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

[Out]

-2/3*(a*d*(-B*d^2+2*C*c*d-3*D*c^2)-b*(A*d^3-2*B*c*d^2+3*C*c^2*d-4*D*c^3))*(d*x+c)^(3/2)/d^5+2/5*(a*d*(C*d-3*D*
c)-b*(-B*d^2+3*C*c*d-6*D*c^2))*(d*x+c)^(5/2)/d^5+2/7*(C*b*d+D*a*d-4*D*b*c)*(d*x+c)^(7/2)/d^5+2/9*b*D*(d*x+c)^(
9/2)/d^5-2*(-a*d+b*c)*(A*d^3-B*c*d^2+C*c^2*d-D*c^3)*(d*x+c)^(1/2)/d^5

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Rubi [A]  time = 0.17, antiderivative size = 212, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.31, size = 184, normalized size = 0.87 \[ \frac {2 \sqrt {c+d x} \left (3 a d \left (d^3 (105 A+x (35 B+3 x (7 C+5 D x)))-2 c d^2 (35 B+x (14 C+9 D x))-48 c^3 D+8 c^2 d (7 C+3 D x)\right )+b \left (-2 c d^3 (105 A+x (42 B+x (27 C+20 D x)))+d^4 x (105 A+x (63 B+5 x (9 C+7 D x)))+24 c^2 d^2 (7 B+x (3 C+2 D x))+128 c^4 D-16 c^3 d (9 C+4 D x)\right )\right )}{315 d^5} \]

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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fricas [A]  time = 0.69, size = 204, normalized size = 0.96 \[ \frac {2 \, {\left (35 \, D b d^{4} x^{4} + 128 \, D b c^{4} + 315 \, A a d^{4} + 168 \, {\left (C a + B b\right )} c^{2} d^{2} - 210 \, {\left (B a + A b\right )} c d^{3} - 5 \, {\left (8 \, D b c d^{3} - 9 \, {\left (D a + C b\right )} d^{4}\right )} x^{3} + 3 \, {\left (16 \, D b c^{2} d^{2} + 21 \, {\left (C a + B b\right )} d^{4} - 18 \, {\left (D a c + C b c\right )} d^{3}\right )} x^{2} - 144 \, {\left (D a c^{3} + C b c^{3}\right )} d - {\left (64 \, D b c^{3} d + 84 \, {\left (C a + B b\right )} c d^{3} - 105 \, {\left (B a + A b\right )} d^{4} - 72 \, {\left (D a c^{2} + C b c^{2}\right )} d^{2}\right )} x\right )} \sqrt {d x + c}}{315 \, d^{5}} \]

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]

2/315*(35*D*b*d^4*x^4 + 128*D*b*c^4 + 315*A*a*d^4 + 168*(C*a + B*b)*c^2*d^2 - 210*(B*a + A*b)*c*d^3 - 5*(8*D*b
*c*d^3 - 9*(D*a + C*b)*d^4)*x^3 + 3*(16*D*b*c^2*d^2 + 21*(C*a + B*b)*d^4 - 18*(D*a*c + C*b*c)*d^3)*x^2 - 144*(
D*a*c^3 + C*b*c^3)*d - (64*D*b*c^3*d + 84*(C*a + B*b)*c*d^3 - 105*(B*a + A*b)*d^4 - 72*(D*a*c^2 + C*b*c^2)*d^2
)*x)*sqrt(d*x + c)/d^5

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giac [A]  time = 1.25, size = 309, normalized size = 1.46 \[ \frac {2 \, {\left (315 \, \sqrt {d x + c} A a + \frac {105 \, {\left ({\left (d x + c\right )}^{\frac {3}{2}} - 3 \, \sqrt {d x + c} c\right )} B a}{d} + \frac {105 \, {\left ({\left (d x + c\right )}^{\frac {3}{2}} - 3 \, \sqrt {d x + c} c\right )} A b}{d} + \frac {21 \, {\left (3 \, {\left (d x + c\right )}^{\frac {5}{2}} - 10 \, {\left (d x + c\right )}^{\frac {3}{2}} c + 15 \, \sqrt {d x + c} c^{2}\right )} C a}{d^{2}} + \frac {21 \, {\left (3 \, {\left (d x + c\right )}^{\frac {5}{2}} - 10 \, {\left (d x + c\right )}^{\frac {3}{2}} c + 15 \, \sqrt {d x + c} c^{2}\right )} B b}{d^{2}} + \frac {9 \, {\left (5 \, {\left (d x + c\right )}^{\frac {7}{2}} - 21 \, {\left (d x + c\right )}^{\frac {5}{2}} c + 35 \, {\left (d x + c\right )}^{\frac {3}{2}} c^{2} - 35 \, \sqrt {d x + c} c^{3}\right )} D a}{d^{3}} + \frac {9 \, {\left (5 \, {\left (d x + c\right )}^{\frac {7}{2}} - 21 \, {\left (d x + c\right )}^{\frac {5}{2}} c + 35 \, {\left (d x + c\right )}^{\frac {3}{2}} c^{2} - 35 \, \sqrt {d x + c} c^{3}\right )} C b}{d^{3}} + \frac {{\left (35 \, {\left (d x + c\right )}^{\frac {9}{2}} - 180 \, {\left (d x + c\right )}^{\frac {7}{2}} c + 378 \, {\left (d x + c\right )}^{\frac {5}{2}} c^{2} - 420 \, {\left (d x + c\right )}^{\frac {3}{2}} c^{3} + 315 \, \sqrt {d x + c} c^{4}\right )} D b}{d^{4}}\right )}}{315 \, d} \]

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

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maple [A]  time = 0.01, size = 241, normalized size = 1.14 \[ \frac {2 \sqrt {d x +c}\, \left (35 D b \,x^{4} d^{4}+45 C b \,d^{4} x^{3}+45 D a \,d^{4} x^{3}-40 D b c \,d^{3} x^{3}+63 B b \,d^{4} x^{2}+63 C a \,d^{4} x^{2}-54 C b c \,d^{3} x^{2}-54 D a c \,d^{3} x^{2}+48 D b \,c^{2} d^{2} x^{2}+105 A b \,d^{4} x +105 B a \,d^{4} x -84 B b c \,d^{3} x -84 C a c \,d^{3} x +72 C b \,c^{2} d^{2} x +72 D a \,c^{2} d^{2} x -64 D b \,c^{3} d x +315 A a \,d^{4}-210 A b c \,d^{3}-210 B a c \,d^{3}+168 B b \,c^{2} d^{2}+168 C a \,c^{2} d^{2}-144 C b \,c^{3} d -144 D a \,c^{3} d +128 D b \,c^{4}\right )}{315 d^{5}} \]

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 = 0.44, size = 198, normalized size = 0.93 \[ \frac {2 \, {\left (35 \, {\left (d x + c\right )}^{\frac {9}{2}} D b - 45 \, {\left (4 \, D b c - {\left (D a + C b\right )} d\right )} {\left (d x + c\right )}^{\frac {7}{2}} + 63 \, {\left (6 \, D b c^{2} - 3 \, {\left (D a + C b\right )} c d + {\left (C a + B b\right )} d^{2}\right )} {\left (d x + c\right )}^{\frac {5}{2}} - 105 \, {\left (4 \, D b c^{3} - 3 \, {\left (D a + C b\right )} c^{2} d + 2 \, {\left (C a + B b\right )} c d^{2} - {\left (B a + A b\right )} d^{3}\right )} {\left (d x + c\right )}^{\frac {3}{2}} + 315 \, {\left (D b c^{4} + A a d^{4} - {\left (D a + C b\right )} c^{3} d + {\left (C a + B b\right )} c^{2} d^{2} - {\left (B a + A b\right )} c d^{3}\right )} \sqrt {d x + c}\right )}}{315 \, d^{5}} \]

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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mupad [B]  time = 3.30, size = 351, normalized size = 1.66 \[ \frac {2\,A\,b\,{\left (c+d\,x\right )}^{3/2}-6\,A\,b\,c\,\sqrt {c+d\,x}}{3\,d^2}+\frac {2\,B\,a\,{\left (c+d\,x\right )}^{3/2}-6\,B\,a\,c\,\sqrt {c+d\,x}}{3\,d^2}+\frac {6\,B\,b\,{\left (c+d\,x\right )}^{5/2}+30\,B\,b\,c^2\,\sqrt {c+d\,x}-20\,B\,b\,c\,{\left (c+d\,x\right )}^{3/2}}{15\,d^3}+\frac {6\,C\,a\,{\left (c+d\,x\right )}^{5/2}+30\,C\,a\,c^2\,\sqrt {c+d\,x}-20\,C\,a\,c\,{\left (c+d\,x\right )}^{3/2}}{15\,d^3}+\frac {2\,A\,a\,\sqrt {c+d\,x}}{d}+\frac {2\,C\,b\,{\left (c+d\,x\right )}^{7/2}}{7\,d^4}-\frac {2\,a\,\sqrt {c+d\,x}\,D\,\left (6\,c\,{\left (c+d\,x\right )}^2-20\,c^2\,\left (c+d\,x\right )+30\,c^3-5\,d^3\,x^3\right )}{35\,d^4}+\frac {2\,b\,x^4\,\sqrt {c+d\,x}\,D}{9\,d}-\frac {6\,C\,b\,c\,{\left (c+d\,x\right )}^{5/2}}{5\,d^4}-\frac {8\,b\,c\,D\,\left (\frac {2\,{\left (c+d\,x\right )}^{7/2}}{7\,d^4}-\frac {2\,c^3\,\sqrt {c+d\,x}}{d^4}+\frac {2\,c^2\,{\left (c+d\,x\right )}^{3/2}}{d^4}-\frac {6\,c\,{\left (c+d\,x\right )}^{5/2}}{5\,d^4}\right )}{9\,d}-\frac {2\,C\,b\,c^3\,\sqrt {c+d\,x}}{d^4}+\frac {2\,C\,b\,c^2\,{\left (c+d\,x\right )}^{3/2}}{d^4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + b*x)*(A + B*x + C*x^2 + x^3*D))/(c + d*x)^(1/2),x)

[Out]

(2*A*b*(c + d*x)^(3/2) - 6*A*b*c*(c + d*x)^(1/2))/(3*d^2) + (2*B*a*(c + d*x)^(3/2) - 6*B*a*c*(c + d*x)^(1/2))/
(3*d^2) + (6*B*b*(c + d*x)^(5/2) + 30*B*b*c^2*(c + d*x)^(1/2) - 20*B*b*c*(c + d*x)^(3/2))/(15*d^3) + (6*C*a*(c
+ d*x)^(5/2) + 30*C*a*c^2*(c + d*x)^(1/2) - 20*C*a*c*(c + d*x)^(3/2))/(15*d^3) + (2*A*a*(c + d*x)^(1/2))/d +
(2*C*b*(c + d*x)^(7/2))/(7*d^4) - (2*a*(c + d*x)^(1/2)*D*(6*c*(c + d*x)^2 - 20*c^2*(c + d*x) + 30*c^3 - 5*d^3*
x^3))/(35*d^4) + (2*b*x^4*(c + d*x)^(1/2)*D)/(9*d) - (6*C*b*c*(c + d*x)^(5/2))/(5*d^4) - (8*b*c*D*((2*(c + d*x
)^(7/2))/(7*d^4) - (2*c^3*(c + d*x)^(1/2))/d^4 + (2*c^2*(c + d*x)^(3/2))/d^4 - (6*c*(c + d*x)^(5/2))/(5*d^4)))
/(9*d) - (2*C*b*c^3*(c + d*x)^(1/2))/d^4 + (2*C*b*c^2*(c + d*x)^(3/2))/d^4

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sympy [A]  time = 87.37, size = 848, normalized size = 4.00 \[ \begin {cases} \frac {- \frac {2 A a c}{\sqrt {c + d x}} - 2 A a \left (- \frac {c}{\sqrt {c + d x}} - \sqrt {c + d x}\right ) - \frac {2 A b c \left (- \frac {c}{\sqrt {c + d x}} - \sqrt {c + d x}\right )}{d} - \frac {2 A b \left (\frac {c^{2}}{\sqrt {c + d x}} + 2 c \sqrt {c + d x} - \frac {\left (c + d x\right )^{\frac {3}{2}}}{3}\right )}{d} - \frac {2 B a c \left (- \frac {c}{\sqrt {c + d x}} - \sqrt {c + d x}\right )}{d} - \frac {2 B a \left (\frac {c^{2}}{\sqrt {c + d x}} + 2 c \sqrt {c + d x} - \frac {\left (c + d x\right )^{\frac {3}{2}}}{3}\right )}{d} - \frac {2 B b c \left (\frac {c^{2}}{\sqrt {c + d x}} + 2 c \sqrt {c + d x} - \frac {\left (c + d x\right )^{\frac {3}{2}}}{3}\right )}{d^{2}} - \frac {2 B b \left (- \frac {c^{3}}{\sqrt {c + d x}} - 3 c^{2} \sqrt {c + d x} + c \left (c + d x\right )^{\frac {3}{2}} - \frac {\left (c + d x\right )^{\frac {5}{2}}}{5}\right )}{d^{2}} - \frac {2 C a c \left (\frac {c^{2}}{\sqrt {c + d x}} + 2 c \sqrt {c + d x} - \frac {\left (c + d x\right )^{\frac {3}{2}}}{3}\right )}{d^{2}} - \frac {2 C a \left (- \frac {c^{3}}{\sqrt {c + d x}} - 3 c^{2} \sqrt {c + d x} + c \left (c + d x\right )^{\frac {3}{2}} - \frac {\left (c + d x\right )^{\frac {5}{2}}}{5}\right )}{d^{2}} - \frac {2 C b c \left (- \frac {c^{3}}{\sqrt {c + d x}} - 3 c^{2} \sqrt {c + d x} + c \left (c + d x\right )^{\frac {3}{2}} - \frac {\left (c + d x\right )^{\frac {5}{2}}}{5}\right )}{d^{3}} - \frac {2 C b \left (\frac {c^{4}}{\sqrt {c + d x}} + 4 c^{3} \sqrt {c + d x} - 2 c^{2} \left (c + d x\right )^{\frac {3}{2}} + \frac {4 c \left (c + d x\right )^{\frac {5}{2}}}{5} - \frac {\left (c + d x\right )^{\frac {7}{2}}}{7}\right )}{d^{3}} - \frac {2 D a c \left (- \frac {c^{3}}{\sqrt {c + d x}} - 3 c^{2} \sqrt {c + d x} + c \left (c + d x\right )^{\frac {3}{2}} - \frac {\left (c + d x\right )^{\frac {5}{2}}}{5}\right )}{d^{3}} - \frac {2 D a \left (\frac {c^{4}}{\sqrt {c + d x}} + 4 c^{3} \sqrt {c + d x} - 2 c^{2} \left (c + d x\right )^{\frac {3}{2}} + \frac {4 c \left (c + d x\right )^{\frac {5}{2}}}{5} - \frac {\left (c + d x\right )^{\frac {7}{2}}}{7}\right )}{d^{3}} - \frac {2 D b c \left (\frac {c^{4}}{\sqrt {c + d x}} + 4 c^{3} \sqrt {c + d x} - 2 c^{2} \left (c + d x\right )^{\frac {3}{2}} + \frac {4 c \left (c + d x\right )^{\frac {5}{2}}}{5} - \frac {\left (c + d x\right )^{\frac {7}{2}}}{7}\right )}{d^{4}} - \frac {2 D b \left (- \frac {c^{5}}{\sqrt {c + d x}} - 5 c^{4} \sqrt {c + d x} + \frac {10 c^{3} \left (c + d x\right )^{\frac {3}{2}}}{3} - 2 c^{2} \left (c + d x\right )^{\frac {5}{2}} + \frac {5 c \left (c + d x\right )^{\frac {7}{2}}}{7} - \frac {\left (c + d x\right )^{\frac {9}{2}}}{9}\right )}{d^{4}}}{d} & \text {for}\: d \neq 0 \\\frac {A a x + \frac {D b x^{5}}{5} + \frac {x^{4} \left (C b + D a\right )}{4} + \frac {x^{3} \left (B b + C a\right )}{3} + \frac {x^{2} \left (A b + B a\right )}{2}}{\sqrt {c}} & \text {otherwise} \end {cases} \]

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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