Optimal. Leaf size=420 \[ -\frac {2 \left (4 B c d^3-A c d^2 e+2 a B d e^2+a A e^3+e \left (5 B c d^2-2 A c d e+3 a B e^2\right ) x\right ) \sqrt {a+c x^2}}{3 e^2 \left (c d^2+a e^2\right ) (d+e x)^{3/2}}-\frac {4 \sqrt {-a} \sqrt {c} \left (4 B c d^2-A c d e+3 a B e^2\right ) \sqrt {d+e x} \sqrt {1+\frac {c x^2}{a}} E\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{3 e^3 \left (c d^2+a e^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {a+c x^2}}+\frac {4 \sqrt {-a} \sqrt {c} (4 B d-A e) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {1+\frac {c x^2}{a}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{3 e^3 \sqrt {d+e x} \sqrt {a+c x^2}} \]
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Rubi [A]
time = 0.22, antiderivative size = 420, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {825, 858, 733,
435, 430} \begin {gather*} -\frac {4 \sqrt {-a} \sqrt {c} \sqrt {\frac {c x^2}{a}+1} \sqrt {d+e x} \left (3 a B e^2-A c d e+4 B c d^2\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 )}{3 e^3 \sqrt {a+c x^2} \left (a e^2+c d^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}}}+\frac {4 \sqrt {-a} \sqrt {c} \sqrt {\frac {c x^2}{a}+1} (4 B d-A e) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}} 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 )}{3 e^3 \sqrt {a+c x^2} \sqrt {d+e x}}-\frac {2 \sqrt {a+c x^2} \left (e x \left (3 a B e^2-2 A c d e+5 B c d^2\right )+a A e^3+2 a B d e^2-A c d^2 e+4 B c d^3\right )}{3 e^2 (d+e x)^{3/2} \left (a e^2+c d^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 430
Rule 435
Rule 733
Rule 825
Rule 858
Rubi steps
\begin {align*} \int \frac {(A+B x) \sqrt {a+c x^2}}{(d+e x)^{5/2}} \, dx &=-\frac {2 \left (4 B c d^3-A c d^2 e+2 a B d e^2+a A e^3+e \left (5 B c d^2-2 A c d e+3 a B e^2\right ) x\right ) \sqrt {a+c x^2}}{3 e^2 \left (c d^2+a e^2\right ) (d+e x)^{3/2}}-\frac {2 \int \frac {a c e (B d-A e)-c \left (4 B c d^2-A c d e+3 a B e^2\right ) x}{\sqrt {d+e x} \sqrt {a+c x^2}} \, dx}{3 e^2 \left (c d^2+a e^2\right )}\\ &=-\frac {2 \left (4 B c d^3-A c d^2 e+2 a B d e^2+a A e^3+e \left (5 B c d^2-2 A c d e+3 a B e^2\right ) x\right ) \sqrt {a+c x^2}}{3 e^2 \left (c d^2+a e^2\right ) (d+e x)^{3/2}}-\frac {(2 c (4 B d-A e)) \int \frac {1}{\sqrt {d+e x} \sqrt {a+c x^2}} \, dx}{3 e^3}+\frac {\left (2 c \left (4 B c d^2-A c d e+3 a B e^2\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {a+c x^2}} \, dx}{3 e^3 \left (c d^2+a e^2\right )}\\ &=-\frac {2 \left (4 B c d^3-A c d^2 e+2 a B d e^2+a A e^3+e \left (5 B c d^2-2 A c d e+3 a B e^2\right ) x\right ) \sqrt {a+c x^2}}{3 e^2 \left (c d^2+a e^2\right ) (d+e x)^{3/2}}+\frac {\left (4 a \sqrt {c} \left (4 B c d^2-A c d e+3 a B e^2\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 )}{3 \sqrt {-a} e^3 \left (c d^2+a e^2\right ) \sqrt {\frac {c (d+e x)}{c d-\frac {a \sqrt {c} e}{\sqrt {-a}}}} \sqrt {a+c x^2}}-\frac {\left (4 a \sqrt {c} (4 B d-A e) \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 )}{3 \sqrt {-a} e^3 \sqrt {d+e x} \sqrt {a+c x^2}}\\ &=-\frac {2 \left (4 B c d^3-A c d^2 e+2 a B d e^2+a A e^3+e \left (5 B c d^2-2 A c d e+3 a B e^2\right ) x\right ) \sqrt {a+c x^2}}{3 e^2 \left (c d^2+a e^2\right ) (d+e x)^{3/2}}-\frac {4 \sqrt {-a} \sqrt {c} \left (4 B c d^2-A c d e+3 a B e^2\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 )}{3 e^3 \left (c d^2+a e^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {a+c x^2}}+\frac {4 \sqrt {-a} \sqrt {c} (4 B d-A e) \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 )}{3 e^3 \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 = 23.25, size = 609, normalized size = 1.45 \begin {gather*} -\frac {2 \sqrt {a+c x^2} \left (a A e^3-A c d e (d+2 e x)+a B e^2 (2 d+3 e x)+B c d^2 (4 d+5 e x)\right )}{3 e^2 \left (c d^2+a e^2\right ) (d+e x)^{3/2}}+\frac {4 \left (e^2 \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}} \left (4 B c d^2-A c d e+3 a B e^2\right ) \left (a+c x^2\right )-\sqrt {c} \left (-i \sqrt {c} d+\sqrt {a} e\right ) \left (-4 B c d^2+A c d e-3 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 (\sqrt {c} d+i \sqrt {a} e\right ) \left (-4 B \sqrt {c} d+3 i \sqrt {a} B e+A \sqrt {c} e\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 )}{3 e^4 \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}} \left (c d^2+a e^2\right ) \sqrt {d+e x} \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. \(3551\) vs.
\(2(354)=708\).
time = 0.78, size = 3552, normalized size = 8.46
method | result | size |
elliptic | \(\frac {\sqrt {\left (e x +d \right ) \left (c \,x^{2}+a \right )}\, \left (-\frac {2 \left (A e -B d \right ) \sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}{3 e^{4} \left (x +\frac {d}{e}\right )^{2}}+\frac {2 \left (c e \,x^{2}+a e \right ) \left (2 A c d e -3 B \,e^{2} a -5 B c \,d^{2}\right )}{3 e^{3} \left (e^{2} a +c \,d^{2}\right ) \sqrt {\left (x +\frac {d}{e}\right ) \left (c e \,x^{2}+a e \right )}}+\frac {2 \left (\frac {\left (A e -2 B d \right ) c}{e^{3}}-\frac {\left (A e -B d \right ) c}{3 e^{3}}-\frac {c d \left (2 A c d e -3 B \,e^{2} a -5 B c \,d^{2}\right )}{3 e^{3} \left (e^{2} a +c \,d^{2}\right )}\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 {B c}{e^{2}}-\frac {c \left (2 A c d e -3 B \,e^{2} a -5 B c \,d^{2}\right )}{3 e^{2} \left (e^{2} a +c \,d^{2}\right )}\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}}\) | \(783\) |
default | \(\text {Expression too large to display}\) | \(3552\) |
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 = 0.88, size = 476, normalized size = 1.13 \begin {gather*} -\frac {2 \, {\left (2 \, {\left (4 \, B c d^{5} - 3 \, A a x^{2} e^{5} + 6 \, {\left (B a d x^{2} - A a d x\right )} e^{4} - {\left (A c d^{2} x^{2} - 12 \, B a d^{2} x + 3 \, A a d^{2}\right )} e^{3} + 2 \, {\left (2 \, B c d^{3} x^{2} - A c d^{3} x + 3 \, B a d^{3}\right )} e^{2} + {\left (8 \, B c d^{4} x - 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 ) + 6 \, {\left (4 \, B c d^{4} e + 3 \, B a x^{2} e^{5} - {\left (A c d x^{2} - 6 \, B a d x\right )} e^{4} + {\left (4 \, B c d^{2} x^{2} - 2 \, A c d^{2} x + 3 \, B a d^{2}\right )} e^{3} + {\left (8 \, B c d^{3} x - 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 (4 \, B c d^{3} e^{2} + {\left (3 \, B a x + A a\right )} e^{5} - 2 \, {\left (A c d x - B a d\right )} e^{4} + {\left (5 \, B c d^{2} x - A c d^{2}\right )} e^{3}\right )} \sqrt {c x^{2} + a} \sqrt {x e + d}\right )}}{9 \, {\left (2 \, c d^{3} x e^{5} + c d^{4} e^{4} + a x^{2} e^{8} + 2 \, a d x e^{7} + {\left (c d^{2} x^{2} + a d^{2}\right )} 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 ) \sqrt {a + c x^{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 {\sqrt {c\,x^2+a}\,\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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