Optimal. Leaf size=336 \[ \frac {3 a b^2 B d^3-3 a^2 b C d^3+3 a^3 d^3 D-b^3 \left (2 c^2 C d-2 B c d^2+5 A d^3-2 c^3 D\right )}{3 b^3 d^2 (b c-a d)^2 (c+d x)^{3/2}}-\frac {A-\frac {a \left (b^2 B-a b C+a^2 D\right )}{b^3}}{(b c-a d) (a+b x) (c+d x)^{3/2}}-\frac {a^2 b C d^3-a^3 d^3 D+a b^2 d \left (4 c C d-3 B d^2-6 c^2 D\right )-b^3 \left (2 B c d^2-5 A d^3-2 c^3 D\right )}{b^2 d^2 (b c-a d)^3 \sqrt {c+d x}}-\frac {\left (b^3 (2 B c-5 A d)-a b^2 (4 c C-3 B d)-a^3 d D-a^2 b (C d-6 c D)\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{b^{3/2} (b c-a d)^{7/2}} \]
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Rubi [A]
time = 0.53, antiderivative size = 336, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {1635, 911,
1275, 214} \begin {gather*} -\frac {A-\frac {a \left (a^2 D-a b C+b^2 B\right )}{b^3}}{(a+b x) (c+d x)^{3/2} (b c-a d)}-\frac {-a^3 d^3 D+a^2 b C d^3+a b^2 d \left (-3 B d^2-6 c^2 D+4 c C d\right )-\left (b^3 \left (-5 A d^3+2 B c d^2-2 c^3 D\right )\right )}{b^2 d^2 \sqrt {c+d x} (b c-a d)^3}+\frac {3 a^3 d^3 D-3 a^2 b C d^3+3 a b^2 B d^3-\left (b^3 \left (5 A d^3-2 B c d^2-2 c^3 D+2 c^2 C d\right )\right )}{3 b^3 d^2 (c+d x)^{3/2} (b c-a d)^2}-\frac {\tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right ) \left (a^3 (-d) D-a^2 b (C d-6 c D)-a b^2 (4 c C-3 B d)+b^3 (2 B c-5 A d)\right )}{b^{3/2} (b c-a d)^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 214
Rule 911
Rule 1275
Rule 1635
Rubi steps
\begin {align*} \int \frac {A+B x+C x^2+D x^3}{(a+b x)^2 (c+d x)^{5/2}} \, dx &=-\frac {A-\frac {a \left (b^2 B-a b C+a^2 D\right )}{b^3}}{(b c-a d) (a+b x) (c+d x)^{3/2}}+\frac {\int \frac {-\frac {b^3 (2 B c-5 A d)-a b^2 (2 c C-3 B d)+3 a^3 d D-a^2 b (3 C d-2 c D)}{2 b^3}-\frac {(b c-a d) (b C-a D) x}{b^2}-\left (c-\frac {a d}{b}\right ) D x^2}{(a+b x) (c+d x)^{5/2}} \, dx}{-b c+a d}\\ &=-\frac {A-\frac {a \left (b^2 B-a b C+a^2 D\right )}{b^3}}{(b c-a d) (a+b x) (c+d x)^{3/2}}-\frac {2 \text {Subst}\left (\int \frac {\frac {-c^2 \left (c-\frac {a d}{b}\right ) D+\frac {c d (b c-a d) (b C-a D)}{b^2}-\frac {d^2 \left (b^3 (2 B c-5 A d)-a b^2 (2 c C-3 B d)+3 a^3 d D-a^2 b (3 C d-2 c D)\right )}{2 b^3}}{d^2}-\frac {\left (-2 c \left (c-\frac {a d}{b}\right ) D+\frac {d (b c-a d) (b C-a D)}{b^2}\right ) x^2}{d^2}-\frac {\left (c-\frac {a d}{b}\right ) D x^4}{d^2}}{x^4 \left (\frac {-b c+a d}{d}+\frac {b x^2}{d}\right )} \, dx,x,\sqrt {c+d x}\right )}{d (b c-a d)}\\ &=-\frac {A-\frac {a \left (b^2 B-a b C+a^2 D\right )}{b^3}}{(b c-a d) (a+b x) (c+d x)^{3/2}}-\frac {2 \text {Subst}\left (\int \left (\frac {3 a b^2 B d^3-3 a^2 b C d^3+3 a^3 d^3 D-b^3 \left (2 c^2 C d-2 B c d^2+5 A d^3-2 c^3 D\right )}{2 b^3 d (b c-a d) x^4}+\frac {-a^2 b C d^3+a^3 d^3 D-a b^2 d \left (4 c C d-3 B d^2-6 c^2 D\right )+b^3 \left (2 B c d^2-5 A d^3-2 c^3 D\right )}{2 b^2 d (b c-a d)^2 x^2}+\frac {d \left (b^3 (2 B c-5 A d)-a b^2 (4 c C-3 B d)-a^3 d D-a^2 b (C d-6 c D)\right )}{2 b (b c-a d)^2 \left (b c-a d-b x^2\right )}\right ) \, dx,x,\sqrt {c+d x}\right )}{d (b c-a d)}\\ &=\frac {3 a b^2 B d^3-3 a^2 b C d^3+3 a^3 d^3 D-b^3 \left (2 c^2 C d-2 B c d^2+5 A d^3-2 c^3 D\right )}{3 b^3 d^2 (b c-a d)^2 (c+d x)^{3/2}}-\frac {A-\frac {a \left (b^2 B-a b C+a^2 D\right )}{b^3}}{(b c-a d) (a+b x) (c+d x)^{3/2}}-\frac {a^2 b C d^3-a^3 d^3 D+a b^2 d \left (4 c C d-3 B d^2-6 c^2 D\right )-b^3 \left (2 B c d^2-5 A d^3-2 c^3 D\right )}{b^2 d^2 (b c-a d)^3 \sqrt {c+d x}}-\frac {\left (b^3 (2 B c-5 A d)-a b^2 (4 c C-3 B d)-a^3 d D-a^2 b (C d-6 c D)\right ) \text {Subst}\left (\int \frac {1}{b c-a d-b x^2} \, dx,x,\sqrt {c+d x}\right )}{b (b c-a d)^3}\\ &=\frac {3 a b^2 B d^3-3 a^2 b C d^3+3 a^3 d^3 D-b^3 \left (2 c^2 C d-2 B c d^2+5 A d^3-2 c^3 D\right )}{3 b^3 d^2 (b c-a d)^2 (c+d x)^{3/2}}-\frac {A-\frac {a \left (b^2 B-a b C+a^2 D\right )}{b^3}}{(b c-a d) (a+b x) (c+d x)^{3/2}}-\frac {a^2 b C d^3-a^3 d^3 D+a b^2 d \left (4 c C d-3 B d^2-6 c^2 D\right )-b^3 \left (2 B c d^2-5 A d^3-2 c^3 D\right )}{b^2 d^2 (b c-a d)^3 \sqrt {c+d x}}-\frac {\left (b^3 (2 B c-5 A d)-a b^2 (4 c C-3 B d)-a^3 d D-a^2 b (C d-6 c D)\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{b^{3/2} (b c-a d)^{7/2}}\\ \end {align*}
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Mathematica [A]
time = 1.16, size = 354, normalized size = 1.05 \begin {gather*} \frac {-3 a^3 d^2 D (c+d x)^2-a^2 b d \left (16 c^3 D+2 c d^2 (2 B-9 C x)+c^2 (-13 C d+18 d D x)+d^3 \left (2 A+6 B x-3 C x^2\right )\right )+a b^2 \left (4 c^4 D+d^4 x (10 A-9 B x)+2 c^3 d (C-5 D x)+2 c d^3 \left (7 A-8 B x+6 C x^2\right )+c^2 d^2 (-11 B+2 x (5 C-9 D x))\right )+b^3 \left (A d^2 \left (3 c^2+20 c d x+15 d^2 x^2\right )+2 c x \left (-4 B c d^2+2 c^3 D-3 B d^3 x+c^2 d (C+3 D x)\right )\right )}{3 b d^2 (-b c+a d)^3 (a+b x) (c+d x)^{3/2}}-\frac {\left (b^3 (2 B c-5 A d)+a b^2 (-4 c C+3 B d)-a^3 d D+a^2 b (-C d+6 c D)\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {-b c+a d}}\right )}{b^{3/2} (-b c+a d)^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 275, normalized size = 0.82
method | result | size |
derivativedivides | \(\frac {-\frac {2 \left (A \,d^{3}-B c \,d^{2}+C \,c^{2} d -D c^{3}\right )}{3 \left (a d -b c \right )^{2} \left (d x +c \right )^{\frac {3}{2}}}-\frac {2 \left (-2 A b \,d^{3}+B a \,d^{3}+B b c \,d^{2}-2 C a c \,d^{2}+3 D a \,c^{2} d -D b \,c^{3}\right )}{\left (a d -b c \right )^{3} \sqrt {d x +c}}+\frac {2 d^{2} \left (\frac {d \left (b^{3} A -a \,b^{2} B +C \,a^{2} b -D a^{3}\right ) \sqrt {d x +c}}{2 b \left (b \left (d x +c \right )+a d -b c \right )}+\frac {\left (5 A \,b^{3} d -3 B a \,b^{2} d -2 B \,b^{3} c +C \,a^{2} b d +4 C a \,b^{2} c +a^{3} d D-6 D a^{2} b c \right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{2 b \sqrt {\left (a d -b c \right ) b}}\right )}{\left (a d -b c \right )^{3}}}{d^{2}}\) | \(275\) |
default | \(\frac {-\frac {2 \left (A \,d^{3}-B c \,d^{2}+C \,c^{2} d -D c^{3}\right )}{3 \left (a d -b c \right )^{2} \left (d x +c \right )^{\frac {3}{2}}}-\frac {2 \left (-2 A b \,d^{3}+B a \,d^{3}+B b c \,d^{2}-2 C a c \,d^{2}+3 D a \,c^{2} d -D b \,c^{3}\right )}{\left (a d -b c \right )^{3} \sqrt {d x +c}}+\frac {2 d^{2} \left (\frac {d \left (b^{3} A -a \,b^{2} B +C \,a^{2} b -D a^{3}\right ) \sqrt {d x +c}}{2 b \left (b \left (d x +c \right )+a d -b c \right )}+\frac {\left (5 A \,b^{3} d -3 B a \,b^{2} d -2 B \,b^{3} c +C \,a^{2} b d +4 C a \,b^{2} c +a^{3} d D-6 D a^{2} b c \right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{2 b \sqrt {\left (a d -b c \right ) b}}\right )}{\left (a d -b c \right )^{3}}}{d^{2}}\) | \(275\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1215 vs.
\(2 (315) = 630\).
time = 1.22, size = 2444, normalized size = 7.27 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.96, size = 439, normalized size = 1.31 \begin {gather*} \frac {{\left (6 \, D a^{2} b c - 4 \, C a b^{2} c + 2 \, B b^{3} c - D a^{3} d - C a^{2} b d + 3 \, B a b^{2} d - 5 \, A b^{3} d\right )} \arctan \left (\frac {\sqrt {d x + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{{\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} \sqrt {-b^{2} c + a b d}} + \frac {\sqrt {d x + c} D a^{3} d - \sqrt {d x + c} C a^{2} b d + \sqrt {d x + c} B a b^{2} d - \sqrt {d x + c} A b^{3} d}{{\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} {\left ({\left (d x + c\right )} b - b c + a d\right )}} - \frac {2 \, {\left (3 \, {\left (d x + c\right )} D b c^{3} - D b c^{4} - 9 \, {\left (d x + c\right )} D a c^{2} d + D a c^{3} d + C b c^{3} d + 6 \, {\left (d x + c\right )} C a c d^{2} - 3 \, {\left (d x + c\right )} B b c d^{2} - C a c^{2} d^{2} - B b c^{2} d^{2} - 3 \, {\left (d x + c\right )} B a d^{3} + 6 \, {\left (d x + c\right )} A b d^{3} + B a c d^{3} + A b c d^{3} - A a d^{4}\right )}}{3 \, {\left (b^{3} c^{3} d^{2} - 3 \, a b^{2} c^{2} d^{3} + 3 \, a^{2} b c d^{4} - a^{3} d^{5}\right )} {\left (d x + c\right )}^{\frac {3}{2}}} \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 {A+B\,x+C\,x^2+x^3\,D}{{\left (a+b\,x\right )}^2\,{\left (c+d\,x\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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