Optimal. Leaf size=210 \[ \frac {2 (b c-a d) \left (c^2 C d-B c d^2+A d^3-c^3 D\right )}{d^5 \sqrt {c+d x}}-\frac {2 \left (a d \left (2 c C d-B d^2-3 c^2 D\right )-b \left (3 c^2 C d-2 B c d^2+A d^3-4 c^3 D\right )\right ) \sqrt {c+d x}}{d^5}+\frac {2 \left (a d (C d-3 c D)-b \left (3 c C d-B d^2-6 c^2 D\right )\right ) (c+d x)^{3/2}}{3 d^5}+\frac {2 (b C d-4 b c D+a d D) (c+d x)^{5/2}}{5 d^5}+\frac {2 b D (c+d x)^{7/2}}{7 d^5} \]
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Rubi [A]
time = 0.11, antiderivative size = 210, 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 = {1634}
\begin {gather*} -\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^2-4 c^3 D+3 c^2 C d\right )\right )}{d^5}+\frac {2 (b c-a d) \left (A d^3-B c d^2+c^3 (-D)+c^2 C 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} \end {gather*}
Antiderivative was successfully verified.
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Rule 1634
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*}
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Mathematica [A]
time = 0.20, size = 188, normalized size = 0.90 \begin {gather*} \frac {14 a d \left (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 \left (-15 A+x \left (15 B+5 C x+3 D x^2\right )\right )\right )+b \left (-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)))\right )}{105 d^5 \sqrt {c+d x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 294, normalized size = 1.40
method | result | size |
gosper | \(-\frac {2 \left (-15 D b \,x^{4} d^{4}-21 C b \,d^{4} x^{3}-21 D a \,d^{4} x^{3}+24 D b c \,d^{3} x^{3}-35 B b \,d^{4} x^{2}-35 C a \,d^{4} x^{2}+42 C b c \,d^{3} x^{2}+42 D a c \,d^{3} x^{2}-48 D b \,c^{2} d^{2} x^{2}-105 A b \,d^{4} x -105 B a \,d^{4} x +140 B b c \,d^{3} x +140 C a c \,d^{3} x -168 C b \,c^{2} d^{2} x -168 D a \,c^{2} d^{2} x +192 D b \,c^{3} d x +105 A a \,d^{4}-210 A b c \,d^{3}-210 B a c \,d^{3}+280 B b \,c^{2} d^{2}+280 C a \,c^{2} d^{2}-336 C b \,c^{3} d -336 D a \,c^{3} d +384 D b \,c^{4}\right )}{105 \sqrt {d x +c}\, d^{5}}\) | \(241\) |
trager | \(-\frac {2 \left (-15 D b \,x^{4} d^{4}-21 C b \,d^{4} x^{3}-21 D a \,d^{4} x^{3}+24 D b c \,d^{3} x^{3}-35 B b \,d^{4} x^{2}-35 C a \,d^{4} x^{2}+42 C b c \,d^{3} x^{2}+42 D a c \,d^{3} x^{2}-48 D b \,c^{2} d^{2} x^{2}-105 A b \,d^{4} x -105 B a \,d^{4} x +140 B b c \,d^{3} x +140 C a c \,d^{3} x -168 C b \,c^{2} d^{2} x -168 D a \,c^{2} d^{2} x +192 D b \,c^{3} d x +105 A a \,d^{4}-210 A b c \,d^{3}-210 B a c \,d^{3}+280 B b \,c^{2} d^{2}+280 C a \,c^{2} d^{2}-336 C b \,c^{3} d -336 D a \,c^{3} d +384 D b \,c^{4}\right )}{105 \sqrt {d x +c}\, d^{5}}\) | \(241\) |
derivativedivides | \(\frac {\frac {2 D b \left (d x +c \right )^{\frac {7}{2}}}{7}+\frac {2 C b d \left (d x +c \right )^{\frac {5}{2}}}{5}+\frac {2 D a d \left (d x +c \right )^{\frac {5}{2}}}{5}-\frac {8 D b c \left (d x +c \right )^{\frac {5}{2}}}{5}+\frac {2 B b \,d^{2} \left (d x +c \right )^{\frac {3}{2}}}{3}+\frac {2 C a \,d^{2} \left (d x +c \right )^{\frac {3}{2}}}{3}-2 C b c d \left (d x +c \right )^{\frac {3}{2}}-2 D a c d \left (d x +c \right )^{\frac {3}{2}}+4 D b \,c^{2} \left (d x +c \right )^{\frac {3}{2}}+2 A \,d^{3} b \sqrt {d x +c}+2 B a \,d^{3} \sqrt {d x +c}-4 B b c \,d^{2} \sqrt {d x +c}-4 C a c \,d^{2} \sqrt {d x +c}+6 C \,c^{2} d b \sqrt {d x +c}+6 D a \,c^{2} d \sqrt {d x +c}-8 D c^{3} b \sqrt {d x +c}-\frac {2 \left (A a \,d^{4}-A b c \,d^{3}-B a c \,d^{3}+B b \,c^{2} d^{2}+C a \,c^{2} d^{2}-C b \,c^{3} d -D a \,c^{3} d +D b \,c^{4}\right )}{\sqrt {d x +c}}}{d^{5}}\) | \(294\) |
default | \(\frac {\frac {2 D b \left (d x +c \right )^{\frac {7}{2}}}{7}+\frac {2 C b d \left (d x +c \right )^{\frac {5}{2}}}{5}+\frac {2 D a d \left (d x +c \right )^{\frac {5}{2}}}{5}-\frac {8 D b c \left (d x +c \right )^{\frac {5}{2}}}{5}+\frac {2 B b \,d^{2} \left (d x +c \right )^{\frac {3}{2}}}{3}+\frac {2 C a \,d^{2} \left (d x +c \right )^{\frac {3}{2}}}{3}-2 C b c d \left (d x +c \right )^{\frac {3}{2}}-2 D a c d \left (d x +c \right )^{\frac {3}{2}}+4 D b \,c^{2} \left (d x +c \right )^{\frac {3}{2}}+2 A \,d^{3} b \sqrt {d x +c}+2 B a \,d^{3} \sqrt {d x +c}-4 B b c \,d^{2} \sqrt {d x +c}-4 C a c \,d^{2} \sqrt {d x +c}+6 C \,c^{2} d b \sqrt {d x +c}+6 D a \,c^{2} d \sqrt {d x +c}-8 D c^{3} b \sqrt {d x +c}-\frac {2 \left (A a \,d^{4}-A b c \,d^{3}-B a c \,d^{3}+B b \,c^{2} d^{2}+C a \,c^{2} d^{2}-C b \,c^{3} d -D a \,c^{3} d +D b \,c^{4}\right )}{\sqrt {d x +c}}}{d^{5}}\) | \(294\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.33, size = 206, normalized size = 0.98 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.69, size = 213, normalized size = 1.01 \begin {gather*} \frac {2 \, {\left (15 \, D b d^{4} x^{4} - 384 \, D b c^{4} - 105 \, A a d^{4} - 280 \, {\left (C a + B b\right )} c^{2} d^{2} + 210 \, {\left (B a + A b\right )} c d^{3} - 3 \, {\left (8 \, D b c d^{3} - 7 \, {\left (D a + C b\right )} d^{4}\right )} x^{3} + {\left (48 \, D b c^{2} d^{2} + 35 \, {\left (C a + B b\right )} d^{4} - 42 \, {\left (D a c + C b c\right )} d^{3}\right )} x^{2} + 336 \, {\left (D a c^{3} + C b c^{3}\right )} d - {\left (192 \, D b c^{3} d + 140 \, {\left (C a + B b\right )} c d^{3} - 105 \, {\left (B a + A b\right )} d^{4} - 168 \, {\left (D a c^{2} + C b c^{2}\right )} d^{2}\right )} x\right )} \sqrt {d x + c}}{105 \, {\left (d^{6} x + c d^{5}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 21.15, size = 230, normalized size = 1.10 \begin {gather*} \frac {2 D b \left (c + d x\right )^{\frac {7}{2}}}{7 d^{5}} + \frac {\left (c + d x\right )^{\frac {5}{2}} \cdot \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}} \cdot \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.57, size = 323, normalized size = 1.54 \begin {gather*} -\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 {gather*}
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 \begin {gather*} \int \frac {\left (a+b\,x\right )\,\left (A+B\,x+C\,x^2+x^3\,D\right )}{{\left (c+d\,x\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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