\(\int \frac {(a+b x)^2 (A+B x+C x^2+D x^3)}{(c+d x)^{5/2}} \, dx\) [19]

Optimal result

Integrand size = 32, antiderivative size = 322 \[ \int \frac {(a+b x)^2 \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{5/2}} \, dx=-\frac {2 (b c-a d)^2 \left (c^2 C d-B c d^2+A d^3-c^3 D\right )}{3 d^6 (c+d x)^{3/2}}-\frac {2 (b c-a d) \left (a d \left (2 c C d-B d^2-3 c^2 D\right )-b \left (4 c^2 C d-3 B c d^2+2 A d^3-5 c^3 D\right )\right )}{d^6 \sqrt {c+d x}}+\frac {2 \left (a^2 d^2 (C d-3 c D)-2 a b d \left (3 c C d-B d^2-6 c^2 D\right )+b^2 \left (6 c^2 C d-3 B c d^2+A d^3-10 c^3 D\right )\right ) \sqrt {c+d x}}{d^6}+\frac {2 \left (a^2 d^2 D+2 a b d (C d-4 c D)-b^2 \left (4 c C d-B d^2-10 c^2 D\right )\right ) (c+d x)^{3/2}}{3 d^6}+\frac {2 b (b C d-5 b c D+2 a d D) (c+d x)^{5/2}}{5 d^6}+\frac {2 b^2 D (c+d x)^{7/2}}{7 d^6} \]

[Out]

-2/3*(-a*d+b*c)^2*(A*d^3-B*c*d^2+C*c^2*d-D*c^3)/d^6/(d*x+c)^(3/2)+2/3*(a^2*d^2*D+2*a*b*d*(C*d-4*D*c)-b^2*(-B*d
^2+4*C*c*d-10*D*c^2))*(d*x+c)^(3/2)/d^6+2/5*b*(C*b*d+2*D*a*d-5*D*b*c)*(d*x+c)^(5/2)/d^6+2/7*b^2*D*(d*x+c)^(7/2
)/d^6-2*(-a*d+b*c)*(a*d*(-B*d^2+2*C*c*d-3*D*c^2)-b*(2*A*d^3-3*B*c*d^2+4*C*c^2*d-5*D*c^3))/d^6/(d*x+c)^(1/2)+2*
(a^2*d^2*(C*d-3*D*c)-2*a*b*d*(-B*d^2+3*C*c*d-6*D*c^2)+b^2*(A*d^3-3*B*c*d^2+6*C*c^2*d-10*D*c^3))*(d*x+c)^(1/2)/
d^6

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 322, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.031, Rules used = {1634} \[ \int \frac {(a+b x)^2 \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{5/2}} \, dx=\frac {2 \sqrt {c+d x} \left (a^2 d^2 (C d-3 c D)-2 a b d \left (-B d^2-6 c^2 D+3 c C d\right )+b^2 \left (A d^3-3 B c d^2-10 c^3 D+6 c^2 C d\right )\right )}{d^6}+\frac {2 (c+d x)^{3/2} \left (a^2 d^2 D+2 a b d (C d-4 c D)-\left (b^2 \left (-B d^2-10 c^2 D+4 c C d\right )\right )\right )}{3 d^6}-\frac {2 (b c-a d) \left (a d \left (-B d^2-3 c^2 D+2 c C d\right )-b \left (2 A d^3-3 B c d^2-5 c^3 D+4 c^2 C d\right )\right )}{d^6 \sqrt {c+d x}}-\frac {2 (b c-a d)^2 \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{3 d^6 (c+d x)^{3/2}}+\frac {2 b (c+d x)^{5/2} (2 a d D-5 b c D+b C d)}{5 d^6}+\frac {2 b^2 D (c+d x)^{7/2}}{7 d^6} \]

[In]

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

[Out]

(-2*(b*c - a*d)^2*(c^2*C*d - B*c*d^2 + A*d^3 - c^3*D))/(3*d^6*(c + d*x)^(3/2)) - (2*(b*c - a*d)*(a*d*(2*c*C*d
- B*d^2 - 3*c^2*D) - b*(4*c^2*C*d - 3*B*c*d^2 + 2*A*d^3 - 5*c^3*D)))/(d^6*Sqrt[c + d*x]) + (2*(a^2*d^2*(C*d -
3*c*D) - 2*a*b*d*(3*c*C*d - B*d^2 - 6*c^2*D) + b^2*(6*c^2*C*d - 3*B*c*d^2 + A*d^3 - 10*c^3*D))*Sqrt[c + d*x])/
d^6 + (2*(a^2*d^2*D + 2*a*b*d*(C*d - 4*c*D) - b^2*(4*c*C*d - B*d^2 - 10*c^2*D))*(c + d*x)^(3/2))/(3*d^6) + (2*
b*(b*C*d - 5*b*c*D + 2*a*d*D)*(c + d*x)^(5/2))/(5*d^6) + (2*b^2*D*(c + d*x)^(7/2))/(7*d^6)

Rule 1634

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*} \text {integral}& = \int \left (\frac {(-b c+a d)^2 \left (c^2 C d-B c d^2+A d^3-c^3 D\right )}{d^5 (c+d x)^{5/2}}+\frac {(b c-a d) \left (a d \left (2 c C d-B d^2-3 c^2 D\right )-b \left (4 c^2 C d-3 B c d^2+2 A d^3-5 c^3 D\right )\right )}{d^5 (c+d x)^{3/2}}+\frac {a^2 d^2 (C d-3 c D)-2 a b d \left (3 c C d-B d^2-6 c^2 D\right )+b^2 \left (6 c^2 C d-3 B c d^2+A d^3-10 c^3 D\right )}{d^5 \sqrt {c+d x}}+\frac {\left (a^2 d^2 D+2 a b d (C d-4 c D)-b^2 \left (4 c C d-B d^2-10 c^2 D\right )\right ) \sqrt {c+d x}}{d^5}+\frac {b (b C d-5 b c D+2 a d D) (c+d x)^{3/2}}{d^5}+\frac {b^2 D (c+d x)^{5/2}}{d^5}\right ) \, dx \\ & = -\frac {2 (b c-a d)^2 \left (c^2 C d-B c d^2+A d^3-c^3 D\right )}{3 d^6 (c+d x)^{3/2}}-\frac {2 (b c-a d) \left (a d \left (2 c C d-B d^2-3 c^2 D\right )-b \left (4 c^2 C d-3 B c d^2+2 A d^3-5 c^3 D\right )\right )}{d^6 \sqrt {c+d x}}+\frac {2 \left (a^2 d^2 (C d-3 c D)-2 a b d \left (3 c C d-B d^2-6 c^2 D\right )+b^2 \left (6 c^2 C d-3 B c d^2+A d^3-10 c^3 D\right )\right ) \sqrt {c+d x}}{d^6}+\frac {2 \left (a^2 d^2 D+2 a b d (C d-4 c D)-b^2 \left (4 c C d-B d^2-10 c^2 D\right )\right ) (c+d x)^{3/2}}{3 d^6}+\frac {2 b (b C d-5 b c D+2 a d D) (c+d x)^{5/2}}{5 d^6}+\frac {2 b^2 D (c+d x)^{7/2}}{7 d^6} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.36 (sec) , antiderivative size = 317, normalized size of antiderivative = 0.98 \[ \int \frac {(a+b x)^2 \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{5/2}} \, dx=\frac {-70 a^2 d^2 \left (16 c^3 D-8 c^2 d (C-3 D x)+2 c d^2 (B+3 x (-2 C+D x))+d^3 \left (A+3 B x-x^2 (3 C+D x)\right )\right )+28 a b d \left (128 c^4 D+c^3 (-80 C d+192 d D x)+8 c^2 d^2 (5 B+3 x (-5 C+2 D x))+d^4 x \left (-15 A+x \left (15 B+5 C x+3 D x^2\right )\right )-2 c d^3 \left (5 A+x \left (-30 B+15 C x+4 D x^2\right )\right )\right )+2 b^2 \left (-1280 c^5 D+128 c^4 d (7 C-15 D x)-16 c^3 d^2 (35 B+6 x (-14 C+5 D x))+d^5 x^2 (105 A+x (35 B+3 x (7 C+5 D x)))+8 c^2 d^3 (35 A+x (-105 B+2 x (21 C+5 D x)))-2 c d^4 x (-210 A+x (105 B+x (28 C+15 D x)))\right )}{105 d^6 (c+d x)^{3/2}} \]

[In]

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

[Out]

(-70*a^2*d^2*(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))) + 28*a*b*d*(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))) + 2*b^2*(-1280*c^5*D + 128*c^4*d*
(7*C - 15*D*x) - 16*c^3*d^2*(35*B + 6*x*(-14*C + 5*D*x)) + d^5*x^2*(105*A + x*(35*B + 3*x*(7*C + 5*D*x))) + 8*
c^2*d^3*(35*A + x*(-105*B + 2*x*(21*C + 5*D*x))) - 2*c*d^4*x*(-210*A + x*(105*B + x*(28*C + 15*D*x)))))/(105*d
^6*(c + d*x)^(3/2))

Maple [A] (verified)

Time = 1.73 (sec) , antiderivative size = 288, normalized size of antiderivative = 0.89

method	result	size
pseudoelliptic	\(-\frac {2 \left (\left (\left (-\frac {3}{7} D x^{5}-\frac {3}{5} C \,x^{4}-3 A \,x^{2}-x^{3} B \right ) b^{2}+6 a x \left (-\frac {1}{5} D x^{3}-\frac {1}{3} C \,x^{2}-B x +A \right ) b +a^{2} \left (-D x^{3}-3 C \,x^{2}+3 B x +A \right )\right ) d^{5}+4 \left (-3 x \left (-\frac {1}{14} D x^{3}-\frac {2}{15} C \,x^{2}-\frac {1}{2} B x +A \right ) b^{2}+a \left (\frac {4}{5} D x^{3}+3 C \,x^{2}-6 B x +A \right ) b +\frac {a^{2} \left (3 D x^{2}-6 C x +B \right )}{2}\right ) c \,d^{4}-8 \left (\left (\frac {2}{7} D x^{3}+\frac {6}{5} C \,x^{2}-3 B x +A \right ) b^{2}+2 a \left (\frac {6}{5} D x^{2}-3 C x +B \right ) b +a^{2} \left (-3 D x +C \right )\right ) c^{2} d^{3}+16 c^{3} \left (\left (\frac {6}{7} D x^{2}-\frac {12}{5} C x +B \right ) b^{2}+2 \left (-\frac {12 D x}{5}+C \right ) a b +D a^{2}\right ) d^{2}-\frac {128 \left (\left (-\frac {15 D x}{7}+C \right ) b +2 D a \right ) b \,c^{4} d}{5}+\frac {256 D b^{2} c^{5}}{7}\right )}{3 \left (d x +c \right )^{\frac {3}{2}} d^{6}}\)	\(288\)
gosper	\(-\frac {2 \left (-15 D b^{2} x^{5} d^{5}-21 C \,b^{2} d^{5} x^{4}-42 D a b \,d^{5} x^{4}+30 D b^{2} c \,d^{4} x^{4}-35 B \,b^{2} d^{5} x^{3}-70 C a b \,d^{5} x^{3}+56 C \,b^{2} c \,d^{4} x^{3}-35 D a^{2} d^{5} x^{3}+112 D a b c \,d^{4} x^{3}-80 D b^{2} c^{2} d^{3} x^{3}-105 A \,b^{2} d^{5} x^{2}-210 B a b \,d^{5} x^{2}+210 B \,b^{2} c \,d^{4} x^{2}-105 C \,a^{2} d^{5} x^{2}+420 C a b c \,d^{4} x^{2}-336 C \,b^{2} c^{2} d^{3} x^{2}+210 D a^{2} c \,d^{4} x^{2}-672 D a b \,c^{2} d^{3} x^{2}+480 D b^{2} c^{3} d^{2} x^{2}+210 A a b \,d^{5} x -420 A \,b^{2} c \,d^{4} x +105 B \,a^{2} d^{5} x -840 B a b c \,d^{4} x +840 B \,b^{2} c^{2} d^{3} x -420 C \,a^{2} c \,d^{4} x +1680 C a b \,c^{2} d^{3} x -1344 C \,b^{2} c^{3} d^{2} x +840 D a^{2} c^{2} d^{3} x -2688 D a b \,c^{3} d^{2} x +1920 D b^{2} c^{4} d x +35 a^{2} A \,d^{5}+140 A a b c \,d^{4}-280 A \,b^{2} c^{2} d^{3}+70 B \,a^{2} c \,d^{4}-560 B a b \,c^{2} d^{3}+560 B \,b^{2} c^{3} d^{2}-280 C \,a^{2} c^{2} d^{3}+1120 C a b \,c^{3} d^{2}-896 C \,b^{2} c^{4} d +560 D a^{2} c^{3} d^{2}-1792 D a b \,c^{4} d +1280 D b^{2} c^{5}\right )}{105 \left (d x +c \right )^{\frac {3}{2}} d^{6}}\)	\(505\)
trager	\(-\frac {2 \left (-15 D b^{2} x^{5} d^{5}-21 C \,b^{2} d^{5} x^{4}-42 D a b \,d^{5} x^{4}+30 D b^{2} c \,d^{4} x^{4}-35 B \,b^{2} d^{5} x^{3}-70 C a b \,d^{5} x^{3}+56 C \,b^{2} c \,d^{4} x^{3}-35 D a^{2} d^{5} x^{3}+112 D a b c \,d^{4} x^{3}-80 D b^{2} c^{2} d^{3} x^{3}-105 A \,b^{2} d^{5} x^{2}-210 B a b \,d^{5} x^{2}+210 B \,b^{2} c \,d^{4} x^{2}-105 C \,a^{2} d^{5} x^{2}+420 C a b c \,d^{4} x^{2}-336 C \,b^{2} c^{2} d^{3} x^{2}+210 D a^{2} c \,d^{4} x^{2}-672 D a b \,c^{2} d^{3} x^{2}+480 D b^{2} c^{3} d^{2} x^{2}+210 A a b \,d^{5} x -420 A \,b^{2} c \,d^{4} x +105 B \,a^{2} d^{5} x -840 B a b c \,d^{4} x +840 B \,b^{2} c^{2} d^{3} x -420 C \,a^{2} c \,d^{4} x +1680 C a b \,c^{2} d^{3} x -1344 C \,b^{2} c^{3} d^{2} x +840 D a^{2} c^{2} d^{3} x -2688 D a b \,c^{3} d^{2} x +1920 D b^{2} c^{4} d x +35 a^{2} A \,d^{5}+140 A a b c \,d^{4}-280 A \,b^{2} c^{2} d^{3}+70 B \,a^{2} c \,d^{4}-560 B a b \,c^{2} d^{3}+560 B \,b^{2} c^{3} d^{2}-280 C \,a^{2} c^{2} d^{3}+1120 C a b \,c^{3} d^{2}-896 C \,b^{2} c^{4} d +560 D a^{2} c^{3} d^{2}-1792 D a b \,c^{4} d +1280 D b^{2} c^{5}\right )}{105 \left (d x +c \right )^{\frac {3}{2}} d^{6}}\)	\(505\)
derivativedivides	\(\frac {\frac {2 D b^{2} \left (d x +c \right )^{\frac {7}{2}}}{7}+\frac {2 C \,b^{2} d \left (d x +c \right )^{\frac {5}{2}}}{5}+\frac {4 D a b d \left (d x +c \right )^{\frac {5}{2}}}{5}-2 D b^{2} c \left (d x +c \right )^{\frac {5}{2}}+\frac {2 B \,b^{2} d^{2} \left (d x +c \right )^{\frac {3}{2}}}{3}+\frac {4 C a b \,d^{2} \left (d x +c \right )^{\frac {3}{2}}}{3}-\frac {8 C \,b^{2} c d \left (d x +c \right )^{\frac {3}{2}}}{3}+\frac {2 D a^{2} d^{2} \left (d x +c \right )^{\frac {3}{2}}}{3}-\frac {16 D a b c d \left (d x +c \right )^{\frac {3}{2}}}{3}+\frac {20 D b^{2} c^{2} \left (d x +c \right )^{\frac {3}{2}}}{3}+2 A \,b^{2} d^{3} \sqrt {d x +c}+4 B a b \,d^{3} \sqrt {d x +c}-6 B \,b^{2} c \,d^{2} \sqrt {d x +c}+2 C \,a^{2} d^{3} \sqrt {d x +c}-12 C a b c \,d^{2} \sqrt {d x +c}+12 C \,b^{2} c^{2} d \sqrt {d x +c}-6 D a^{2} c \,d^{2} \sqrt {d x +c}+24 D a b \,c^{2} d \sqrt {d x +c}-20 D b^{2} c^{3} \sqrt {d x +c}-\frac {2 \left (2 a b A \,d^{4}-2 A \,b^{2} c \,d^{3}+a^{2} B \,d^{4}-4 B a b c \,d^{3}+3 B \,b^{2} c^{2} d^{2}-2 C \,a^{2} c \,d^{3}+6 C a b \,c^{2} d^{2}-4 C \,b^{2} c^{3} d +3 D a^{2} c^{2} d^{2}-8 D a b \,c^{3} d +5 D b^{2} c^{4}\right )}{\sqrt {d x +c}}-\frac {2 \left (a^{2} A \,d^{5}-2 A a b c \,d^{4}+A \,b^{2} c^{2} d^{3}-B \,a^{2} c \,d^{4}+2 B a b \,c^{2} d^{3}-B \,b^{2} c^{3} d^{2}+C \,a^{2} c^{2} d^{3}-2 C a b \,c^{3} d^{2}+C \,b^{2} c^{4} d -D a^{2} c^{3} d^{2}+2 D a b \,c^{4} d -D b^{2} c^{5}\right )}{3 \left (d x +c \right )^{\frac {3}{2}}}}{d^{6}}\)	\(547\)
default	\(\frac {\frac {2 D b^{2} \left (d x +c \right )^{\frac {7}{2}}}{7}+\frac {2 C \,b^{2} d \left (d x +c \right )^{\frac {5}{2}}}{5}+\frac {4 D a b d \left (d x +c \right )^{\frac {5}{2}}}{5}-2 D b^{2} c \left (d x +c \right )^{\frac {5}{2}}+\frac {2 B \,b^{2} d^{2} \left (d x +c \right )^{\frac {3}{2}}}{3}+\frac {4 C a b \,d^{2} \left (d x +c \right )^{\frac {3}{2}}}{3}-\frac {8 C \,b^{2} c d \left (d x +c \right )^{\frac {3}{2}}}{3}+\frac {2 D a^{2} d^{2} \left (d x +c \right )^{\frac {3}{2}}}{3}-\frac {16 D a b c d \left (d x +c \right )^{\frac {3}{2}}}{3}+\frac {20 D b^{2} c^{2} \left (d x +c \right )^{\frac {3}{2}}}{3}+2 A \,b^{2} d^{3} \sqrt {d x +c}+4 B a b \,d^{3} \sqrt {d x +c}-6 B \,b^{2} c \,d^{2} \sqrt {d x +c}+2 C \,a^{2} d^{3} \sqrt {d x +c}-12 C a b c \,d^{2} \sqrt {d x +c}+12 C \,b^{2} c^{2} d \sqrt {d x +c}-6 D a^{2} c \,d^{2} \sqrt {d x +c}+24 D a b \,c^{2} d \sqrt {d x +c}-20 D b^{2} c^{3} \sqrt {d x +c}-\frac {2 \left (2 a b A \,d^{4}-2 A \,b^{2} c \,d^{3}+a^{2} B \,d^{4}-4 B a b c \,d^{3}+3 B \,b^{2} c^{2} d^{2}-2 C \,a^{2} c \,d^{3}+6 C a b \,c^{2} d^{2}-4 C \,b^{2} c^{3} d +3 D a^{2} c^{2} d^{2}-8 D a b \,c^{3} d +5 D b^{2} c^{4}\right )}{\sqrt {d x +c}}-\frac {2 \left (a^{2} A \,d^{5}-2 A a b c \,d^{4}+A \,b^{2} c^{2} d^{3}-B \,a^{2} c \,d^{4}+2 B a b \,c^{2} d^{3}-B \,b^{2} c^{3} d^{2}+C \,a^{2} c^{2} d^{3}-2 C a b \,c^{3} d^{2}+C \,b^{2} c^{4} d -D a^{2} c^{3} d^{2}+2 D a b \,c^{4} d -D b^{2} c^{5}\right )}{3 \left (d x +c \right )^{\frac {3}{2}}}}{d^{6}}\)	\(547\)

[In]

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

[Out]

-2/3/(d*x+c)^(3/2)*(((-3/7*D*x^5-3/5*C*x^4-3*A*x^2-x^3*B)*b^2+6*a*x*(-1/5*D*x^3-1/3*C*x^2-B*x+A)*b+a^2*(-D*x^3
-3*C*x^2+3*B*x+A))*d^5+4*(-3*x*(-1/14*D*x^3-2/15*C*x^2-1/2*B*x+A)*b^2+a*(4/5*D*x^3+3*C*x^2-6*B*x+A)*b+1/2*a^2*
(3*D*x^2-6*C*x+B))*c*d^4-8*((2/7*D*x^3+6/5*C*x^2-3*B*x+A)*b^2+2*a*(6/5*D*x^2-3*C*x+B)*b+a^2*(-3*D*x+C))*c^2*d^
3+16*c^3*((6/7*D*x^2-12/5*C*x+B)*b^2+2*(-12/5*D*x+C)*a*b+D*a^2)*d^2-128/5*((-15/7*D*x+C)*b+2*D*a)*b*c^4*d+256/
7*D*b^2*c^5)/d^6

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 431, normalized size of antiderivative = 1.34 \[ \int \frac {(a+b x)^2 \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{5/2}} \, dx=\frac {2 \, {\left (15 \, D b^{2} d^{5} x^{5} - 1280 \, D b^{2} c^{5} - 35 \, A a^{2} d^{5} + 280 \, {\left (C a^{2} + 2 \, B a b + A b^{2}\right )} c^{2} d^{3} - 70 \, {\left (B a^{2} + 2 \, A a b\right )} c d^{4} - 3 \, {\left (10 \, D b^{2} c d^{4} - 7 \, {\left (2 \, D a b + C b^{2}\right )} d^{5}\right )} x^{4} + {\left (80 \, D b^{2} c^{2} d^{3} + 35 \, {\left (D a^{2} + 2 \, C a b + B b^{2}\right )} d^{5} - 56 \, {\left (2 \, D a b c + C b^{2} c\right )} d^{4}\right )} x^{3} - 560 \, {\left (D a^{2} c^{3} + {\left (2 \, C a b + B b^{2}\right )} c^{3}\right )} d^{2} - 3 \, {\left (160 \, D b^{2} c^{3} d^{2} - 35 \, {\left (C a^{2} + 2 \, B a b + A b^{2}\right )} d^{5} + 70 \, {\left (D a^{2} c + {\left (2 \, C a b + B b^{2}\right )} c\right )} d^{4} - 112 \, {\left (2 \, D a b c^{2} + C b^{2} c^{2}\right )} d^{3}\right )} x^{2} + 896 \, {\left (2 \, D a b c^{4} + C b^{2} c^{4}\right )} d - 3 \, {\left (640 \, D b^{2} c^{4} d - 140 \, {\left (C a^{2} + 2 \, B a b + A b^{2}\right )} c d^{4} + 35 \, {\left (B a^{2} + 2 \, A a b\right )} d^{5} + 280 \, {\left (D a^{2} c^{2} + {\left (2 \, C a b + B b^{2}\right )} c^{2}\right )} d^{3} - 448 \, {\left (2 \, D a b c^{3} + C b^{2} c^{3}\right )} d^{2}\right )} x\right )} \sqrt {d x + c}}{105 \, {\left (d^{8} x^{2} + 2 \, c d^{7} x + c^{2} d^{6}\right )}} \]

[In]

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

[Out]

2/105*(15*D*b^2*d^5*x^5 - 1280*D*b^2*c^5 - 35*A*a^2*d^5 + 280*(C*a^2 + 2*B*a*b + A*b^2)*c^2*d^3 - 70*(B*a^2 +
2*A*a*b)*c*d^4 - 3*(10*D*b^2*c*d^4 - 7*(2*D*a*b + C*b^2)*d^5)*x^4 + (80*D*b^2*c^2*d^3 + 35*(D*a^2 + 2*C*a*b +
B*b^2)*d^5 - 56*(2*D*a*b*c + C*b^2*c)*d^4)*x^3 - 560*(D*a^2*c^3 + (2*C*a*b + B*b^2)*c^3)*d^2 - 3*(160*D*b^2*c^
3*d^2 - 35*(C*a^2 + 2*B*a*b + A*b^2)*d^5 + 70*(D*a^2*c + (2*C*a*b + B*b^2)*c)*d^4 - 112*(2*D*a*b*c^2 + C*b^2*c
^2)*d^3)*x^2 + 896*(2*D*a*b*c^4 + C*b^2*c^4)*d - 3*(640*D*b^2*c^4*d - 140*(C*a^2 + 2*B*a*b + A*b^2)*c*d^4 + 35
*(B*a^2 + 2*A*a*b)*d^5 + 280*(D*a^2*c^2 + (2*C*a*b + B*b^2)*c^2)*d^3 - 448*(2*D*a*b*c^3 + C*b^2*c^3)*d^2)*x)*s
qrt(d*x + c)/(d^8*x^2 + 2*c*d^7*x + c^2*d^6)

Sympy [A] (verification not implemented)

Time = 29.35 (sec) , antiderivative size = 476, normalized size of antiderivative = 1.48 \[ \int \frac {(a+b x)^2 \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{5/2}} \, dx=\begin {cases} \frac {2 \left (\frac {D b^{2} \left (c + d x\right )^{\frac {7}{2}}}{7 d^{5}} + \frac {\left (c + d x\right )^{\frac {5}{2}} \left (C b^{2} d + 2 D a b d - 5 D b^{2} c\right )}{5 d^{5}} + \frac {\left (c + d x\right )^{\frac {3}{2}} \left (B b^{2} d^{2} + 2 C a b d^{2} - 4 C b^{2} c d + D a^{2} d^{2} - 8 D a b c d + 10 D b^{2} c^{2}\right )}{3 d^{5}} + \frac {\sqrt {c + d x} \left (A b^{2} d^{3} + 2 B a b d^{3} - 3 B b^{2} c d^{2} + C a^{2} d^{3} - 6 C a b c d^{2} + 6 C b^{2} c^{2} d - 3 D a^{2} c d^{2} + 12 D a b c^{2} d - 10 D b^{2} c^{3}\right )}{d^{5}} - \frac {\left (a d - b c\right ) \left (2 A b d^{3} + B a d^{3} - 3 B b c d^{2} - 2 C a c d^{2} + 4 C b c^{2} d + 3 D a c^{2} d - 5 D b c^{3}\right )}{d^{5} \sqrt {c + d x}} + \frac {\left (a d - b c\right )^{2} \left (- A d^{3} + B c d^{2} - C c^{2} d + D c^{3}\right )}{3 d^{5} \left (c + d x\right )^{\frac {3}{2}}}\right )}{d} & \text {for}\: d \neq 0 \\\frac {A a^{2} x + \frac {D b^{2} x^{6}}{6} + \frac {x^{5} \left (C b^{2} + 2 D a b\right )}{5} + \frac {x^{4} \left (B b^{2} + 2 C a b + D a^{2}\right )}{4} + \frac {x^{3} \left (A b^{2} + 2 B a b + C a^{2}\right )}{3} + \frac {x^{2} \cdot \left (2 A a b + B a^{2}\right )}{2}}{c^{\frac {5}{2}}} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((2*(D*b**2*(c + d*x)**(7/2)/(7*d**5) + (c + d*x)**(5/2)*(C*b**2*d + 2*D*a*b*d - 5*D*b**2*c)/(5*d**5)
+ (c + d*x)**(3/2)*(B*b**2*d**2 + 2*C*a*b*d**2 - 4*C*b**2*c*d + D*a**2*d**2 - 8*D*a*b*c*d + 10*D*b**2*c**2)/(
3*d**5) + sqrt(c + d*x)*(A*b**2*d**3 + 2*B*a*b*d**3 - 3*B*b**2*c*d**2 + C*a**2*d**3 - 6*C*a*b*c*d**2 + 6*C*b**
2*c**2*d - 3*D*a**2*c*d**2 + 12*D*a*b*c**2*d - 10*D*b**2*c**3)/d**5 - (a*d - b*c)*(2*A*b*d**3 + B*a*d**3 - 3*B
*b*c*d**2 - 2*C*a*c*d**2 + 4*C*b*c**2*d + 3*D*a*c**2*d - 5*D*b*c**3)/(d**5*sqrt(c + d*x)) + (a*d - b*c)**2*(-A
*d**3 + B*c*d**2 - C*c**2*d + D*c**3)/(3*d**5*(c + d*x)**(3/2)))/d, Ne(d, 0)), ((A*a**2*x + D*b**2*x**6/6 + x*
*5*(C*b**2 + 2*D*a*b)/5 + x**4*(B*b**2 + 2*C*a*b + D*a**2)/4 + x**3*(A*b**2 + 2*B*a*b + C*a**2)/3 + x**2*(2*A*
a*b + B*a**2)/2)/c**(5/2), True))

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 393, normalized size of antiderivative = 1.22 \[ \int \frac {(a+b x)^2 \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{5/2}} \, dx=\frac {2 \, {\left (\frac {15 \, {\left (d x + c\right )}^{\frac {7}{2}} D b^{2} - 21 \, {\left (5 \, D b^{2} c - {\left (2 \, D a b + C b^{2}\right )} d\right )} {\left (d x + c\right )}^{\frac {5}{2}} + 35 \, {\left (10 \, D b^{2} c^{2} - 4 \, {\left (2 \, D a b + C b^{2}\right )} c d + {\left (D a^{2} + 2 \, C a b + B b^{2}\right )} d^{2}\right )} {\left (d x + c\right )}^{\frac {3}{2}} - 105 \, {\left (10 \, D b^{2} c^{3} - 6 \, {\left (2 \, D a b + C b^{2}\right )} c^{2} d + 3 \, {\left (D a^{2} + 2 \, C a b + B b^{2}\right )} c d^{2} - {\left (C a^{2} + 2 \, B a b + A b^{2}\right )} d^{3}\right )} \sqrt {d x + c}}{d^{5}} + \frac {35 \, {\left (D b^{2} c^{5} - A a^{2} d^{5} - {\left (2 \, D a b + C b^{2}\right )} c^{4} d + {\left (D a^{2} + 2 \, C a b + B b^{2}\right )} c^{3} d^{2} - {\left (C a^{2} + 2 \, B a b + A b^{2}\right )} c^{2} d^{3} + {\left (B a^{2} + 2 \, A a b\right )} c d^{4} - 3 \, {\left (5 \, D b^{2} c^{4} - 4 \, {\left (2 \, D a b + C b^{2}\right )} c^{3} d + 3 \, {\left (D a^{2} + 2 \, C a b + B b^{2}\right )} c^{2} d^{2} - 2 \, {\left (C a^{2} + 2 \, B a b + A b^{2}\right )} c d^{3} + {\left (B a^{2} + 2 \, A a b\right )} d^{4}\right )} {\left (d x + c\right )}\right )}}{{\left (d x + c\right )}^{\frac {3}{2}} d^{5}}\right )}}{105 \, d} \]

[In]

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

[Out]

2/105*((15*(d*x + c)^(7/2)*D*b^2 - 21*(5*D*b^2*c - (2*D*a*b + C*b^2)*d)*(d*x + c)^(5/2) + 35*(10*D*b^2*c^2 - 4
*(2*D*a*b + C*b^2)*c*d + (D*a^2 + 2*C*a*b + B*b^2)*d^2)*(d*x + c)^(3/2) - 105*(10*D*b^2*c^3 - 6*(2*D*a*b + C*b
^2)*c^2*d + 3*(D*a^2 + 2*C*a*b + B*b^2)*c*d^2 - (C*a^2 + 2*B*a*b + A*b^2)*d^3)*sqrt(d*x + c))/d^5 + 35*(D*b^2*
c^5 - A*a^2*d^5 - (2*D*a*b + C*b^2)*c^4*d + (D*a^2 + 2*C*a*b + B*b^2)*c^3*d^2 - (C*a^2 + 2*B*a*b + A*b^2)*c^2*
d^3 + (B*a^2 + 2*A*a*b)*c*d^4 - 3*(5*D*b^2*c^4 - 4*(2*D*a*b + C*b^2)*c^3*d + 3*(D*a^2 + 2*C*a*b + B*b^2)*c^2*d
^2 - 2*(C*a^2 + 2*B*a*b + A*b^2)*c*d^3 + (B*a^2 + 2*A*a*b)*d^4)*(d*x + c))/((d*x + c)^(3/2)*d^5))/d

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 622 vs. \(2 (302) = 604\).

Time = 0.30 (sec) , antiderivative size = 622, normalized size of antiderivative = 1.93 \[ \int \frac {(a+b x)^2 \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{5/2}} \, dx=-\frac {2 \, {\left (15 \, {\left (d x + c\right )} D b^{2} c^{4} - D b^{2} c^{5} - 24 \, {\left (d x + c\right )} D a b c^{3} d - 12 \, {\left (d x + c\right )} C b^{2} c^{3} d + 2 \, D a b c^{4} d + C b^{2} c^{4} d + 9 \, {\left (d x + c\right )} D a^{2} c^{2} d^{2} + 18 \, {\left (d x + c\right )} C a b c^{2} d^{2} + 9 \, {\left (d x + c\right )} B b^{2} c^{2} d^{2} - D a^{2} c^{3} d^{2} - 2 \, C a b c^{3} d^{2} - B b^{2} c^{3} d^{2} - 6 \, {\left (d x + c\right )} C a^{2} c d^{3} - 12 \, {\left (d x + c\right )} B a b c d^{3} - 6 \, {\left (d x + c\right )} A b^{2} c d^{3} + C a^{2} c^{2} d^{3} + 2 \, B a b c^{2} d^{3} + A b^{2} c^{2} d^{3} + 3 \, {\left (d x + c\right )} B a^{2} d^{4} + 6 \, {\left (d x + c\right )} A a b d^{4} - B a^{2} c d^{4} - 2 \, A a b c d^{4} + A a^{2} d^{5}\right )}}{3 \, {\left (d x + c\right )}^{\frac {3}{2}} d^{6}} + \frac {2 \, {\left (15 \, {\left (d x + c\right )}^{\frac {7}{2}} D b^{2} d^{36} - 105 \, {\left (d x + c\right )}^{\frac {5}{2}} D b^{2} c d^{36} + 350 \, {\left (d x + c\right )}^{\frac {3}{2}} D b^{2} c^{2} d^{36} - 1050 \, \sqrt {d x + c} D b^{2} c^{3} d^{36} + 42 \, {\left (d x + c\right )}^{\frac {5}{2}} D a b d^{37} + 21 \, {\left (d x + c\right )}^{\frac {5}{2}} C b^{2} d^{37} - 280 \, {\left (d x + c\right )}^{\frac {3}{2}} D a b c d^{37} - 140 \, {\left (d x + c\right )}^{\frac {3}{2}} C b^{2} c d^{37} + 1260 \, \sqrt {d x + c} D a b c^{2} d^{37} + 630 \, \sqrt {d x + c} C b^{2} c^{2} d^{37} + 35 \, {\left (d x + c\right )}^{\frac {3}{2}} D a^{2} d^{38} + 70 \, {\left (d x + c\right )}^{\frac {3}{2}} C a b d^{38} + 35 \, {\left (d x + c\right )}^{\frac {3}{2}} B b^{2} d^{38} - 315 \, \sqrt {d x + c} D a^{2} c d^{38} - 630 \, \sqrt {d x + c} C a b c d^{38} - 315 \, \sqrt {d x + c} B b^{2} c d^{38} + 105 \, \sqrt {d x + c} C a^{2} d^{39} + 210 \, \sqrt {d x + c} B a b d^{39} + 105 \, \sqrt {d x + c} A b^{2} d^{39}\right )}}{105 \, d^{42}} \]

[In]

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

[Out]

-2/3*(15*(d*x + c)*D*b^2*c^4 - D*b^2*c^5 - 24*(d*x + c)*D*a*b*c^3*d - 12*(d*x + c)*C*b^2*c^3*d + 2*D*a*b*c^4*d
+ C*b^2*c^4*d + 9*(d*x + c)*D*a^2*c^2*d^2 + 18*(d*x + c)*C*a*b*c^2*d^2 + 9*(d*x + c)*B*b^2*c^2*d^2 - D*a^2*c^
3*d^2 - 2*C*a*b*c^3*d^2 - B*b^2*c^3*d^2 - 6*(d*x + c)*C*a^2*c*d^3 - 12*(d*x + c)*B*a*b*c*d^3 - 6*(d*x + c)*A*b
^2*c*d^3 + C*a^2*c^2*d^3 + 2*B*a*b*c^2*d^3 + A*b^2*c^2*d^3 + 3*(d*x + c)*B*a^2*d^4 + 6*(d*x + c)*A*a*b*d^4 - B
*a^2*c*d^4 - 2*A*a*b*c*d^4 + A*a^2*d^5)/((d*x + c)^(3/2)*d^6) + 2/105*(15*(d*x + c)^(7/2)*D*b^2*d^36 - 105*(d*
x + c)^(5/2)*D*b^2*c*d^36 + 350*(d*x + c)^(3/2)*D*b^2*c^2*d^36 - 1050*sqrt(d*x + c)*D*b^2*c^3*d^36 + 42*(d*x +
c)^(5/2)*D*a*b*d^37 + 21*(d*x + c)^(5/2)*C*b^2*d^37 - 280*(d*x + c)^(3/2)*D*a*b*c*d^37 - 140*(d*x + c)^(3/2)*
C*b^2*c*d^37 + 1260*sqrt(d*x + c)*D*a*b*c^2*d^37 + 630*sqrt(d*x + c)*C*b^2*c^2*d^37 + 35*(d*x + c)^(3/2)*D*a^2
*d^38 + 70*(d*x + c)^(3/2)*C*a*b*d^38 + 35*(d*x + c)^(3/2)*B*b^2*d^38 - 315*sqrt(d*x + c)*D*a^2*c*d^38 - 630*s
qrt(d*x + c)*C*a*b*c*d^38 - 315*sqrt(d*x + c)*B*b^2*c*d^38 + 105*sqrt(d*x + c)*C*a^2*d^39 + 210*sqrt(d*x + c)*
B*a*b*d^39 + 105*sqrt(d*x + c)*A*b^2*d^39)/d^42

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b x)^2 \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{5/2}} \, dx=\int \frac {{\left (a+b\,x\right )}^2\,\left (A+B\,x+C\,x^2+x^3\,D\right )}{{\left (c+d\,x\right )}^{5/2}} \,d x \]

[In]

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

[Out]

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