Optimal. Leaf size=437 \[ -\frac {4 \left (9 a B e^2+4 c d (8 B d-5 A e)+c e (8 B d-5 A e) x\right ) \sqrt {a+c x^2}}{15 e^4 \sqrt {d+e x}}+\frac {2 (8 B d-5 A e+3 B e x) \left (a+c x^2\right )^{3/2}}{15 e^2 (d+e x)^{3/2}}-\frac {8 \sqrt {-a} \sqrt {c} \left (9 a B e^2+4 c d (8 B d-5 A e)\right ) \sqrt {d+e x} \sqrt {1+\frac {c x^2}{a}} E\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{15 e^5 \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {a+c x^2}}+\frac {8 \sqrt {-a} \sqrt {c} \left (32 B c d^3-20 A c d^2 e+17 a B d e^2-5 a A e^3\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {1+\frac {c x^2}{a}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{15 e^5 \sqrt {d+e x} \sqrt {a+c x^2}} \]
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Rubi [A]
time = 0.30, antiderivative size = 437, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {827, 858, 733,
435, 430} \begin {gather*} \frac {8 \sqrt {-a} \sqrt {c} \sqrt {\frac {c x^2}{a}+1} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}} \left (-5 a A e^3+17 a B d e^2-20 A c d^2 e+32 B c d^3\right ) F\left (\text {ArcSin}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{15 e^5 \sqrt {a+c x^2} \sqrt {d+e x}}-\frac {8 \sqrt {-a} \sqrt {c} \sqrt {\frac {c x^2}{a}+1} \sqrt {d+e x} \left (9 a B e^2+4 c d (8 B d-5 A e)\right ) E\left (\text {ArcSin}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{15 e^5 \sqrt {a+c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}}}+\frac {2 \left (a+c x^2\right )^{3/2} (-5 A e+8 B d+3 B e x)}{15 e^2 (d+e x)^{3/2}}-\frac {4 \sqrt {a+c x^2} \left (9 a B e^2+c e x (8 B d-5 A e)+4 c d (8 B d-5 A e)\right )}{15 e^4 \sqrt {d+e x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 430
Rule 435
Rule 733
Rule 827
Rule 858
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (a+c x^2\right )^{3/2}}{(d+e x)^{5/2}} \, dx &=\frac {2 (8 B d-5 A e+3 B e x) \left (a+c x^2\right )^{3/2}}{15 e^2 (d+e x)^{3/2}}-\frac {2 \int \frac {(-3 a B e+c (8 B d-5 A e) x) \sqrt {a+c x^2}}{(d+e x)^{3/2}} \, dx}{5 e^2}\\ &=-\frac {4 \left (9 a B e^2+4 c d (8 B d-5 A e)+c e (8 B d-5 A e) x\right ) \sqrt {a+c x^2}}{15 e^4 \sqrt {d+e x}}+\frac {2 (8 B d-5 A e+3 B e x) \left (a+c x^2\right )^{3/2}}{15 e^2 (d+e x)^{3/2}}+\frac {4 \int \frac {-a c e (8 B d-5 A e)+c \left (9 a B e^2+4 c d (8 B d-5 A e)\right ) x}{\sqrt {d+e x} \sqrt {a+c x^2}} \, dx}{15 e^4}\\ &=-\frac {4 \left (9 a B e^2+4 c d (8 B d-5 A e)+c e (8 B d-5 A e) x\right ) \sqrt {a+c x^2}}{15 e^4 \sqrt {d+e x}}+\frac {2 (8 B d-5 A e+3 B e x) \left (a+c x^2\right )^{3/2}}{15 e^2 (d+e x)^{3/2}}-\frac {\left (4 c \left (32 B c d^3-20 A c d^2 e+17 a B d e^2-5 a A e^3\right )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a+c x^2}} \, dx}{15 e^5}+\frac {\left (4 c \left (9 a B e^2+4 c d (8 B d-5 A e)\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {a+c x^2}} \, dx}{15 e^5}\\ &=-\frac {4 \left (9 a B e^2+4 c d (8 B d-5 A e)+c e (8 B d-5 A e) x\right ) \sqrt {a+c x^2}}{15 e^4 \sqrt {d+e x}}+\frac {2 (8 B d-5 A e+3 B e x) \left (a+c x^2\right )^{3/2}}{15 e^2 (d+e x)^{3/2}}+\frac {\left (8 a \sqrt {c} \left (9 a B e^2+4 c d (8 B d-5 A e)\right ) \sqrt {d+e x} \sqrt {1+\frac {c x^2}{a}}\right ) \text {Subst}\left (\int \frac {\sqrt {1+\frac {2 a \sqrt {c} e x^2}{\sqrt {-a} \left (c d-\frac {a \sqrt {c} e}{\sqrt {-a}}\right )}}}{\sqrt {1-x^2}} \, dx,x,\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )}{15 \sqrt {-a} e^5 \sqrt {\frac {c (d+e x)}{c d-\frac {a \sqrt {c} e}{\sqrt {-a}}}} \sqrt {a+c x^2}}-\frac {\left (8 a \sqrt {c} \left (32 B c d^3-20 A c d^2 e+17 a B d e^2-5 a A e^3\right ) \sqrt {\frac {c (d+e x)}{c d-\frac {a \sqrt {c} e}{\sqrt {-a}}}} \sqrt {1+\frac {c x^2}{a}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+\frac {2 a \sqrt {c} e x^2}{\sqrt {-a} \left (c d-\frac {a \sqrt {c} e}{\sqrt {-a}}\right )}}} \, dx,x,\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )}{15 \sqrt {-a} e^5 \sqrt {d+e x} \sqrt {a+c x^2}}\\ &=-\frac {4 \left (9 a B e^2+4 c d (8 B d-5 A e)+c e (8 B d-5 A e) x\right ) \sqrt {a+c x^2}}{15 e^4 \sqrt {d+e x}}+\frac {2 (8 B d-5 A e+3 B e x) \left (a+c x^2\right )^{3/2}}{15 e^2 (d+e x)^{3/2}}-\frac {8 \sqrt {-a} \sqrt {c} \left (9 a B e^2+4 c d (8 B d-5 A e)\right ) \sqrt {d+e x} \sqrt {1+\frac {c x^2}{a}} E\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{15 e^5 \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {a+c x^2}}+\frac {8 \sqrt {-a} \sqrt {c} \left (32 B c d^3-20 A c d^2 e+17 a B d e^2-5 a A e^3\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {1+\frac {c x^2}{a}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{15 e^5 \sqrt {d+e x} \sqrt {a+c x^2}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 23.07, size = 628, normalized size = 1.44 \begin {gather*} \frac {\sqrt {d+e x} \left (-\frac {2 \left (a+c x^2\right ) \left (5 a A e^3+5 a B e^2 (2 d+3 e x)-5 A c e \left (8 d^2+10 d e x+e^2 x^2\right )+B c \left (64 d^3+80 d^2 e x+8 d e^2 x^2-3 e^3 x^3\right )\right )}{e^4 (d+e x)^2}-\frac {8 \left (-e^2 \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}} \left (32 B c d^2-20 A c d e+9 a B e^2\right ) \left (a+c x^2\right )+\sqrt {c} \left (-i \sqrt {c} d+\sqrt {a} e\right ) \left (-32 B c d^2+20 A c d e-9 a B e^2\right ) \sqrt {\frac {e \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {i \sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x)^{3/2} E\left (i \sinh ^{-1}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right )|\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )+\sqrt {a} \sqrt {c} e \left (B \left (32 c d^2+8 i \sqrt {a} \sqrt {c} d e+9 a e^2\right )-5 A \left (4 c d e+i \sqrt {a} \sqrt {c} e^2\right )\right ) \sqrt {\frac {e \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {i \sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x)^{3/2} F\left (i \sinh ^{-1}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right )|\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )\right )}{e^6 \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}} (d+e x)}\right )}{15 \sqrt {a+c x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(3867\) vs.
\(2(365)=730\).
time = 0.76, size = 3868, normalized size = 8.85
method | result | size |
elliptic | \(\frac {\sqrt {\left (e x +d \right ) \left (c \,x^{2}+a \right )}\, \left (-\frac {2 \left (A a \,e^{3}+A c \,d^{2} e -a B d \,e^{2}-B c \,d^{3}\right ) \sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}{3 e^{6} \left (x +\frac {d}{e}\right )^{2}}+\frac {2 \left (c e \,x^{2}+a e \right ) \left (8 A c d e -3 B \,e^{2} a -11 B c \,d^{2}\right )}{3 e^{5} \sqrt {\left (x +\frac {d}{e}\right ) \left (c e \,x^{2}+a e \right )}}+\frac {2 B c x \sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}{5 e^{3}}+\frac {2 \left (\frac {c^{2} \left (A e -2 B d \right )}{e^{3}}-\frac {4 B \,c^{2} d}{5 e^{3}}\right ) \sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}{3 c e}+\frac {2 \left (\frac {c \left (2 A a \,e^{3}+3 A c \,d^{2} e -4 a B d \,e^{2}-4 B c \,d^{3}\right )}{e^{5}}-\frac {\left (A a \,e^{3}+A c \,d^{2} e -a B d \,e^{2}-B c \,d^{3}\right ) c}{3 e^{5}}-\frac {\left (8 A c d e -3 B \,e^{2} a -11 B c \,d^{2}\right ) c d}{3 e^{5}}-\frac {2 d a B c}{5 e^{3}}-\frac {\left (\frac {c^{2} \left (A e -2 B d \right )}{e^{3}}-\frac {4 B \,c^{2} d}{5 e^{3}}\right ) a}{3 c}\right ) \left (\frac {d}{e}-\frac {\sqrt {-a c}}{c}\right ) \sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}+\frac {\sqrt {-a c}}{c}}}\, \EllipticF \left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\right )}{\sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}+\frac {2 \left (-\frac {c \left (2 A c d e -2 B \,e^{2} a -3 B c \,d^{2}\right )}{e^{4}}-\frac {\left (8 A c d e -3 B \,e^{2} a -11 B c \,d^{2}\right ) c}{3 e^{4}}-\frac {3 a B c}{5 e^{2}}-\frac {2 \left (\frac {c^{2} \left (A e -2 B d \right )}{e^{3}}-\frac {4 B \,c^{2} d}{5 e^{3}}\right ) d}{3 e}\right ) \left (\frac {d}{e}-\frac {\sqrt {-a c}}{c}\right ) \sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}+\frac {\sqrt {-a c}}{c}}}\, \left (\left (-\frac {d}{e}-\frac {\sqrt {-a c}}{c}\right ) \EllipticE \left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\right )+\frac {\sqrt {-a c}\, \EllipticF \left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\right )}{c}\right )}{\sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}\right )}{\sqrt {e x +d}\, \sqrt {c \,x^{2}+a}}\) | \(997\) |
risch | \(\text {Expression too large to display}\) | \(2664\) |
default | \(\text {Expression too large to display}\) | \(3868\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.87, size = 477, normalized size = 1.09 \begin {gather*} -\frac {2 \, {\left (4 \, {\left (32 \, B c d^{5} - 15 \, A a x^{2} e^{5} + 3 \, {\left (11 \, B a d x^{2} - 10 \, A a d x\right )} e^{4} - {\left (20 \, A c d^{2} x^{2} - 66 \, B a d^{2} x + 15 \, A a d^{2}\right )} e^{3} + {\left (32 \, B c d^{3} x^{2} - 40 \, A c d^{3} x + 33 \, B a d^{3}\right )} e^{2} + 4 \, {\left (16 \, B c d^{4} x - 5 \, A c d^{4}\right )} e\right )} \sqrt {c} e^{\frac {1}{2}} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )} e^{\left (-2\right )}}{3 \, c}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )} e^{\left (-3\right )}}{27 \, c}, \frac {1}{3} \, {\left (3 \, x e + d\right )} e^{\left (-1\right )}\right ) + 12 \, {\left (32 \, B c d^{4} e + 9 \, B a x^{2} e^{5} - 2 \, {\left (10 \, A c d x^{2} - 9 \, B a d x\right )} e^{4} + {\left (32 \, B c d^{2} x^{2} - 40 \, A c d^{2} x + 9 \, B a d^{2}\right )} e^{3} + 4 \, {\left (16 \, B c d^{3} x - 5 \, A c d^{3}\right )} e^{2}\right )} \sqrt {c} e^{\frac {1}{2}} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )} e^{\left (-2\right )}}{3 \, c}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )} e^{\left (-3\right )}}{27 \, c}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )} e^{\left (-2\right )}}{3 \, c}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )} e^{\left (-3\right )}}{27 \, c}, \frac {1}{3} \, {\left (3 \, x e + d\right )} e^{\left (-1\right )}\right )\right ) + 3 \, {\left (64 \, B c d^{3} e^{2} - {\left (3 \, B c x^{3} + 5 \, A c x^{2} - 15 \, B a x - 5 \, A a\right )} e^{5} + 2 \, {\left (4 \, B c d x^{2} - 25 \, A c d x + 5 \, B a d\right )} e^{4} + 40 \, {\left (2 \, B c d^{2} x - A c d^{2}\right )} e^{3}\right )} \sqrt {c x^{2} + a} \sqrt {x e + d}\right )}}{45 \, {\left (x^{2} e^{8} + 2 \, d x e^{7} + d^{2} e^{6}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (A + B x\right ) \left (a + c x^{2}\right )^{\frac {3}{2}}}{\left (d + e x\right )^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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 (c\,x^2+a\right )}^{3/2}\,\left (A+B\,x\right )}{{\left (d+e\,x\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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