + c)*c^2)*C*a/d^2 + 21*(3*(d*x
+ c)^(5/2) - 10*(d*x + c)^(3/2)*c + 15*sqrt(d*x + c)*c^2)*B*b/d^2 + 9*(5*(d*x + c)^(7/2) - 21*(d*x + c)^(5/2)
*c + 35*(d*x + c)^(3/2)*c^2 - 35*sqrt(d*x + c)*c^3)*D*a/d^3 + 9*(5*(d*x + c)^(7/2) - 21*(d*x + c)^(5/2)*c + 35
*(d*x + c)^(3/2)*c^2 - 35*sqrt(d*x + c)*c^3)*C*b/d^3 + (35*(d*x + c)^(9/2) - 180*(d*x + c)^(7/2)*c + 378*(d*x
+ c)^(5/2)*c^2 - 420*(d*x + c)^(3/2)*c^3 + 315*sqrt(d*x + c)*c^4)*D*b/d^4)/d