Optimal. Leaf size=322 \[ \frac {2 \sqrt {c+d x} \left (a^2 d^2 (C d-3 c D)-2 a b d \left (-B d^2-6 c^2 D+3 c C d\right )+b^2 \left (A d^3-3 B c d^2-10 c^3 D+6 c^2 C d\right )\right )}{d^6}+\frac {2 (c+d x)^{3/2} \left (a^2 d^2 D+2 a b d (C d-4 c D)-\left (b^2 \left (-B d^2-10 c^2 D+4 c C d\right )\right )\right )}{3 d^6}-\frac {2 (b c-a d) \left (a d \left (-B d^2-3 c^2 D+2 c C d\right )-b \left (2 A d^3-3 B c d^2-5 c^3 D+4 c^2 C d\right )\right )}{d^6 \sqrt {c+d x}}-\frac {2 (b c-a d)^2 \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{3 d^6 (c+d x)^{3/2}}+\frac {2 b (c+d x)^{5/2} (2 a d D-5 b c D+b C d)}{5 d^6}+\frac {2 b^2 D (c+d x)^{7/2}}{7 d^6} \]
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Rubi [A] time = 0.24, antiderivative size = 322, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.031, Rules used = {1620} \[ \frac {2 \sqrt {c+d x} \left (a^2 d^2 (C d-3 c D)-2 a b d \left (-B d^2-6 c^2 D+3 c C d\right )+b^2 \left (A d^3-3 B c d^2+6 c^2 C d-10 c^3 D\right )\right )}{d^6}+\frac {2 (c+d x)^{3/2} \left (a^2 d^2 D+2 a b d (C d-4 c D)+b^2 \left (-\left (-B d^2-10 c^2 D+4 c C d\right )\right )\right )}{3 d^6}-\frac {2 (b c-a d) \left (a d \left (-B d^2-3 c^2 D+2 c C d\right )-b \left (2 A d^3-3 B c d^2+4 c^2 C d-5 c^3 D\right )\right )}{d^6 \sqrt {c+d x}}-\frac {2 (b c-a d)^2 \left (A d^3-B c d^2+c^2 C d+c^3 (-D)\right )}{3 d^6 (c+d x)^{3/2}}+\frac {2 b (c+d x)^{5/2} (2 a d D-5 b c D+b C d)}{5 d^6}+\frac {2 b^2 D (c+d x)^{7/2}}{7 d^6} \]
Antiderivative was successfully verified.
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Rule 1620
Rubi steps
\begin {align*} \int \frac {(a+b x)^2 \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{5/2}} \, dx &=\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*}
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Mathematica [A] time = 0.70, size = 287, normalized size = 0.89 \[ \frac {2 \left (105 (c+d x)^2 \left (a^2 d^2 (C d-3 c D)+2 a b d \left (B d^2+6 c^2 D-3 c C d\right )+b^2 \left (A d^3-3 B c d^2-10 c^3 D+6 c^2 C d\right )\right )+35 (c+d x)^3 \left (a^2 d^2 D+2 a b d (C d-4 c D)+b^2 \left (B d^2+10 c^2 D-4 c C d\right )\right )-105 (c+d x) (b c-a d) \left (b \left (-2 A d^3+3 B c d^2+5 c^3 D-4 c^2 C d\right )-a d \left (B d^2+3 c^2 D-2 c C d\right )\right )+35 (b c-a d)^2 \left (-A d^3+B c d^2+c^3 D-c^2 C d\right )+21 b (c+d x)^4 (2 a d D-5 b c D+b C d)+15 b^2 D (c+d x)^5\right )}{105 d^6 (c+d x)^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.78, size = 431, normalized size = 1.34 \[ \frac {2 \, {\left (15 \, D b^{2} d^{5} x^{5} - 1280 \, D b^{2} c^{5} - 35 \, A a^{2} d^{5} + 280 \, {\left (C a^{2} + 2 \, B a b + A b^{2}\right )} c^{2} d^{3} - 70 \, {\left (B a^{2} + 2 \, A a b\right )} c d^{4} - 3 \, {\left (10 \, D b^{2} c d^{4} - 7 \, {\left (2 \, D a b + C b^{2}\right )} d^{5}\right )} x^{4} + {\left (80 \, D b^{2} c^{2} d^{3} + 35 \, {\left (D a^{2} + 2 \, C a b + B b^{2}\right )} d^{5} - 56 \, {\left (2 \, D a b c + C b^{2} c\right )} d^{4}\right )} x^{3} - 560 \, {\left (D a^{2} c^{3} + {\left (2 \, C a b + B b^{2}\right )} c^{3}\right )} d^{2} - 3 \, {\left (160 \, D b^{2} c^{3} d^{2} - 35 \, {\left (C a^{2} + 2 \, B a b + A b^{2}\right )} d^{5} + 70 \, {\left (D a^{2} c + {\left (2 \, C a b + B b^{2}\right )} c\right )} d^{4} - 112 \, {\left (2 \, D a b c^{2} + C b^{2} c^{2}\right )} d^{3}\right )} x^{2} + 896 \, {\left (2 \, D a b c^{4} + C b^{2} c^{4}\right )} d - 3 \, {\left (640 \, D b^{2} c^{4} d - 140 \, {\left (C a^{2} + 2 \, B a b + A b^{2}\right )} c d^{4} + 35 \, {\left (B a^{2} + 2 \, A a b\right )} d^{5} + 280 \, {\left (D a^{2} c^{2} + {\left (2 \, C a b + B b^{2}\right )} c^{2}\right )} d^{3} - 448 \, {\left (2 \, D a b c^{3} + C b^{2} c^{3}\right )} d^{2}\right )} x\right )} \sqrt {d x + c}}{105 \, {\left (d^{8} x^{2} + 2 \, c d^{7} x + c^{2} d^{6}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.39, size = 622, normalized size = 1.93 \[ -\frac {2 \, {\left (15 \, {\left (d x + c\right )} D b^{2} c^{4} - D b^{2} c^{5} - 24 \, {\left (d x + c\right )} D a b c^{3} d - 12 \, {\left (d x + c\right )} C b^{2} c^{3} d + 2 \, D a b c^{4} d + C b^{2} c^{4} d + 9 \, {\left (d x + c\right )} D a^{2} c^{2} d^{2} + 18 \, {\left (d x + c\right )} C a b c^{2} d^{2} + 9 \, {\left (d x + c\right )} B b^{2} c^{2} d^{2} - D a^{2} c^{3} d^{2} - 2 \, C a b c^{3} d^{2} - B b^{2} c^{3} d^{2} - 6 \, {\left (d x + c\right )} C a^{2} c d^{3} - 12 \, {\left (d x + c\right )} B a b c d^{3} - 6 \, {\left (d x + c\right )} A b^{2} c d^{3} + C a^{2} c^{2} d^{3} + 2 \, B a b c^{2} d^{3} + A b^{2} c^{2} d^{3} + 3 \, {\left (d x + c\right )} B a^{2} d^{4} + 6 \, {\left (d x + c\right )} A a b d^{4} - B a^{2} c d^{4} - 2 \, A a b c d^{4} + A a^{2} d^{5}\right )}}{3 \, {\left (d x + c\right )}^{\frac {3}{2}} d^{6}} + \frac {2 \, {\left (15 \, {\left (d x + c\right )}^{\frac {7}{2}} D b^{2} d^{36} - 105 \, {\left (d x + c\right )}^{\frac {5}{2}} D b^{2} c d^{36} + 350 \, {\left (d x + c\right )}^{\frac {3}{2}} D b^{2} c^{2} d^{36} - 1050 \, \sqrt {d x + c} D b^{2} c^{3} d^{36} + 42 \, {\left (d x + c\right )}^{\frac {5}{2}} D a b d^{37} + 21 \, {\left (d x + c\right )}^{\frac {5}{2}} C b^{2} d^{37} - 280 \, {\left (d x + c\right )}^{\frac {3}{2}} D a b c d^{37} - 140 \, {\left (d x + c\right )}^{\frac {3}{2}} C b^{2} c d^{37} + 1260 \, \sqrt {d x + c} D a b c^{2} d^{37} + 630 \, \sqrt {d x + c} C b^{2} c^{2} d^{37} + 35 \, {\left (d x + c\right )}^{\frac {3}{2}} D a^{2} d^{38} + 70 \, {\left (d x + c\right )}^{\frac {3}{2}} C a b d^{38} + 35 \, {\left (d x + c\right )}^{\frac {3}{2}} B b^{2} d^{38} - 315 \, \sqrt {d x + c} D a^{2} c d^{38} - 630 \, \sqrt {d x + c} C a b c d^{38} - 315 \, \sqrt {d x + c} B b^{2} c d^{38} + 105 \, \sqrt {d x + c} C a^{2} d^{39} + 210 \, \sqrt {d x + c} B a b d^{39} + 105 \, \sqrt {d x + c} A b^{2} d^{39}\right )}}{105 \, d^{42}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 505, normalized size = 1.57 \[ -\frac {2 \left (-15 b^{2} D x^{5} d^{5}-21 C \,b^{2} d^{5} x^{4}-42 D a b \,d^{5} x^{4}+30 D b^{2} c \,d^{4} x^{4}-35 B \,b^{2} d^{5} x^{3}-70 C a b \,d^{5} x^{3}+56 C \,b^{2} c \,d^{4} x^{3}-35 D a^{2} d^{5} x^{3}+112 D a b c \,d^{4} x^{3}-80 D b^{2} c^{2} d^{3} x^{3}-105 A \,b^{2} d^{5} x^{2}-210 B a b \,d^{5} x^{2}+210 B \,b^{2} c \,d^{4} x^{2}-105 C \,a^{2} d^{5} x^{2}+420 C a b c \,d^{4} x^{2}-336 C \,b^{2} c^{2} d^{3} x^{2}+210 D a^{2} c \,d^{4} x^{2}-672 D a b \,c^{2} d^{3} x^{2}+480 D b^{2} c^{3} d^{2} x^{2}+210 A a b \,d^{5} x -420 A \,b^{2} c \,d^{4} x +105 B \,a^{2} d^{5} x -840 B a b c \,d^{4} x +840 B \,b^{2} c^{2} d^{3} x -420 C \,a^{2} c \,d^{4} x +1680 C a b \,c^{2} d^{3} x -1344 C \,b^{2} c^{3} d^{2} x +840 D a^{2} c^{2} d^{3} x -2688 D a b \,c^{3} d^{2} x +1920 D b^{2} c^{4} d x +35 a^{2} A \,d^{5}+140 A a b c \,d^{4}-280 A \,b^{2} c^{2} d^{3}+70 B \,a^{2} c \,d^{4}-560 B a b \,c^{2} d^{3}+560 B \,b^{2} c^{3} d^{2}-280 C \,a^{2} c^{2} d^{3}+1120 C a b \,c^{3} d^{2}-896 C \,b^{2} c^{4} d +560 D a^{2} c^{3} d^{2}-1792 D a b \,c^{4} d +1280 D b^{2} c^{5}\right )}{105 \left (d x +c \right )^{\frac {3}{2}} d^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.47, size = 393, normalized size = 1.22 \[ \frac {2 \, {\left (\frac {15 \, {\left (d x + c\right )}^{\frac {7}{2}} D b^{2} - 21 \, {\left (5 \, D b^{2} c - {\left (2 \, D a b + C b^{2}\right )} d\right )} {\left (d x + c\right )}^{\frac {5}{2}} + 35 \, {\left (10 \, D b^{2} c^{2} - 4 \, {\left (2 \, D a b + C b^{2}\right )} c d + {\left (D a^{2} + 2 \, C a b + B b^{2}\right )} d^{2}\right )} {\left (d x + c\right )}^{\frac {3}{2}} - 105 \, {\left (10 \, D b^{2} c^{3} - 6 \, {\left (2 \, D a b + C b^{2}\right )} c^{2} d + 3 \, {\left (D a^{2} + 2 \, C a b + B b^{2}\right )} c d^{2} - {\left (C a^{2} + 2 \, B a b + A b^{2}\right )} d^{3}\right )} \sqrt {d x + c}}{d^{5}} + \frac {35 \, {\left (D b^{2} c^{5} - A a^{2} d^{5} - {\left (2 \, D a b + C b^{2}\right )} c^{4} d + {\left (D a^{2} + 2 \, C a b + B b^{2}\right )} c^{3} d^{2} - {\left (C a^{2} + 2 \, B a b + A b^{2}\right )} c^{2} d^{3} + {\left (B a^{2} + 2 \, A a b\right )} c d^{4} - 3 \, {\left (5 \, D b^{2} c^{4} - 4 \, {\left (2 \, D a b + C b^{2}\right )} c^{3} d + 3 \, {\left (D a^{2} + 2 \, C a b + B b^{2}\right )} c^{2} d^{2} - 2 \, {\left (C a^{2} + 2 \, B a b + A b^{2}\right )} c d^{3} + {\left (B a^{2} + 2 \, A a b\right )} d^{4}\right )} {\left (d x + c\right )}\right )}}{{\left (d x + c\right )}^{\frac {3}{2}} d^{5}}\right )}}{105 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a+b\,x\right )}^2\,\left (A+B\,x+C\,x^2+x^3\,D\right )}{{\left (c+d\,x\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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